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The main object of this surveycum-expository article is to present an overview of some recent developments involving the Riemann Zeta function, which have their roots in the works of the great eighteenth-century Swiss mathematician, Leonhard Euler (1707 1783) and the Russian mathematician, Christian Goldbach (1690 1764).
❱❖▲❯▼❊✿ ✸ | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ ❚❤❡ ❩❡t❛ ❛♥❞ ❘❡❧❛t❡❞ ❋✉♥❝t✐♦♥s✿ ❘❡❝❡♥t ❉❡✈❡❧♦♣♠❡♥ts 1,2,∗ ❍✳ ▼✳ ❙❘■❱❆❙❚❆❱❆ ❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❙t❛t✐st✐❝s✱ ❯♥✐✈❡rs✐t② ♦❢ ❱✐❝t♦r✐❛✱ ❱✐❝t♦r✐❛✱ ❇r✐t✐s❤ ❈♦❧✉♠❜✐❛ ❱✽❲ ✸❘✹✱ ❈❛♥❛❞❛ ❉❡♣❛rt♠❡♥t ♦❢ ▼❡❞✐❝❛❧ ❘❡s❡❛r❝❤✱ ❈❤✐♥❛ ▼❡❞✐❝❛❧ ❯♥✐✈❡rs✐t② ❍♦s♣✐t❛❧✱ ❈❤✐♥❛ ▼❡❞✐❝❛❧ ❯♥✐✈❡rs✐t②✱ ❚❛✐❝❤✉♥❣ ✹✵✹✵✷✱ ❚❛✐✇❛♥✱ ❘❡♣✉❜❧✐❝ ♦❢ ❈❤✐♥❛ ✯❈♦rr❡s♣♦♥❞✐♥❣ ❆✉t❤♦r✿ ❍✳ ▼✳ ❙❘■❱❆❙❚❆❱❆ ✭❊♠❛✐❧✿ ❤❛r✐♠sr✐❅♠❛t❤✳✉✈✐❝✳❝❛✮ ✭❘❡❝❡✐✈❡❞✿ ✷✶✲❏❛♥✲✷✵✶✾❀ ❛❝❝❡♣t❡❞✿ ✷✷✲❋❡❜✲✷✵✶✾❀ ♣✉❜❧✐s❤❡❞✿ ✸✶✲▼❛r✲✷✵✶✾✮ ❉❖■✿ ❤tt♣✿✴✴❞①✳❞♦✐✳♦r❣✴✶✵✳✷✺✵✼✸✴❥❛❡❝✳✷✵✶✾✸✶✳✷✷✾ ❚❤❡ ♠❛✐♥ ♦❜❥❡❝t ♦❢ t❤✐s s✉r✈❡②✲ ❝✉♠✲❡①♣♦s✐t♦r② ❛rt✐❝❧❡ ✐s t♦ ♣r❡s❡♥t ❛♥ ♦✈❡r✈✐❡✇ ♦❢ s♦♠❡ r❡❝❡♥t ❞❡✈❡❧♦♣♠❡♥ts ✐♥✈♦❧✈✐♥❣ t❤❡ ❘✐❡✲ ♠❛♥♥ ❩❡t❛ ❢✉♥❝t✐♦♥ ζ(s)✱ t❤❡ ❍✉r✇✐t③ ✭♦r ❣❡♥✲ ❡r❛❧✐③❡❞✮ ❩❡t❛ ❢✉♥❝t✐♦♥ ζ(s, a)✱ ❛♥❞ t❤❡ ❍✉r✇✐t③✲ ▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ Φ(z, s, a), ✇❤✐❝❤ ❤❛✈❡ t❤❡✐r r♦♦ts ✐♥ t❤❡ ✇♦r❦s ♦❢ t❤❡ ❣r❡❛t ❡✐❣❤t❡❡♥t❤✲❝❡♥t✉r② ❙✇✐ss ♠❛t❤❡♠❛t✐❝✐❛♥✱ ▲❡♦♥❤❛r❞ ❊✉❧❡r ✭✶✼✵✼✕ ✶✼✽✸✮ ❛♥❞ t❤❡ ❘✉ss✐❛♥ ♠❛t❤❡♠❛t✐❝✐❛♥✱ ❈❤r✐s✲ t✐❛♥ ●♦❧❞❜❛❝❤ ✭✶✻✾✵✕✶✼✻✹✮✳ ❲❡ ❛✐♠ ❛t ❝♦♥✲ s✐❞❡r✐♥❣ t❤❡ ♣r♦❜❧❡♠s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❡✈❛❧✲ ✉❛t✐♦♥s ❛♥❞ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ζ(s) ✇❤❡♥ s ∈ N \ {1}✱ N ❜❡✐♥❣ t❤❡ s❡t ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs✱ ✇✐t❤ ❡♠♣❤❛s✐s ✉♣♦♥ s❡✈❡r❛❧ ✐♥t❡r❡st✐♥❣ ❝❧❛ss❡s ♦❢ r❛♣✐❞❧② ❝♦♥✈❡r❣❡♥t s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r ζ(2n+1) (n ∈ N)✳ ❙②♠❜♦❧✐❝ ❛♥❞ ♥✉♠❡r✐❝❛❧ ❝♦♠✲ ♣✉t❛t✐♦♥s ✉s✐♥❣ ▼❛t❤❡♠❛t✐❝❛ ✭❱❡rs✐♦♥ ✹✳✵✮ ❢♦r ▲✐♥✉① ✇✐❧❧ ❛❧s♦ ❜❡ ♣r♦✈✐❞❡❞ ❢♦r s✉♣♣♦rt✐♥❣ t❤❡✐r ❝♦♠♣✉t❛t✐♦♥❛❧ ✉s❡❢✉❧♥❡ss✳ ❆❜str❛❝t✳ ❞✉❝t✐✈❡ ❆r❣✉♠❡♥t❀ ❙②♠❜♦❧✐❝ ❛♥❞ ◆✉♠❡r✲ ✐❝❛❧ ❈♦♠♣✉t❛t✐♦♥s❀ ❊✉❧❡r ❙✉♠s✳ ✶✳ ❚❤r♦✉❣❤♦✉t t❤✐s ❧❡❝t✉r❡✱ ✇❡ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛♥❞❛r❞ ♥♦t❛t✐♦♥s✿ N := {1, 2, 3, · · · }, N0 := {0, 1, 2, · · · } = N∪{0} ❛♥❞ Z− := {−1, −2, −3, · · · } = Z− \ {0} ❑❡②✇♦r❞s ❘✐❡♠❛♥♥ ❩❡t❛ ❋✉♥❝t✐♦♥❀ ❍✉r✇✐t③ ✭♦r ❣❡♥✲ ❡r❛❧✐③❡❞✮ ❩❡t❛ ❋✉♥❝t✐♦♥❀ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❋✉♥❝t✐♦♥❀ ❆ss♦❝✐❛t❡❞ ❙❡r✐❡s ❛♥❞ ■♥✲ t❡❣r❛❧s❀ r✐❡s ❜❡rs❀ ■♥tr♦❞✉❝t✐♦♥✱ ❉❡✜♥✐t✐♦♥s ❛♥❞ Pr❡❧✐♠✐♥❛r✐❡s ❆♥❛❧②t✐❝ ◆✉♠❜❡r ❘❡♣r❡s❡♥t❛t✐♦♥s❀ ❚❤❡♦r②❀ ❍❛r♠♦♥✐❝ ❙❡✲ ◆✉♠✲ ❇❡r♥♦✉❧❧✐ ◆✉♠❜❡rs ❛♥❞ ❇❡r♥♦✉❧❧✐ P♦❧②♥♦♠✐❛❧s❀ ●❡♥❡r❛t✐♥❣ ❋✉♥❝t✐♦♥s❀ ❊✉✲ ❧❡r ◆✉♠❜❡rs ❛♥❞ ❊✉❧❡r P♦❧②♥♦♠✐❛❧s❀ ■♥✲ ❆❧s♦✱ ❛s ✉s✉❛❧✱ Z ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ✐♥t❡❣❡rs✱ R ❞❡♥♦t❡s t❤❡ s❡t ♦❢ r❡❛❧ ♥✉♠❜❡rs✱ R+ ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs ❛♥❞ C ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✳ ❙♦♠❡ r❛t❤❡r ✐♠♣♦rt❛♥t ❛♥❞ ♣♦t❡♥t✐❛❧❧② ✉s❡❢✉❧ ❢✉♥❝t✐♦♥s ✐♥ ❆♥❛❧②t✐❝ ◆✉♠❜❡r ❚❤❡♦r② ✐♥❝❧✉❞❡ ✭❢♦r ❡①❛♠♣❧❡✮ t❤❡ ❘✐❡♠❛♥♥ ❩❡t❛ ❢✉♥❝t✐♦♥ ζ(s) ❛♥❞ t❤❡ ❍✉r✇✐t③ ✭♦r ❣❡♥❡r❛❧✐③❡❞✮ ❩❡t❛ ❢✉♥❝t✐♦♥ ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✸✷✾ ❱❖▲❯▼❊✿ ✸ ζ(s, a)✱ ✇❤✐❝❤ ❛r❡ ❞❡✜♥❡❞ ✭❢♦r ζ (s) := (s) > 1✮ ❜② ∞ ∞ 1 = s n − 2−s n=1 ( (s) > 1) s (2n − 1) n=1 ∞ n−1 (−1) − 21−s ns n=1 ( (s) > 0; s = 1) ✭✶✮ ❛♥❞ ∞ ζ (s, a) := s, (n + a) n=0 (s) > 1; a ∈ C\Z− , ✭✷✮ ❛♥❞ ❢♦r (s) 1; s = ❜② t❤❡✐r ♠❡r♦♠♦r✲ ♣❤✐❝ ❝♦♥t✐♥✉❛t✐♦♥s ✭s❡❡✱ ❢♦r ❞❡t❛✐❧s✱ t❤❡ ❡①❝❡❧❧❡♥t ✇♦r❦s ❜② ❚✐t❝❤♠❛rs❤ ❬✼✷❪ ❛♥❞ ❆♣♦st♦❧ ❬✹❪ ❛s ✇❡❧❧ ❛s t❤❡ ♠♦♥✉♠❡♥t❛❧ tr❡❛t✐s❡ ❜② ❲❤✐tt❛❦❡r ❛♥❞ ❲❛ts♦♥ ❬✼✺❪❀ s❡❡ ❛❧s♦ ❬✶✱ ❈❤❛♣t❡r ✷✸❪ ❛♥❞ ❬✺✼✱ ❈❤❛♣t❡r ✷❪✮✱ s♦ t❤❛t ✭♦❜✈✐♦✉s❧②✮ −1 ζ (s, 1) = ζ (s) = (2s − 1) ζ s, ❛♥❞ ❆ ❝❧❛ss✐❝❛❧ ❛❜♦✉t t❤r❡❡✲❝❡♥t✉r②✲♦❧❞ t❤❡♦r❡♠ ♦❢ ❈❤r✐st✐❛♥ ●♦❧❞❜❛❝❤ ✭✶✻✾✵✕✶✼✻✹✮ ✇❛s st❛t❡❞ ✐♥ ❛ ❧❡tt❡r ❞❛t❡❞ ✶✼✷✾ ❢r♦♠ ●♦❧❞❜❛❝❤ t♦ ❉❛♥✐❡❧ ❇❡r♥♦✉❧❧✐ ✭✶✼✵✵✕✶✼✽✷✮✳ ●♦❧❞❜❛❝❤✬s ❚❤❡♦r❡♠ ✭♦r✱ ❡q✉✐✈❛❧❡♥t❧②✱ t❤❡ ●♦❧❞❜❛❝❤✲❊✉❧❡r ❚❤❡♦r❡♠✮ ✇❛s r❡✈✐✈❡❞ ❛♥❞ r❡✈✐s✐t❡❞ r❡❝❡♥t❧② ❛s t❤❡ ❢♦❧❧♦✇✲ ✐♥❣ ♣r♦❜❧❡♠ ✐♥ ♠❛♥② ♣✉❜❧✐❝❛t✐♦♥s s✉❝❤ ❛s ✭❢♦r ❡①❛♠♣❧❡✮ ❬❛❪ ❏✳ ❉✳ ❙❤❛❧❧✐t ❛♥❞ ❑✳ ❩✐❦❛♥✱ ❆ t❤❡♦r❡♠ ♦❢ ●♦❧❞❜❛❝❤✱ ❆♠❡r✳ ▼❛t❤✳ ▼♦♥t❤❧② ✾✸ ✭✶✾✽✻✮✱ ✹✵✷✕✹✵✸ ❬❜❪ ❏✳ ❑✳ ❍❛✉❣❤❧✉♥❞✱ ❉✳ ❚❥❛❞❡♥ ❛♥❞ ❏✳ ●r♦❡♥✲ ❡✈❡❧❞✱ Pr♦❜❧❡♠ ✸✽✱ ◆✐❡✉✇ ❆r❝❤✳ ❲✐s❦✳ (❙❡r✳ 5) ✹ ✭✷✵✵✸✮✱ ✾✺✕✾✻✳ ✇✐t❤✱ ♦❢ ❝♦✉rs❡✱ ζ(0, a) = − a, ζ(0) = − ω∈S ζ(−2n) = (n ∈ N) ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✲ s❤✐♣s✿ ms − m−1 ζ s, j=1 j m , (m ∈ N \ {1}) ✭✸✮ ❛♥❞ ζ (s, ma) = s m m−1 j=0 j ζ s, a + m = 1, ω−1 ✭✺✮ ✇❤❡r❡ S ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ❛❧❧ ♥♦♥tr✐✈✐❛❧ ✐♥t❡❣❡r k t❤ ♣♦✇❡rs✱ t❤❛t ✐s✱ ❛♥❞ , (m ∈ N) ✭✹✮ ✸✸✵ ●❡♥❡r❛❧❧② s♣❡❛❦✐♥❣✱ ▼❛t❤❡♠❛t✐❝s ❛♣♣❡❛❧s t♦ t❤❡ ✐♥t❡❧❧❡❝t✳ ■♥ ❛❞❞✐t✐♦♥✱ ❤♦✇❡✈❡r✱ ❣r❡❛t ♠❛t❤✲ ❡♠❛t✐❝s ♣♦ss❡ss❡s ❛ ❦✐♥❞ ♦❢ ♣❡r❝❡♣t✉❛❧ q✉❛❧✐t② ✇❤✐❝❤ ❡♥❞♦✇s ✐t ✇✐t❤ ❛ ❜❡❛✉t② ❝♦♠♣❛r❛❜❧❡ t♦ t❤❛t ♦❢ ❣r❡❛t ❛rt ♦r ❣r❡❛t ♠✉s✐❝✳ ▼✉❝❤ ♦❢ t❤❡ ✇♦r❦ ♦❢ t❤❡ ✶✽t❤ ❝❡♥t✉r② ❙✇✐ss ♠❛t❤❡♠❛t✐❝✐❛♥✱ ▲❡♦♥❤❛r❞ ❊✉❧❡r ✭✶✼✵✼✕✶✼✽✸✮✱ ❜❡❧♦♥❣s ✐♥ t❤✐s ❝❛t✲ ❡❣♦r②✳ ❊✉❧❡r✬s ✇♦r❦ ♦♥ ζ(s) ❜❡❣❛♥ ❛r♦✉♥❞ ✶✼✸✵ ✇✐t❤ ❛♣♣r♦①✐♠❛t✐♦♥s t♦ t❤❡ ✈❛❧✉❡ ♦❢ ζ(2)✱ ❝♦♥✲ t✐♥✉❡❞ ✇✐t❤ t❤❡ ❡✈❛❧✉❛t✐♦♥ ♦❢ ζ(2n) (n ∈ N)✱ ❛♥❞ r❡s✉❧t❡❞ ❛r♦✉♥❞ ✶✼✹✾ ✐♥ t❤❡ ❞✐s❝♦✈❡r② ♦❢ t❤❡ ❝❡❧❡❜r❛t❡❞ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥ ❢♦r ζ(s) ❛❧♠♦st ✶✶✵ ②❡❛rs ❜❡❢♦r❡ t❤❡ r❡♠❛r❦❛❜❧② ✐♥✢✉❡♥t✐❛❧ ●❡r✲ ♠❛♥ ♠❛t❤❡♠❛t✐❝✐❛♥✱ ●❡♦r❣ ❋r✐❡❞r✐❝❤ ❇❡r♥❤❛r❞ ❘✐❡♠❛♥♥ ✭✶✽✷✻✕✶✽✻✻✮✳ ❛♥❞ ζ (s, 2) = ζ (s) − ζ (s) = | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ S := nk : n, k ∈ N \ {1} = {4, 8, 9, 16, 25, 27, 32, 36, · · · } ❲❤❛t ❞♦❡s ●♦❧❞❜❛❝❤✬s ❚❤❡♦r❡♠ ✭✺✮ ❤❛✈❡ t♦ ❞♦ ✇✐t❤ t❤❡ ❘✐❡♠❛♥♥ ❩❡t❛ ❢✉♥❝t✐♦♥ ζ(s) ❞❡✜♥❡❞ ❜② ✭✶✮❄ ■♥ ♦r❞❡r t♦ ❛♥s✇❡r t❤✐s q✉❡st✐♦♥✱ ❧❡t T ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ❛❧❧ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs t❤❛t ❛r❡ ♥♦t ✐♥ S ♦t❤❡r t❤❛♥ 1✱ t❤❛t ✐s✱ T := {τ : τ ∈ /S ❛♥❞ τ ∈ N \ {1}} ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✸ ❲❡ t❤❡♥ ✜♥❞ t❤❛t ω∈S ✇❤❡r❡ = ω−1 ∞ τk − k=2 τ ∈T ∞ f(x) := x − [x] = ❚❤❡ ❢r❛❝t✐♦♥❛❧ ♣❛rt ♦❢ x ∈ R −1 ❆s ❛ ♠❛tt❡r ♦❢ ❢❛❝t✱ ✐t ✐s ❢❛✐r❧② str❛✐❣❤t❢♦r✇❛r❞ t♦ s❤♦✇ ❛❧s♦ t❤❛t ∞ = τ −jk ∞ k=2 τ ∈T j=1 ∞ ∞ = τ (−1)k f ζ(k) = j −k k=2 ∞ k=2 j=1 τ ∈T ∞ ∞ = k=2 n=2 ∞ f ζ(2k) = nk k=1 ∞ k=1 ∞ k=2 f ζ(4k) = s✐♥❝❡ ✐t ✐s ❡❛s✐❧② s❡❡♥ t❤❛t ∞ τj k=1 , , f ζ(2k + 1) = ζ(k) − , = ∞ | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ , (7 − coth π) , ❛♥❞ s♦ ♦♥✳ −k ❙❡✈❡r❛❧ ❡①t❡♥s✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ♦❢ ●♦❧❞❜❛❝❤✬s ❚❤❡♦r❡♠ ✭✺✮ ❤❛✈❡ ❜❡❡♥ ✐♥✈❡st✐❣❛t❡❞✳ j −k j −k j −k + + ❋♦r ❡①❛♠♣❧❡✱ ✇❡ r❡❝❛❧❧ t❤❡ ❢♦❧❧♦✇✐♥❣ ❣❡♥❡r❛❧✐③❛✲ ∞ t✐♦♥s ❣✐✈❡♥ ✐♥ + 6j −k + 7j −k + 10j −k ❬❝❪ ❏✳ ❈❤♦✐ ❛♥❞ ❍✳ ▼✳ ❙r✐✈❛st❛✈❛✱ ❙❡r✐❡s ✐♥✈♦❧✈✐♥❣ j=1 j −k j −k + 11 + 12 + ··· t❤❡ ❩❡t❛ ❢✉♥❝t✐♦♥s ❛♥❞ ❛ ❢❛♠✐❧② ♦❢ ❣❡♥❡r❛❧✐③❡❞ −k ●♦❧❞❜❛❝❤✲❊✉❧❡r s❡r✐❡s✱ ❆♠❡r✳ ▼❛t❤✳ ▼♦♥t❤❧② −k −k −k −k + + + + + ··· ✶✷✶ ✭✷✵✶✹✮✱ ✷✷✾✕✷✸✻✳ k=2 j=1 τ ∈T ∞ = k=2 ∞ + 4−k + 9−k + 25−k + 36−k + · · · = k=2 ∞ −k + + 27 ∞ −k + 125 −k + ··· + ··· ∞ n−k = = k=2 n=2 ω∈Sp,0 ζ(k) − = ω−1 ❛♥❞ ω∈Sp,1 ∞ ζ(k) − = ✭✻✮ k=2 ❙✐♥❝❡ ζ(s) ✐s ❛ ❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ ✐ts ❛r❣✉✲ ♠❡♥t s ❢♦r s 2✱ ✇❡ ❤❛✈❡ < ζ(n) , (p ∈ N \ {1}) k=2 ❚❤✉s✱ ✐♥ t❡r♠s ♦❢ t❤❡ ❘✐❡♠❛♥♥ ❩❡t❛ ❢✉♥❝t✐♦♥ ζ(s) ❞❡✜♥❡❞ ❜② ✭✶✮✱ ●♦❧❞❜❛❝❤✬s ❚❤❡♦r❡♠ ✭✺✮ ✐s ❡❛s✐❧② s❡❡♥ t♦ ❛ss✉♠❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡❧❡❣❛♥t ❢♦r♠✿ ω∈S 1 = ψ (1) − ψ − ω−1 p p ζ(2) = π2 < 2, 1 = 1+ ψ ω−1 p p − ψ (1) , (p ∈ N), ✇❤❡r❡ t❤❡ s❡t Sp,0 ✐s ❞❡✜♥❡❞ (❢♦r ✜①❡❞ p ∈ N \ {1}) ❜② Sp,0 := (pn)k : n ∈ N ❛♥❞ k ∈ N \ {1} ❛♥❞ t❤❡ s❡t Sp,1 ✐s ❞❡✜♥❡❞ (❢♦r ✜①❡❞ p ∈ N) ❜② ✭✼✮ t❤❡ ❛❜♦✈❡ ❛❧t❡r♥❛t✐✈❡ ❢♦r♠ ✭✻✮ ♦❢ ●♦❧❞❜❛❝❤✬s ❚❤❡♦r❡♠ ✭✺✮ ❝❛♥ ❛❧s♦ ❜❡ r❡✇r✐tt❡♥ ❛s ❢♦❧❧♦✇s✿ Sp,1 := (pn + 1)k : n ∈ N ❛♥❞ k ∈ N \ {1} , ❛♥❞ t❤❡ Ps✐ ✭♦r ❉✐❣❛♠♠❛✮ ❢✉♥❝t✐♦♥ ψ(z) ✐s ❞❡✲ ✜♥❡❞ ✭❛s ✉s✉❛❧✮ ❜② ∞ f ζ(k) = 1, k=2 ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ψ(z) := d Γ (z) {log Γ(z)} = dz Γ(z) ✸✸✶ ❱❖▲❯▼❊✿ ✸ ♦r ✜♥❞ ❛ ❝❧♦s❡❞✲❢♦r♠ ❡✈❛❧✉❛t✐♦♥ ♦r ❡①♣r❡ss✐♦♥✳ z ψ(t) dt log Γ(z) = ■♥ ❢❛❝t✱ ✐♥ t❡r♠s ♦❢ t❤❡ ❘✐❡♠❛♥♥ ③❡t❛ ❢✉♥❝t✐♦♥ ζ(s) ❛♥❞ ❍✉r✇✐t③ ✭♦r ❣❡♥❡r❛❧✐③❡❞✮ ③❡t❛ ❢✉♥❝t✐♦♥ ζ(s, a)✱ ✇❡ ❤❛✈❡ ∞ = ω−1 ω∈Sp,0 ζ(k) pk k=2 , ✐s t❤❡ s✉♠ ♦❢ t✇♦ ♣r✐♠❡ ♥✉♠❜❡rs✿ = + = + 3; (p ∈ N \ {1}) ❛♥❞ = ω−1 ω∈Sp,1 ❚❤❡ ♥❛♠❡ ♦❢ ❈❤r✐st✐❛♥ ●♦❧❞❜❛❝❤ ✭✶✻✾✵✕✶✼✻✹✮ ✐s ✉s✉❛❧❧② ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ r❡❧❛t✐✈❡❧② ♠♦r❡ ♣♦♣✉❧❛r ❝♦♥❥❡❝t✉r❡ ✜rst ♣r♦♣♦s❡❞ ✐♥ ❛ ❧❡tt❡r ❞❛t❡❞ ✶✼✹✷ ❢r♦♠ ●♦❧❞❜❛❝❤ t♦ ❊✉❧❡r ✭❦♥♦✇♥ ❛s ●♦❧❞❜❛❝❤✬s ❈♦♥❥❡❝t✉r❡✮✱ t❤❛t ✐s✱ ❊✈❡r② ♣♦s✐t✐✈❡ ❡✈❡♥ ✐♥t❡❣❡r ❣r❡❛t❡r t❤❛♥ 1 ψ (1) − ψ − p p = = + = + 5; ∞ 1 ζ k, + k p p k=2 =1+ ψ p p = + = + 5; et cetera ❏✉st ❛s t❤❡ ❝❡❧❡❜r❛t❡❞ ❘✐❡♠❛♥♥ ❍②♣♦t❤❡s✐s ❞❛t❡❞ ✶✽✺✾ t❤❛t ❛❧❧ ♥♦♥tr✐✈✐❛❧ ③❡r♦s ♦❢ ζ(s) ❧✐❡ − ψ (1) ♦♥ t❤❡ ❝r✐t✐❝❛❧ ❧✐♥❡: (p ∈ N) ♦r✱ ❡q✉✐✈❛❧❡♥t❧②✱ ∞ ∞ (s) = = (pn)k − k=2 n=1 = ω∈Sp,0 + ω−1 ∞ ∞ k=2 j=1 ∞ ∞ k=2 j=2 ζ(kj) pkj ∞ ζ(kj) pkj = k=2 j=1 = ω∈Sp,1 ❯♥❝❧❡ P❡❞r♦s ❛♥❞ ●♦❧❞❜❛❝❤✬s ❈♦♥❥❡❝t✉r❡: ✭❜② ❆♣♦st♦❧♦s ❉♦①✐❛❞✐s✮✱ ❋❛❜❡r ❛♥❞ ❋❛❜❡r✱ ▲♦♥❞♦♥✱ ✷✵✵✶✳ t❤❡ ❇r✐t✐s❤ ♣✉❜❧✐s❤❡r ✭❋❛❜❡r ❛♥❞ ❋❛❜❡r✮ ❤❛❞ ♦✛❡r❡❞ ❛ r❡✇❛r❞ ♦❢ ♦♥❡ ♠✐❧❧✐♦♥ ❯✳❑✳ P♦✉♥❞s t♦ ❛♥②♦♥❡ ✇❤♦ ❝❛♥ ♣r♦✈❡ ●♦❧❞❜❛❝❤✬s ❈♦♥❥❡❝t✉r❡✳ ❆ ◆♦✈❡❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❖❜s❡ss✐♦♥ (pn + 1)k − k=2 n=1 ∞ ∞ 1 ζ kj, + pkj p + ω−1 ∞ ∞ k=2 j=2 1 ζ kj, + pkj p , (p ∈ N)✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤✐s ❧❛st ♣❛✐r ♦❢ t❤❡ ●♦❧❞❜❛❝❤✲❊✉❧❡r t②♣❡ s✉♠s ♣♦s❡s ❛ ♥❛t✉r❛❧ q✉❡s✲ t✐♦♥ ❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ♦♣❡♥ ♣r♦❜❧❡♠✳ ❖♣❡♥ Pr♦❜❧❡♠✳ ❋♦r ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞♦✉✲ ❜❧❡ s✉♠s✿ ∞ ∞ k=2 j=1 ∞ ∞ k=2 j=1 ✸✸✷ , ●♦❧❞❜❛❝❤✬s ❝♦♥❥❡❝t✉r❡ ❤❛s ♥♦t ❜❡❡♥ ♣r♦✈❡♥ ❛s ②❡t✳ ■♥t❡r❡st✐♥❣❧②✱ ♥♦t t♦♦ ❧♦♥❣ ❛❣♦ ✐♥ t❤❡ ②❡❛r ✷✵✵✶✱ ♦♥ t❤❡ ♦❝❝❛s✐♦♥ ♦❢ t❤❡ ♣✉❜❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✭✑✈❡r② ❢✉♥♥②✱ t❡♥❞❡r✱ ❝❤❛r♠✐♥❣✱ ❛♥❞ ✐rr❡s✐st✐❜❧❡✑✮ ♥♦✈❡❧✿ (p ∈ N \ {1})✱ ❛♥❞ ∞ | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ ∞ ζ(2) := k=1 ζ(kj) pkj 1 ζ kj, + kj p p ❆♥♦t❤❡r r❡s✉❧t t❤❛t ❤❛s ❛ttr❛❝t❡❞ ❢❛s❝✐♥❛t✲ ✐♥❣❧② ❛♥❞ t❛♥t❛❧✐③✐♥❣❧② ❧❛r❣❡ ♥✉♠❜❡r ♦❢ s❡❡♠✲ ✐♥❣❧② ✐♥❞❡♣❡♥❞❡♥t s♦❧✉t✐♦♥s ✐s t❤❡ s♦✲❝❛❧❧❡❞ ❇❛s❧❡r Pr♦❜❧❡♠ ♦r ❇❛s❡❧ Pr♦❜❧❡♠✿ , π2 = , k2 ✭✽✮ ✇❤✐❝❤ ✇❛s ✉s❡❞ ❛❜♦✈❡ ✐♥ ✭✼✮✳ ■t ✇❛s ♦❢ ✈✐t❛❧ ✐♠♣♦rt❛♥❝❡ t♦ ▲❡♦♥❤❛r❞ ❊✉❧❡r ✭✶✼✵✼✕✶✼✽✸✮ ❛♥❞ t❤❡ ❇❡r♥♦✉❧❧✐ ❜r♦t❤❡rs ❬❏❛❦♦❜ ❇❡r♥♦✉❧❧✐ ✭✶✻✺✹✕ ✶✼✵✺✮ ❛♥❞ ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐ ✭✶✻✻✼✕✶✼✹✽✮❪✳ ❘❡✲ ♠❛r❦❛❜❧② ♠❛♥② ✭♦✈❡r ❛ ❝♦✉♣❧❡ ♦❢ ❞♦③❡♥✮ ❡ss❡♥✲ ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✸ t✐❛❧❧② ✐♥❞❡♣❡♥❞❡♥t s♦❧✉t✐♦♥s t❤❡ ❇❛s❧❡r Pr♦❜✲ ❧❡♠ ✭✽✮ ❤❛✈❡ ❛♣♣❡❛r❡❞ ✐♥ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ❧✐t✲ ❡r❛t✉r❡ ❡✈❡r s✐♥❝❡ ❊✉❧❡r ✜rst s♦❧✈❡❞ t❤✐s ♣r♦❜❧❡♠ ✐♥ t❤❡ ②❡❛r ✶✼✸✻✳ | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ t♦ ❛ ●❡r♠❛♥ ♣r✐♥❝❡ss✳ ❚❤❡ q✉❛❧✐t② ♦❢ ❛❧❧ ❤✐s ❧❡t✲ t❡rs r❡✢❡❝ts ❊✉❧❡r✬s ♣❧❡❛s❛♥t ♣❡rs♦♥❛❧✐t②✳ ■♥ t❤❡ ❛❜♦✈❡ ❝♦♥t❡①t✱ ♦♥❡ ♦t❤❡r r❡♠❛r❦❛❜❧❡ ❝❧❛ss✐❝❛❧ r❡s✉❧t ✐♥✈♦❧✈✐♥❣ ❘✐❡♠❛♥♥✬s ❩❡t❛ ❢✉♥❝✲ t✐♦♥ ζ(s) ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ ❡❧❡❣❛♥t s❡r✐❡s r❡♣r❡s❡♥✲ ❚❤❡ ❝✐t② ♦❢ ❇❛s❡❧ ✐♥ ❙✇✐t③❡r❧❛♥❞ ✇❛s ♦♥❡ ♦❢ t❛t✐♦♥ ❢♦r ζ (3)✿ ♠❛♥② ❢r❡❡ ❝✐t✐❡s ✐♥ ❊✉r♦♣❡✳ ❇② t❤❡ ✶✼t❤ ❝❡♥✲ t✉r②✱ ✐t ❤❛❞ ❜❡❝♦♠❡ ❛♥ ✐♠♣♦rt❛♥t ❝❡♥t❡r ♦❢ tr❛❞❡ ❛♥❞ ❝♦♠♠❡r❝❡✳ ❚❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❇❛s❡❧ ❜❡❝❛♠❡ ❛ ♥♦t❡❞ ✐♥st✐t✉t✐♦♥ ✐♥ ❊✉r♦♣❡✱ ❧❛r❣❡❧② t❤r♦✉❣❤ t❤❡ ❢❛♠❡ ♦❢ ❛♥ ❡①tr❛♦r❞✐♥❛r② ❢❛♠✐❧②✱ ♥❛♠❡❧②✱ t❤❡ ❇❡r♥♦✉❧❧✐s✳ ❚❤✐s ❢❛♠✐❧② ❤❛❞ ❝♦♠❡ ❢r♦♠ ❆♥t✇❡r♣ t♦ ❇❛s❡❧✳ ❚❤❡ ❢♦✉♥❞❡r ♦❢ t❤❡ ❇❡r♥♦✉❧❧✐ ♠❛t❤❡♠❛t✐❝❛❧ ❞②♥❛st② ✇❛s ◆✐❝♦❧❛✉s ✭◆✐❝❤♦❧❛s✮ ❇❡r♥♦✉❧❧✐ ✇❤♦ ✇❛s ❛ ♣❛✐♥t❡r ❛♥❞ ❆❧❞❡r♠❛♥ ♦❢ ❇❛s❡❧✳ ❍❡ ❤❛❞ ✸ s♦♥s❀ t✇♦ ♦❢ ❤✐s s♦♥s✱ ❏❛❦♦❜ ❬♦❢t❡♥ r❡❢❡rr❡❞ t♦ ❛s ❏❛♠❡s❪ ✭✶✻✺✹✕✶✼✵✺✮ ❛♥❞ ❏♦✲ ❤❛♥♥ ❬♦❢t❡♥ r❡❢❡rr❡❞ t♦ ❛s ❏♦❤♥❪ ✭✶✻✻✼✕✶✼✹✽✮✱ ❜❡✲ ❝❛♠❡ ♥♦t❡❞ ♠❛t❤❡♠❛t✐❝✐❛♥s✳ ❇♦t❤ ✇❡r❡ ♣✉♣✐❧s ♦❢ ●♦tt❢r✐❡❞ ❲✐❧❤❡❧♠ ✈♦♥ ▲❡✐❜♥✐③ ✭✶✻✹✻✕✶✼✶✻✮ ✇✐t❤ ✇❤♦♠ ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐ ❝❛rr✐❡❞ ♦♥ ❛♥ ❡①✲ t❡♥s✐✈❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❛♥❞ ✇✐t❤ ✇❤♦s❡ ✇♦r❦ ❜♦t❤ ❏❛❝♦❜ ❇❡r♥♦✉❧❧✐ ❛♥❞ ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐ ❜❡✲ ❝❛♠❡ ❢❛♠✐❧✐❛r✳ ❏❛❝♦❜ ❇❡r♥♦✉❧❧✐ ✇❛s ❛ ♣r♦❢❡s✲ s♦r ❛t t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❇❛s❡❧ ✉♥t✐❧ ❤✐s ❞❡❛t❤ ✐♥ ✶✼✵✺✳ ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐✱ ✇❤♦ ❤❛❞ ❜❡❡♥ ❛ ♣r♦❢❡ss♦r ❛t t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ●r♦♥✐♥❣❡♥ ✐♥ t❤❡ ✭♣r❡s❡♥t✲❞❛②✮ ◆❡t❤❡r❧❛♥❞s✱ r❡♣❧❛❝❡❞ ❤✐s ❜r♦t❤❡r ❛t t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❇❛s❡❧✳ ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐ ❤❛❞ ✸ s♦♥s✳ ❚✇♦ ♦❢ t❤❡♠✱ ◆✐❝❤♦❧❛s ■■ ✭✶✻✾✺✕✶✼✷✻✮ ❛♥❞ ❉❛♥✐❡❧ ✭✶✼✵✵✕✶✼✽✷✮✱ ✇❡r❡ ♠❛t❤❡♠❛t✐❝✐❛♥s ✇❤♦ ❜❡❢r✐❡♥❞❡❞ ❊✉❧❡r✳ ❚❤❡② ❜♦t❤ ✇❡♥t t♦ t❤❡ ❆❝❛❞❡♠② ✐♥ ❙t✳ P❡t❡rs❜✉r❣ ✐♥ ✶✼✷✺ ❛♥❞ t❤❡② ❜♦t❤ ❤❛❞ ❛ ❤✐❣❤ r❡❣❛r❞ ❢♦r t❤❡✐r ②♦✉♥❣❡r ❝♦❧✲ ❧❡❛❣✉❡✱ ❊✉❧❡r✳ ❆❢t❡r s♦♠❡ ❡✛♦rt✱ ❉❛♥✐❡❧ ✇r♦t❡ t♦ ❊✉❧❡r t❤❛t ❤❡ ❤❛❞ s❡❝✉r❡❞ ❢♦r ❤✐♠ ❛ st✐♣❡♥❞ ✐♥ t❤❡ ❆❝❛❞❡♠②✳ ❚❤❡ ❛♣♣♦✐♥t♠❡♥t ❢♦r ❊✉❧❡r ✇❛s ❛❝t✉❛❧❧② ✐♥ t❤❡ ♣❤②s✐♦❧♦❣② s❡❝t✐♦♥✱ ❜✉t ❊✉❧❡r q✉✐❝❦❧② ❞r✐❢t❡❞ ✐♥t♦ t❤❡ ♠❛t❤❡♠❛t✐❝s s❡❝t✐♦♥✳ ❍❡ t❤✉s ❧❡❢t ❇❛s❡❧ ❢♦r ❙t✳ P❡t❡rs❜✉r❣ ✐♥ ✶✼✷✼ ❛♥❞ r❡✲ ♠❛✐♥❡❞ t❤❡r❡ ✉♥t✐❧ ✶✼✹✶ ✇❤❡♥ ❤❡ ✇❛s s✉♠♠♦♥❡❞ ❜② ❋r❡❞❡r✐❝❦ t❤❡ ●r❡❛t ♦❢ Pr✉ss✐❛ t♦ t❤❡ ❇❡r❧✐♥ ❆❝❛❞❡♠②✳ ❊✉❧❡r ✇❛s ✐♥ ❇❡r❧✐♥ ✉♥t✐❧ ✶✼✻✻ ✇❤❡♥ ❤❡ ✇❛s s✉♠♠♦♥❡❞ ❜❛❝❦ t♦ t❤❡ ❆❝❛❞❡♠② ✐♥ ❙t✳ P❡t❡rs❜✉r❣ ✇❤❡r❡ ❤❡ r❡♠❛✐♥❡❞ ✉♥t✐❧ ❤✐s ❞❡❛t❤ ✐♥ ✶✼✽✸✳ ❊✉❧❡r ❝❛rr✐❡❞ ♦♥ ❛♥ ❡①t❡♥s✐✈❡ ❝♦rr❡s♣♦♥✲ ❞❡♥❝❡ ✇✐t❤ ✈❛r✐♦✉s ♠❛t❤❡♠❛t✐❝✐❛♥s✱ ❡s♣❡❝✐❛❧❧② ✇✐t❤ ❈❤r✐st✐❛♥ ●♦❧❞❜❛❝❤ ✭✶✻✾✵✕✶✼✻✹✮✳ ❍❡ ❛❧s♦ ✇r♦t❡ ❛ s❡r✐❡s ♦❢ ❧❡tt❡rs ♦♥ ✈❛r✐♦✉s s✉❜❥❡❝ts ✐♥ ♥❛t✉r❛❧ ♣❤✐❧♦s♦♣❤② ❛♥❞ ❛❞❞r❡ss❡❞ t❤❡s❡ ❧❡tt❡rs 4π ζ (3) = − ∞ k=0 ζ (2k) , ✭✾✮ (2k + 1) (2k + 2) · 22k ✇❤✐❝❤ ✇❛s ❛❝t✉❛❧❧② ❝♦♥t❛✐♥❡❞ ✐♥ ❊✉❧❡r✬s ✶✼✼✷ ♣❛♣❡r ❡♥t✐t❧❡❞ ✏ ❊①❡r❝✐t❛t✐♦♥❡s ❆♥❛❧②t✐❝❛❡ ✑ ✭❝❢✳✱ ❡✳❣✳✱ ❆②♦✉❜ ❬✺✱ ♣♣✳ ✶✵✽✹✕✶✵✽✺❪✮✳ ■♥ ❢❛❝t✱ ❚❤✐s ✶✼✼✷ r❡s✉❧t ♦❢ ❊✉❧❡r ✇❛s r❡❞✐s❝♦✈❡r❡❞ ✭❛♠♦♥❣ ♦t❤❡rs✮ ❜② ❘❛♠❛s✇❛♠✐ ❬✹✹❪ ✭s❡❡ ❛❧s♦ ❛ ♣❛♣❡r ❜② ❙r✐✈❛st❛✈❛ ❬✹✼✱ ♣✳ ✼✱ ❊q✉❛t✐♦♥ ✭✸✺✮❪✮ ❛♥❞ ✭♠♦r❡ r❡❝❡♥t❧②✮ ❜② ❊✇❡❧❧ ❬✶✾❪✳ ▼♦r❡♦✈❡r✱ ❥✉st ❛s ♣♦✐♥t❡❞ ♦✉t ❜② ✭❢♦r ❡①❛♠♣❧❡✮ ❈❤❡♥ ❛♥❞ ❙r✐✲ ✈❛st❛✈❛ ❬✼✱ ♣♣✳ ✶✽✵✲✶✽✶❪✱ ❛♥♦t❤❡r s❡r✐❡s r❡♣r❡✲ s❡♥t❛t✐♦♥✿ ζ (3) = ∞ k−1 (−1) , 2k k=1 k k ✭✶✵✮ ✇❤✐❝❤ ♣❧❛②❡❞ ❛ ❦❡② rô❧❡ ✐♥ t❤❡ ❝❡❧❡❜r❛t❡❞ ♣r♦♦❢ ✭s❡❡✱ ❢♦r ❞❡t❛✐❧s✱ ❬✸❪✮ ♦❢ t❤❡ ✐rr❛t✐♦♥❛❧✐t② ♦❢ ζ (3) ❜② ❘♦❣❡r ❆♣ér② ✭✶✾✶✻✲✶✾✾✹✮✱ ✇❛s ❞❡r✐✈❡❞ ✐♥✲ ❞❡♣❡♥❞❡♥t❧② ❜② ✭❛♠♦♥❣ ♦t❤❡rs✮ ❍❥♦rt♥❛❡s ❬✷✽❪✱ ●♦s♣❡r ❬✷✹❪✱ ❛♥❞ ❆♣ér② ❬✸❪✳ ❙✉❝❤ ❡❧❡❣❛♥t ❡①✲ ♣r❡ss✐♦♥s ❛s ✐♥ ✭✶✵✮ ❛r❡ ❦♥♦✇♥ ❛❧s♦ ❢♦r ζ(2) ❛♥❞ ζ(4)✿ ∞ ζ (2) = ❛♥❞ ζ (4) = 36 17 2k k=1 k k ∞ 2k k=1 k k ◆♦ s✉❝❤ s✐♥❣❧❡✲t❡r♠ s✉♠ ❡①♣r❡ss✐♦♥s ❛r❡ ❦♥♦✇♥ ❢♦r ζ(n) ✇❤❡♥ n 5✳ ■t ✐s ❡❛s✐❧② ♦❜s❡r✈❡❞ t❤❛t ❊✉❧❡r✬s s❡r✐❡s ✐♥ ✭✾✮ ❝♦♥✈❡r❣❡s ❢❛st❡r t❤❛♥ t❤❡ ❞❡✜♥✐♥❣ s❡r✐❡s ❢♦r ζ (3)✱ ❜✉t ♦❜✈✐♦✉s❧② ♥♦t ❛s ❢❛st ❛s t❤❡ s❡r✐❡s ✐♥ ✭✶✵✮✳ ■♥ ❢❛❝t✱ t❤❡ ♦r❞❡r ❡st✐♠❛t❡s ❢♦r t❤❡✐r ❣❡♥❡r❛❧ t❡r♠s ❛r❡ ❣✐✈❡♥ ❛s ❢♦❧❧♦✇s✿ ζ (3) = − ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ 4π ∞ k=0 ζ (2k) , (2k + 1) (2k + 2) 22k ✸✸✸ ❱❖▲❯▼❊✿ ✸ O k −2 · 2−2k (k → ∞) ❛♥❞ ζ (3) = ∞ k−1 (−1) , 2k k=1 k k O k − · 2−2k (k → ∞) ■t ✐s ❡s♣❡❝✐❛❧❧② r❡♠❛r❦❛❜❧❡ t❤❛t ❊✉❧❡r ✇❛s ❛❧✲ r❡❛❞② ❜❧✐♥❞ ✇❤❡♥ ❤❡ ♣❡r❢♦r♠❡❞ t❤❡ ❜r❡❛t❤t❛❦✐♥❣ ❝❛❧❝✉❧❛t✐♦♥s ❧❡❛❞✐♥❣ t♦ ❤✐s r❡s✉❧t ✭✾✮ r❛t❤❡r ♠❡♥✲ t❛❧❧②✳ ❊✈❛❧✉❛t✐♦♥s ♦❢ s✉❝❤ ❩❡t❛ ✈❛❧✉❡s ❛s ζ (3)✱ ζ (5)✱ ❡t ❝❡t❡r❛ ❛r❡ ❦♥♦✇♥ t♦ ❛r✐s❡ ♥❛t✉r❛❧❧② ✐♥ ❛ ✇✐❞❡ ✈❛r✐❡t② ♦❢ ❛♣♣❧✐❝❛t✐♦♥s s✉❝❤ ❛s t❤♦s❡ ✐♥ ❊❧❛st♦✲ st❛t✐❝s✱ ◗✉❛♥t✉♠ ❋✐❡❧❞ ❚❤❡♦r②✱ ❡t ❝❡t❡r❛ ✭s❡❡✱ ❢♦r ❡①❛♠♣❧❡✱ ❚r✐❝♦♠✐ ❬✼✸❪✱ ❲✐tt❡♥ ❬✼✼❪✱ ❛♥❞ ◆❛s❤ ❛♥❞ ❖✬❈♦♥♥♦r ❬✸✾❪✱ ❬✹✵❪✮✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❡✈❡♥ ✐♥t❡❣❡r ❛r❣✉♠❡♥ts✱ ✇❡ ❛❧r❡❛❞② ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣✉t❛t✐♦♥❛❧❧② ✉s❡❢✉❧ r❡❧❛✲ t✐♦♥s❤✐♣✿ 2n ζ (2n) = (−1) n−1 (2π) B2n · (2n)! ✭✶✶✮ (n ∈ N0 := N ∪ {0}) ✇✐t❤ t❤❡ ✇❡❧❧✲t❛❜✉❧❛t❡❞ ❇❡r♥♦✉❧❧✐ ♥✉♠❜❡rs ❞❡✜♥❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❣❡♥✲ ❡r❛t✐♥❣ ❢✉♥❝t✐♦♥✿ ∞ ez z zn = Bn − n=0 n! (|z| < 2π) , ✭✶✷✮ ♠❛♥② ❝♦♠♣✉t❛t✐♦♥❛❧❧② ✉s❡❢✉❧ s♣❡❝✐❛❧ ❝❛s❡s ❝♦♥✲ s✐❞❡r❡❞ ❤❡r❡✱ ✇❡ ♦❜s❡r✈❡ t❤❛t ζ (3) ❝❛♥ ❜❡ r❡♣r❡✲ s❡♥t❡❞ ❜② ♠❡❛♥s ♦❢ s❡r✐❡s ✇❤✐❝❤ ❝♦♥✈❡r❣❡ ♠✉❝❤ ♠♦r❡ r❛♣✐❞❧② t❤❛♥ t❤❛t ✐♥ ❊✉❧❡r✬s ❝❡❧❡❜r❛t❡❞ ❢♦r✲ ♠✉❧❛ ✭✾✮ ❛s ✇❡❧❧ ❛s t❤❛t ✐♥ t❤❡ s❡r✐❡s ✭✶✵✮ ✇❤✐❝❤ ✇❛s ✉s❡❞ r❡❝❡♥t❧② ❜② ❆♣ér② ❬✸❪ ✐♥ ❤✐s ♣r♦♦❢ ♦❢ t❤❡ ✐rr❛t✐♦♥❛❧✐t② ♦❢ ζ (3)✳ ❙②♠❜♦❧✐❝ ❛♥❞ ♥✉♠❡r✲ ✐❝❛❧ ❝♦♠♣✉t❛t✐♦♥s ✉s✐♥❣ ▼❛t❤❡♠❛t✐❝❛ ✭❱❡rs✐♦♥ ✹✳✵✮ ❢♦r ▲✐♥✉① s❤♦✇✱ ❛♠♦♥❣ ♦t❤❡r t❤✐♥❣s✱ t❤❛t ♦♥❧② ✺✵ t❡r♠s ♦❢ ♦♥❡ ♦❢ t❤❡s❡ s❡r✐❡s ❛r❡ ❝❛♣❛✲ ❜❧❡ ♦❢ ♣r♦❞✉❝✐♥❣ ❛♥ ❛❝❝✉r❛❝② ♦❢ s❡✈❡♥ ❞❡❝✐♠❛❧ ♣❧❛❝❡s✳ ■♥ t❤❡ s❡❝♦♥❞ ♣❛rt ♦❢ t❤✐s ❧❡❝t✉r❡✱ ✇❡ ❝♦♥✲ s✐❞❡r ❛ ✈❛r✐❡t② ♦❢ s❡r✐❡s ❛♥❞ ✐♥t❡❣r❛❧s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ Φ(z, s, a) ❛s ✇❡❧❧ ❛s ✐ts ✈❛r✐♦✉s ✐♥t❡r❡st✐♥❣ ❡①t❡♥s✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ✭s❡❡ ❙❡❝t✐♦♥ ✻✮✳ ✷✳ ∞ k=0 ζ (2n) = k=0 n−1 ♥ (s)k ζ (s + k, a) tk = ζ (s, a − t) k! (|t| < |a|) , (s)2k ζ (s + 2k) (2k)! m2k ζ (2k) ζ (2n − 2k) , k=1 ✭✶✸✮ (n ∈ N \ {1})✳ ❖✉r ♣r❡s❡♥t❛t✐♦♥ ✐♥ t❤✐s ❧❡❝t✉r❡ ❝♦♥s✐sts ♦❢ t✇♦ ♠❛❥♦r ♣❛rts✳ ❋✐rst ♦❢ ❛❧❧✱ ♠♦t✐✲ ✈❛t❡❞ ❡ss❡♥t✐❛❧❧② ❜② ❛ ❣❡♥✉✐♥❡ ♥❡❡❞ ✭❢♦r ❝♦♠♣✉✲ t❛t✐♦♥❛❧ ♣✉r♣♦s❡s✮ ❢♦r ❡①♣r❡ss✐♥❣ ζ (2n + 1) ❛s ❛ r❛♣✐❞❧② ❝♦♥✈❡r❣✐♥❣ s❡r✐❡s ❢♦r ❛❧❧ n ∈ N✱ ✇❡ ♣r♦✲ ♣♦s❡ t♦ ♣r❡s❡♥t ❛ r❛t❤❡r s②st❡♠❛t✐❝ ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ t❤❡ ✈❛r✐♦✉s ✐♥t❡r❡st✐♥❣ ❢❛♠✐❧✐❡s ♦❢ r❛♣✐❞❧② ❝♦♥✲ ✈❡r❣❡♥t s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r t❤❡ ❘✐❡♠❛♥♥ ζ (2n + 1) (n ∈ N)✳ ❘❡❧❡✈❛♥t ❝♦♥♥❡❝t✐♦♥s ♦❢ t❤❡ r❡s✉❧ts ♣r❡s❡♥t❡❞ ❤❡r❡ ✇✐t❤ ♠❛♥② ♦t❤❡r ❦♥♦✇♥ s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r ζ (2n + 1) (n ∈ N) ❛r❡ ❛❧s♦ ❜r✐❡✢② ✐♥❞✐❝❛t❡❞✳ ■♥ ❢❛❝t✱ ❢♦r t✇♦ ♦❢ t❤❡ ✸✸✹ ♥ ✭✶✹✮ ②✐❡❧❞s✱ ❢♦r a = ❛♥❞ t = ±1/m✱ ❛ ✉s❡❢✉❧ t❤❡ s❡r✐❡s ✐❞❡♥t✐t② ✐♥ t❤❡ ❢♦r♠✿ ∞ −1 ❙❡r✐❡s ❘❡♣r❡s❡♥t❛t✐♦♥s ❢♦r ζ (2 + 1) ( ∈ N) ❚❤❡ ❢♦❧❧♦✇✐♥❣ s✐♠♣❧❡ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❜✐♥♦✲ ♠✐❛❧ t❤❡♦r❡♠ ❛♥❞ t❤❡ ❞❡✜♥✐t✐♦♥ ✭✶✮✿ ❛s ✇❡❧❧ ❛s ❜② t❤❡ ❢❛♠✐❧✐❛r r❡❝✉rs✐♦♥ ❢♦r♠✉❧❛✿ n+ | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ s s−1 (2 − 1) ζ (s) − (m = 2) m−2 = j s s (m − 1) ζ (s) − m − ζ s, m j=2 (m ∈ N \ {1, 2}) ✭✶✺✮ ✇❤❡r❡ (λ)ν ❞❡♥♦t❡s t❤❡ ❣❡♥❡r❛❧ P♦❝❤❤❛♠♠❡r s②♠❜♦❧ ♦r t❤❡ s❤✐❢t❡❞ ❢❛❝t♦r✐❛❧✱ s✐♥❝❡ (1)n = n! (n ∈ N0 ), ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✸ ✇❤✐❝❤ ✐s ❞❡✜♥❡❞✱ ✐♥ t❡r♠s ♦❢ t❤❡ ❢❛♠✐❧✐❛r ●❛♠♠❛ ❢✉♥❝t✐♦♥✱ ❜② (λ)ν := Γ(λ + ν) Γ(λ) = 1, ✐t ❜❡✐♥❣ ✉♥❞❡rst♦♦❞ ❝♦♥✈❡♥t✐♦♥❛❧❧② t❤❛t (0)0 := ❛♥❞ ❛ss✉♠❡❞ t❛❝✐t❧② t❤❛t t❤❡ Γ✲q✉♦t✐❡♥t ❡①✐sts ✭❙❡❡✱ ❢♦r ❞❡t❛✐❧s✱ ❬✹✽❪ ❛♥❞ ❬✺✼❪✮✳ ▼❛❦✐♥❣ ✉s❡ ♦❢ t❤❡ ❢❛♠✐❧✐❛r ❤❛r♠♦♥✐❝ ♥✉♠❜❡rs Hn ❣✐✈❡♥ ❜② ζ (0) = − ; ζ (−2n) = (n ∈ N) ; ζ (0) = − log (2π) , Hn := j=1 j (n ∈ N) , ✭✶✻✮ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t ♦❢ s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r ζ (2n + 1) (n ∈ N) ✇❡r❡ ♣r♦✈❡♥ ❜② ❙r✐✈❛st❛✈❛ ❬✺✶❪ ❜② ❛♣♣❡❛❧✐♥❣ ❛♣♣r♦♣r✐❛t❡❧② t♦ t❤❡ s❡r✐❡s ✐❞❡♥t✐t② ✭✶✺✮ ✐♥ ✐ts s♣❡❝✐❛❧ ❝❛s❡s ✇❤❡♥ m = 2, 3, 4, ❛♥❞ 6✱ ❛♥❞ ❛❧s♦ t♦ ♠❛♥② ♦t❤❡r ♣r♦♣❡rt✐❡s ❛♥❞ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ t❤❡ ❘✐❡♠❛♥♥ ❩❡t❛ ❢✉♥❝t✐♦♥ s✉❝❤ ❛s t❤❡ ❢❛♠✐❧✐❛r ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥ ❢♦r ζ(s) ✭✇❤✐❝❤ ✇❛s ❞✐s❝♦✈❡r❡❞ ❜② ❊✉❧❡r ❛r♦✉♥❞ ✶✼✹✾✱ t❤❛t ✐s✱ ❛❧♠♦st ✶✶✵ ②❡❛rs ❜❡❢♦r❡ ❘✐❡♠❛♥♥✮✿ s−1 ζ (s) = · (2π) sin πs Γ (1 − s) ζ (1 − s) ✭✶✼✮ ♦r✱ ❡q✉✐✈❛❧❡♥t❧②✱ −s ζ (1 − s) = · (2π) cos πs Γ (s) ζ (s) , ✭✶✽✮ t❤❡ ❢❛♠✐❧✐❛r ❞❡r✐✈❛t✐✈❡ ❢♦r♠✉❧❛✿ ζ (−2n + ε) − ζ(−2n) ε ζ (−2n + ε) = lim ε→0 ε n (−1) = 2n (2n)! ζ (2n + 1) , (n ∈ N) , · (2π) ζ (−2n) = lim ε→0 ✭✶✾✮ s♦ t❤❛t ζ(3) ζ(5) , ζ (−4) = , 4π 4π 45 ζ(7) 315 ζ(9) ζ (−6) = , ζ (−8) = , ··· 4π 4π ✭✷✵✮ ❛♥❞ ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♠✐t r❡❧❛t✐♦♥s❤✐♣s✿ sin 12 πs s + 2n lim s→−2n = (−1) n π (n ∈ N) ✭✷✶✮ ❛♥❞ ζ (s + 2k) s + 2n lim s→−2n n ζ (−2) = − ✇✐t❤✱ ♦❢ ❝♦✉rs❡✱ (ν = 0; λ ∈ C \ {✵}) λ(λ + 1) · · · (λ + n − 1) (ν = n ∈ N; λ ∈ C), | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ (−1) n−k (2n − 2k)! ζ (2n − 2k + 1) 2(n−k) · (2π) (k = 1, , n − 1; n ∈ N \ {1}) = ❋✐rst ❙❡r✐❡s ❘❡♣r❡s❡♥t❛t✐♦♥✿ 2n H2n − log π 22n+1 − (2n)! n−1 k (−1) ζ (2k + 1) + (2n − 2k)! π 2k k=1 , (n ∈ N) ✭✷✷✮ ∞ (2k − 1)! ζ (2k) +2 (2n + 2k)! 22k ζ (2n + 1) = (−1) (2π) n−1 k=1 ❙❡❝♦♥❞ ❙❡r✐❡s ❘❡♣r❡s❡♥t❛t✐♦♥✿ 2n H2n − log 32 π · (2π) 32n+1 − (2n)! ∞ (2k − 1)! ζ (2k) +2 (2n + 2k)! 32k k=1 , (n ∈ N) ✭✷✸✮ n−1 k (−1) ζ (2k + 1) + 2k (2n − 2k)! π ζ (2n + 1) = (−1) k=1 n−1 ❚❤✐r❞ ❙❡r✐❡s ❘❡♣r❡s❡♥t❛t✐♦♥✿ 2n · (2π) n−1 ζ (2n + 1) = (−1) 4n+1 + 22n − H2n − log π (2n)! n−1 k (−1) ζ (2k + 1) + · 2k , (n ∈ N) k=1 (2n − 2k)! π ∞ (2k − 1)! ζ (2k) +2 (2n + 2k)! 42k ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ k=1 ✭✷✹✮ ✸✸✺ ❱❖▲❯▼❊✿ ✸ ✭✶✷✮ ❛♥❞ ❋♦✉rt❤ ❙❡r✐❡s ❘❡♣r❡s❡♥t❛t✐♦♥✿ 2n n−1 ζ (2n + 1) = (−1) · (2π) 2n 2n (2 + 1) + 22n − H2n − log 31 π (2n)! n−1 k ζ (2k + 1) (−1) + · , (n ∈ N) 2k k=1 (2n − 2k)! π ∞ (2k − 1)! ζ (2k) +2 (2n + 2k)! 62k k=1 ✭✷✺✮ ❍❡r❡✱ ❛s ✇❡❧❧ ❛s ❡❧s❡✇❤❡r❡ ✐♥ t❤✐s ♣r❡s❡♥t❛t✐♦♥✱ ❛♥ ❡♠♣t② s✉♠ ✐s ✉♥❞❡rst♦♦❞ ✭❛s ✉s✉❛❧✮ t♦ ❜❡ ③❡r♦✳ ❚❤❡ ✜rst s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥ ✭✷✷✮ ✐s ♠❛r❦❡❞❧② ❞✐✛❡r❡♥t ❢r♦♠ ❡❛❝❤ ♦❢ t❤❡ s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r ζ (2n + 1)✱ ✇❤✐❝❤ ✇❡r❡ ❣✐✈❡♥ ❡❛r❧✐❡r ❜② ❩❤❛♥❣ ❛♥❞ ❲✐❧❧✐❛♠s ❬✼✾✱ ♣✳ ✶✺✾✵✱ ❊q✉❛t✐♦♥ ✭✺✹✮❪ ❛♥❞ ✭s✉❜s❡q✉❡♥t❧②✮ ❜② ❈✈✐❥♦✈✐➣ ❛♥❞ ❑❧✐♥♦✇s❦✐ ❬✶✹✱ ♣✳ ✶✷✻✺✱ ❚❤❡♦r❡♠ ❆❪ ✭s❡❡ ❛❧s♦ ❬✽✵❪ ❛♥❞ ❬✽✶❪✮✳ ❙✐♥❝❡ ζ (2k) → ❛s k → ∞, t❤❡ ❣❡♥❡r❛❧ t❡r♠ ✐♥ t❤❡ s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥ ✭✷✷✮ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ♦r❞❡r ❡st✐♠❛t❡ ✿ O 2−2k · k −2n−1 ∞ 2exz zn = E (x) n ez + n=0 n! n En (0) = (−1) En (1) = ■♥ ❝❛s❡ ✇❡ s✉✐t❛❜❧② ❝♦♠❜✐♥❡ ✭✷✷✮ ❛♥❞ ✭✷✹✮✱ ✇❡ r❡❛❞✐❧② ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡r✐❡s r❡♣r❡s❡♥✲ t❛t✐♦♥✿ ✭✷✼✮ − 2n+1 Bn+1 , (n ∈ N) n+1 ✭✷✽✮ ❚❤✉s✱ ❜② ❝♦♠❜✐♥✐♥❣ ✭✷✽✮ ✇✐t❤ t❤❡ ✐❞❡♥t✐t② ✭✶✶✮✱ ✇❡ ✜♥❞ t❤❛t E2n−1 (0) = · (−1) (2π) n 2n (2n − 1)! 22n − ζ (2n) , ✭✷✾✮ ✇❤❡r❡ n ∈ N ■❢ ✇❡ ❛♣♣❧② t❤❡ r❡❧❛t✐♦♥s❤✐♣ ✭✷✾✮✱ t❤❡ s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥ ✭✷✻✮ ❝❛♥ ✐♠♠❡❞✐❛t❡❧② ❜❡ ♣✉t ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❧t❡r♥❛t✐✈❡ ❢♦r♠✿ 2n ζ (2n + 1) = (−1) n−1 + n−1 log 2 · (2π) (22n − 1) (22n+1 − 1) (2n)! k (−1) 22k − (2n − 2k)! k=1 ∞ (k → ∞; n ∈ N) , (k → ∞; n ∈ N) (|z| < π) , r❡s♣❡❝t✐✈❡❧②✱ ✐t ✐s ❦♥♦✇♥ t❤❛t ✭❝❢✳✱ ❡✳❣✳✱ ❬✸✼✱ ♣✳ ✷✾❪✮ + ✇❤❡r❡❛s t❤❡ ❣❡♥❡r❛❧ t❡r♠ ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❛❢♦r❡✲ ❝✐t❡❞ ❡❛r❧✐❡r s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ❤❛s t❤❡ ♦r❞❡r ❡st✐♠❛t❡ ❣✐✈❡♥ ❜❡❧♦✇✿ O 2−2k · k −2n | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ k−1 k=1 (−1) (2n + 2k)! ζ (2k + 1) π 2k π 2k E2k−1 (0) , ✭✸✵✮ ✇❤❡r❡ n ∈ N✱ ✇❤✐❝❤ ✐s ❛ s❧✐❣❤t❧② ♠♦❞✐✜❡❞ ❛♥❞ ❝♦rr❡❝t❡❞ ✈❡rs✐♦♥ ♦❢ ❛ r❡s✉❧t ♣r♦✈❡♥✱ ✉s✐♥❣ ❛ s✐❣✲ ♥✐✜❝❛♥t❧② ❞✐✛❡r❡♥t t❡❝❤♥✐q✉❡✱ ❜② ❚s✉♠✉r❛ ❬✼✹✱ ♣✳ ✸✽✸✱ ❚❤❡♦r❡♠ ❇❪✳ ❖♥❡ ♦t❤❡r ✐♥t❡r❡st✐♥❣ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ s❡✲ r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ✭✷✷✮ ❛♥❞ ✭✷✹✮ ❧❡❛❞s ✉s t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ✈❛r✐❛♥t ♦❢ ❚s✉♠✉r❛✬s r❡s✉❧t ✭✷✻✮ ♦r ✭✸✵✮✿ 2n · (2π) (22n − 1) (22n+1 − 1) n−1 k (−1) 22k − ζ (2k + 1) log (2n)! + (2n − 2k)! π 2k k=1 , ∞ 2k (2k − 1)! − ζ (2k) −2 4k (2n + 2k)! ζ (2n + 1) = (−1) n−1 k=1 ✭✷✻✮ ✇❤❡r❡ n ∈ N✳ ▼♦r❡♦✈❡r✱ ✐♥ t❡r♠s ♦❢ t❤❡ ❇❡r♥♦✉❧❧✐ ♥✉♠❜❡rs Bn ❛♥❞ t❤❡ ❊✉❧❡r ♣♦❧②♥♦♠✐✲ ❛❧s En (x) ❞❡✜♥❡❞ ❜② t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ✸✸✻ ζ (2n + 1) = (−1) n−1 + k=1 ∞ n−1 H2n − log π 2n 22n+1 − (2n)! k (−1) 22k+1 − (2n − 2k)! −4 k=1 4π ζ (2k + 1) π 2k (2k − 1)! 22k−1 − (2n + 2k)! ζ (2k) , 24k ✭✸✶✮ ✇❤❡r❡ n ∈ N, ✇❤✐❝❤ ✐s ❡ss❡♥t✐❛❧❧② t❤❡ s❛♠❡ ❛s t❤❡ ❞❡t❡r♠✐♥❛♥t❛❧ ❡①♣r❡ss✐♦♥ ❢♦r ζ (2n + 1) ❞❡r✐✈❡❞ ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✸ ❜② ❊✇❡❧❧ ❬✷✵✱ ♣✳ ✶✵✶✵✱ ❈♦r♦❧❧❛r② ✸❪ ❜② ❡♠♣❧♦②✐♥❣ ❛♥ ❡♥t✐r❡❧② ❞✐✛❡r❡♥t t❡❝❤♥✐q✉❡ ❢r♦♠ ♦✉rs✳ ❆ ♥✉♠❜❡r ♦❢ ♦t❤❡r s✐♠✐❧❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ t❤❡ s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ✭✷✷✮ t♦ ✭✷✺✮ ✇♦✉❧❞ ②✐❡❧❞ s♦♠❡ ✐♥t❡r❡st✐♥❣ ❝♦♠♣❛♥✐♦♥s ♦❢ ❊✇❡❧❧✬s r❡s✉❧t ✭✸✶✮✳ ◆❡①t✱ ❜② s❡tt✐♥❣ t = 1/m ❛♥❞ ❞✐✛❡r❡♥t✐❛t✐♥❣ ❜♦t❤ s✐❞❡s ✇✐t❤ r❡s♣❡❝t t♦ s✱ ✇❡ ✜♥❞ ❢r♦♠ t❤❡ ❢♦❧✲ ❧♦✇✐♥❣ ♦❜✈✐♦✉s ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ s❡r✐❡s ✐❞❡♥t✐t② ✭✶✹✮✿ ∞ k=0 = (s)2k+1 ζ (s + 2k + 1, a) t2k+1 (2k + 1)! [ζ (s, a − t) − ζ (s, a + t)] , (|t| < |a|) ✭✸✷✮ t❤❛t | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ ✇❤✐❝❤✱ ✐♥ ❧✐❣❤t ♦❢ t❤❡ ❡❧❡♠❡♥t❛r② ✐❞❡♥t✐t②✿ (2k)! (2k − 1)! = (2n + 2k)! (2n + 2k − 1)! (2k − 1)! − 2n , (n ∈ N) , (2n + 2k)! ✭✸✻✮ ✇♦✉❧❞ ❝♦♠❜✐♥❡ ✇✐t❤ t❤❡ r❡s✉❧t ✭✷✷✮ t♦ ②✐❡❧❞ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥✿ 2n (2π) n (22n+1 − 1) n−1 k−1 (−1) k ζ (2k + 1) π 2k k=1 (2n − 2k)! , (n ∈ N) ✭✸✼✮ · ∞ (2k)! ζ (2k) + (2n + 2k)! 22k ζ (2n + 1) = (−1) n k=0 ∞ ζ (s + 2k + 1, a) (s)2k+1 2k (2k + 1)! m +ζ (s + 2k + 1, a) k=0 2k j=0 ❚❤✐s ❧❛st s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥ ✭✸✼✮ ✐s ♣r❡❝✐s❡❧② t❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ ♠❛✐♥ r❡s✉❧t ♦❢ ❈✈✐❥♦✈✐➣ ❛♥❞ s + j ❑❧✐♥♦✇s❦✐ ❬✶✹✱ ♣✳ ✶✷✻✺✱ ❚❤❡♦r❡♠ ❆❪✳ ❆s ❛ ♠❛t✲ t❡r ♦❢ ❢❛❝t✱ ✐♥ ✈✐❡✇ ♦❢ ❛ ❦♥♦✇♥ ❞❡r✐✈❛t✐✈❡ ❢♦r✲ ♠✉❧❛ ❬✺✶✱ ♣✳ ✸✽✾✱ ❊q✉❛t✐♦♥ ✭✷✶✮❪✱ t❤❡ s❡r✐❡s r❡♣✲ ✭✸✸✮ r❡s❡♥t❛t✐♦♥ ✭✸✼✮ ✐s ❡ss❡♥t✐❛❧❧② t❤❡ s❛♠❡ ❛s ❛ r❡✲ s✉❧t ❣✐✈❡♥ ❡❛r❧✐❡r ❜② ❩❤❛♥❣ ❛♥❞ ❲✐❧❧✐❛♠s ❬✼✾✱ ✇❤❡r❡ ♠ ∈ N \ {✶} ■♥ t❤❡ ♣❛rt✐❝✉❧❛r ❝❛s❡ ♣✳ ✶✺✾✵✱ ❊q✉❛t✐♦♥ ✭✺✹✮❪ ✭s❡❡ ❛❧s♦ ❩❤❛♥❣ ❛♥❞ ❲✐❧❧✐❛♠s ❬✼✾✱ ♣✳ ✶✺✾✶✱ ❊q✉❛t✐♦♥ ✭✺✼✮❪ ✇❤❡r❡ ✇❤❡♥ m = 2, ✭✸✸✮ ✐♠♠❡❞✐❛t❡❧② ②✐❡❧❞s ❛♥ ♦❜✈✐♦✉s❧② ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ✭❛s②♠♣t♦t✐❝ ✮ ✈❡r✲ ζ (s + 2k + 1, a) s✐♦♥ ♦❢ ✭✸✼✮ ✇❛s ♣r♦✈❡♥ s✐♠✐❧❛r❧②✮✳ ∞ m ∂ = ∂s ζ s, a − m − ζ s, a + m (s)2k+1 2k (2k + 1)! +ζ (s + 2k + 1, a) k=0 2k j=0 =− a− −s log a − , s+j ✭✸✹✮ ❯♣♦♥ ❧❡tt✐♥❣ s → −2n − (n ∈ N) ✐♥ t❤❡ ❢✉rt❤❡r s♣❡❝✐❛❧ ♦❢ t❤✐s ❧❛st ✐❞❡♥t✐t② ✭✸✹✮ ✇❤❡♥ a = 1, ❲✐❧t♦♥ ❬✺✼✱ ♣✳ ✾✷❪ ❞❡❞✉❝❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡✲ r✐❡s r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r ζ (2n + 1) ✭s❡❡ ❛❧s♦ ❬✷✼✱ ♣✳ ✸✺✼✱ ❊♥tr② ✭✺✹✳✻✳✾✮❪✮✿ H2n+1 − log π (2n + 1)! n−1 k (−1) ζ (2k + 1) + (2n − 2k + 1)! π 2k k=1 , (n ∈ N) , ∞ (2k − 1)! ζ (2k) +2 (2n + 2k + 1)! 22k ζ (2n + 1) = (−1) n−1 π 2n k=1 ■♥ ❧✐❣❤t ♦❢ ❛♥♦t❤❡r ❡❧❡♠❡♥t❛r② ✐❞❡♥t✐t②✿ (2k)! (2k − 1)! = (2n + 2k + 1)! (2n + 2k)! − (2n + 1) (2k − 1)! , ✭✸✽✮ (2n + 2k + 1)! ✇❤❡r❡ n, k ∈ N, ✇❡ ❝❛♥ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ②❡t ❛♥♦t❤❡r s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r ζ (2n + 1) ❜② ❛♣♣❧②✐♥❣ ✭✷✷✮ ❛♥❞ ✭✸✺✮✿ 2n · (2π) (2n − 1) 22n + n−1 k−1 (−1) ζ (2k + 1) k π 2k k=1 (2n − 2k + 1)! , (n ∈ N) , · ∞ (2k)! ζ (2k) + (2n + 2k + 1)! 22k ζ (2n + 1) = (−1) ✭✸✺✮ ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ n k=0 ✭✸✾✮ ✸✸✼ ❱❖▲❯▼❊✿ ✸ ✇❤✐❝❤ ♣r♦✈✐❞❡s ❛ s✐❣♥✐✜❝❛♥t❧② s✐♠♣❧❡r ✭❛♥❞ ♠✉❝❤ ♠♦r❡ r❛♣✐❞❧② ❝♦♥✈❡r❣❡♥t✮ ✈❡rs✐♦♥ ♦❢ t❤❡ ❢♦❧❧♦✇✲ ✐♥❣ ♦t❤❡r ♠❛✐♥ r❡s✉❧t ♦❢ ❈✈✐❥♦✈✐➣ ❛♥❞ ❑❧✐♥♦✇s❦✐ ❬✶✹✱ ♣✳ ✶✷✻✺✱ ❚❤❡♦r❡♠ ❇❪✿ 2n ∞ n ζ (2n + 1) = (−1) · (2π) (2n)! Ωn,k k=0 ζ (2k) , 22k | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ t✇♦ ✭♦♥❧② s❡❡♠✐♥❣❧② ❞✐✛❡r❡♥t✮ ✈❡rs✐♦♥s ♦❢ t❤❡ s❡✲ r✐❡s r❡♣r❡s❡♥t❛t✐♦♥ ✭✸✼✮✳ ■♥❞❡❡❞✱ ✐❢ ✇❡ ❛♣♣❡❛❧ t♦ ✭✹✸✮ ✇✐t❤ m = 4✱ ✇❡ ❝❛♥ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠✉❝❤ ♠♦r❡ r❛♣✐❞❧② ❝♦♥✈❡r❣❡♥t s❡r✐❡s r❡♣r❡s❡♥t❛✲ t✐♦♥ ❢♦r ζ (2n + 1) ✭s❡❡ ❬✺✵✱ ♣✳ ✾✱ ❊q✉❛t✐♦♥ ✭✹✶✮❪✮✿ ✭✹✵✮ ✇❤❡r❡ n 2n ∈ N ❛♥❞ t❤❡ ❝♦❡✣❝✐❡♥ts Ωn,k · (2π) n (n ∈ N; k ∈ N0 ) ❛r❡ ❣✐✈❡♥ ❡①♣❧✐❝✐t❧② ❛s ❛ ✜♥✐t❡ ζ (2n + 1) = (−1) n (24n+1 + 22n − 1) s✉♠ ♦❢ ❇❡r♥♦✉❧❧✐ ♥✉♠❜❡rs ❬✶✹✱ ♣✳ ✶✷✻✺✱ ❚❤❡✲ n−1 −1 22n−1 − ♦r❡♠ ❇✭✐✮❪ ✭s❡❡✱ ❢♦r ❞❡t❛✐❧s✱ ❙r✐✈❛st❛✈❛ ❬✺✶✱ ♣♣✳ (2n)! B2n log − (2n − 1)! ζ (1 − 2n) ✸✾✸✲✸✾✹❪✮✿ 42n−1 2n ζ − 2n, − 2n B2n−j (2n − 1)! , Ωn,k := j , (j + 2k + 1) (j + 1) · j n−1 k−1 · j=0 (−1) k ζ (2k + 1) ✭✹✶✮ 2k + (2n − 2k)! π k=1 ✇❤❡r❡ n ∈ N; k ∈ N0 ∞ (2k)! ζ (2k) + (2n + 2k)! 42k ✸✳ ❖t❤❡r ❋❛♠✐❧✐❡s ♦❢ ❙❡r✐❡s ❘❡♣r❡s❡♥t❛t✐♦♥s ❢♦r ♥ ♥ k=0 ✭✹✹✮ ζ (2 + 1) ( ∈ N) ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ st❛rt ♦♥❝❡ ❛❣❛✐♥ ❢r♦♠ t❤❡ ✐❞❡♥t✐t② ✭✶✹✮ ✇✐t❤ ✭♦❢ ❝♦✉rs❡✮ a = 1, t = ±1/m, ❛♥❞ s r❡♣❧❛❝❡❞ ❜② s + ❚❤✉s✱ ❜② ❛♣♣❧②✐♥❣ ✭✶✺✮✱ ✇❡ ✜♥❞ ②❡t ❛♥♦t❤❡r ❝❧❛ss ♦❢ s❡r✐❡s ✐❞❡♥t✐t✐❡s ✐♥✲ ❝❧✉❞✐♥❣✱ ❢♦r ❡①❛♠♣❧❡✱ ∞ k=1 (s + 1)2k ζ (s + 2k) = (2s − 2) ζ (s) ✭✹✷✮ (2k)! 22k ✇❤❡r❡ n ∈ N ❛♥❞ ✭❛♥❞ ✐♥ ✇❤❛t ❢♦❧❧♦✇s ✮ ❛ ♣r✐♠❡ ❞❡♥♦t❡s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ζ (s) ♦r ζ (s, a) ✇✐t❤ r❡s♣❡❝t t♦ s✳ ❇② ✈✐rt✉❡ ♦❢ t❤❡ ✐❞❡♥t✐t✐❡s ✭✸✻✮ ❛♥❞ ✭✸✽✮✱ t❤❡ r❡s✉❧ts ✭✷✹✮ ❛♥❞ ✭✹✹✮ ✇♦✉❧❞ ❧❡❛❞ ✉s ❡✈❡♥t✉❛❧❧② t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❞❞✐t✐♦♥❛❧ s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r ζ (2n + 1) (n ∈ N) ✭s❡❡ ❬✺✵✱ ♣✳ ✶✵✱ ❊q✉❛t✐♦♥s ✭✹✷✮ ❛♥❞ ✭✹✸✮❪✮✿ ❛♥❞ ∞ (s + 1)2k ζ (s + 2k) 2n H2n+1 − log π n−1 π (2k)! m2k ζ (2n + 1) = (−1) k=1 (2n + 1)! s s+1 m (m − 3) ζ (s) + m − ζ (s + 1) (4n − 1) 22n+1 − B2n+2 log − ζ (−2n − 1) −2ζ s + 1, + (2n + 2)! (2n + 1)! m , = 4n+3 2m m−2 − ζ −2n − 1, j j (2n + 1)! − mζ s, + ζ s + 1, m m n−1 j=2 k (−1) ζ (2k + 1) ✭✹✸✮ + 2k (2n − 2k + 1)! π k=1 ✇❤❡r❡ m ∈ N \ {1, 2} ■♥ ❢❛❝t✱ ✐t ✐s t❤❡ s❡r✐❡s ∞ (2k − 1)! ζ (2k) ✐❞❡♥t✐t② ✭✹✷✮ ✇❤✐❝❤ ✇❛s ✜rst ❛♣♣❧✐❡❞ ❜② ❩❤❛♥❣ +2 2k (2n + 2k + 1)! ❛♥❞ ❲✐❧❧✐❛♠s ❬✼✾❪ ✭❛♥❞✱ s✉❜s❡q✉❡♥t❧②✱ ❜② ❈✈✐✲ k=1 ❥♦✈✐➣ ❛♥❞ ❑❧✐♥♦✇s❦✐ ❬✶✹❪✮ ✇✐t❤ ❛ ✈✐❡✇ t♦ ♣r♦✈✐♥❣ ✸✸✽ ✭✹✺✮ ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✸ ❛♥❞ ❬✹✵❪✮ ♦❜t❛✐♥❡❞ ❛ ♥✉♠❜❡r ♦❢ r❡♠❛r❦❛❜❧❡ ✐♥✲ t❡❣r❛❧ ❡①♣r❡ss✐♦♥s ❢♦r ζ (3)✱ ✐♥❝❧✉❞✐♥❣ ✭❢♦r ❡①❛♠✲ ♣❧❡✮ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ❬✷✻✱ ♣✳ ✶✹✽✾ ❡t s❡q✳❪✿ | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ (0 < τ < 2)✱ ❙r✐✈❛st❛✈❛ ❬✸✼❪ ❞❡r✐✈❡❞ t❤❡ ❢♦❧❧♦✇✲ ✐♥❣ s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r ζ (2n + 1) ✭s❡❡ ❛❧s♦ t❤❡ ✇♦r❦ ❜② ❙r✐✈❛st❛✈❛ ❡t ❛❧✳ ❬✻✷❪✮✿ 2n (2π) n−1 z cot z dz ✭✺✶✮ ζ (2n + 1) = (−1) (2n)! (22n+1 − 1) log ■♥ ❢❛❝t✱ ✐♥ ✈✐❡✇ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡r✐❡s ❡①♣❛♥s✐♦♥ n−1 2n (2j)! 22j − j ❬✶✼✱ ♣✳ ✺✶✱ ❊q✉❛t✐♦♥ ✶✳✷✵✭✸✮❪✿ + (−1) ζ (2j + 1) 2j 2j (2π) · j=1 , ∞ ∞ z 2k ζ (2k) z cot z = −2 ζ (2k) , (|z| < π) , ✭✺✷✮ + π 2k k=0 (k + n) · π/2 2π ζ (3) = log − 7 k=0 t❤❡ r❡s✉❧t ✭✺✶✮ ❡q✉✐✈❛❧❡♥t t♦ t❤❡ s❡r✐❡s r❡♣r❡s❡♥✲ t❛t✐♦♥ ✭ ❝❢✳ t❤❡ ✇♦r❦ ❜② ❉☛ ❛❜r♦✇s❦✐ ❬✶✻✱ ♣✳ ✷✵✷❪❀ s❡❡ ❛❧s♦ t❤❡ ♣❛♣❡r ❜② ❈❤❡♥ ❛♥❞ ❙r✐✈❛st❛✈❛ ❬✼✱ ♣✳ ✶✾✶✱ ❊q✉❛t✐♦♥ ✭✻✵✮❪✮✿ 2π ζ (3) = ∞ log + k=0 ζ (2k) (k + 1) · 22k ✭✺✸✮ ▼♦r❡♦✈❡r✱ ✐❢ ✇❡ ✐♥t❡❣r❛t❡ ❜② ♣❛rts✱ ✇❡ ❡❛s✐❧② ✜♥❞ t❤❛t π/2 π/2 z cot z dz = −2 z log sin z dz, ✭✺✹✮ s♦ t❤❛t t❤❡ r❡s✉❧t ✭✺✶✮ ✐s ❡q✉✐✈❛❧❡♥t ❛❧s♦ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡❣r❛❧ r❡♣r❡s❡♥t❛t✐♦♥✿ ζ (3) = 16 2π log + 7 π/2 z log sin z dz, ✭✺✺✮ ❋✉rt❤❡r♠♦r❡✱ s✐♥❝❡ e2z √ + 1, i := −1 , −1 ✭✺✻✮ ❜② r❡♣❧❛❝✐♥❣ z ✐♥ t❤❡ ❦♥♦✇♥ ❡①♣❛♥s✐♦♥ ✭✺✷✮ ❜② iπz, ✐t ✐s ❡❛s✐❧② s❡❡♥ t❤❛t ✭❝❢✳✱ ❡✳❣✳✱ ❬✷✵✱ ♣✳ ✷✺❪❀ s❡❡ ❛❧s♦ ❬✶✼✱ ♣✳ ✺✶✱ ❊q✉❛t✐♦♥ ✶✳✷✵✭✶✮❪✮ πz πz + = eπz − ∞ k=0 k+1 (−1) ζ (2k) 2k z , (|z| < 2) 22k−1 ✭✺✼✮ ❯♣♦♥ s❡tt✐♥❣ z = it ✐♥ ✭✺✼✮✱ ♠✉❧t✐♣❧②✐♥❣ ❜♦t❤ s✐❞❡s ❜② tm−1 (m ∈ N)✱ ❛♥❞ t❤❡♥ ✐♥t❡❣r❛t✐♥❣ t❤❡ r❡s✉❧t✐♥❣ ❡q✉❛t✐♦♥ ❢r♦♠ t = t♦ t = τ ✸✹✵ ❛♥❞ 2n n−1 ζ (2n + 1) = (−1) (2π) (2n + 1)! (22n − 1) log n−1 2n + j + (−1) 2j j=1 , 2j · (2j)! − · ζ (2j + 1) 2j (2π) ∞ ζ (2k) + 2k k + n + · 2 k=0 ✭✺✾✮ ✇❤❡r❡ n ∈ N✳ ❯♣♦♥ s❡tt✐♥❣ n = 1, ✭✺✾✮ ✐♠♠❡❞✐✲ ❛t❡❧② r❡❞✉❝❡s t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡r✐❡s r❡♣r❡s❡♥t❛✲ t✐♦♥ ❢♦r ζ (3)✿ ✇❤✐❝❤ ✇❛s ♣r♦✈❡♥ ✐♥ t❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ ✶✼✼✷ ♣❛♣❡r ❜② ❊✉❧❡r ✭❝❢✳✱ ❡✳❣✳✱ ❬✺✱ ♣✳ ✶✵✽✹❪✮✳ i cot iz = coth z = ✭✺✽✮ ζ (3) = 2π ∞ log + k=0 ζ (2k) (2k + 3) · 22k , ✭✻✵✮ ✇❤✐❝❤ ✇❛s ♣r♦✈❡♥ ✐♥❞❡♣❡♥❞❡♥t❧② ❜② ✭❛♠♦♥❣ ♦t❤✲ ❡rs✮ ●❧❛ss❡r ❬✷✸✱ ♣✳ ✹✹✻✱ ❊q✉❛t✐♦♥ ✭✶✷✮❪✱ ❩❤❛♥❣ ❛♥❞ ❲✐❧❧✐❛♠s ❬✼✾✱ ♣✳ ✶✺✽✺✱ ❊q✉❛t✐♦♥ ✭✷✺✮❪✱ ❛♥❞ ❉☛ ❛❜r♦✇s❦✐ ❬✶✻✱ ♣✳ ✷✵✻❪ ✭s❡❡ ❛❧s♦ t❤❡ ✇♦r❦ ❜② ❈❤❡♥ ❛♥❞ ❙r✐✈❛st❛✈❛ ❬✼✱ ♣✳ ✶✽✸✱ ❊q✉❛t✐♦♥ ✭✷✼✮❪✮✳ ❋✉rt❤❡r♠♦r❡✱ ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ ✭✺✽✮ ✇❤❡♥ n = ②✐❡❧❞s ✭❝❢✳ ❉☛ ❛❜r♦✇s❦✐ ❬✶✻✱ ♣✳ ✷✵✷❪❀ s❡❡ ❛❧s♦ ❈❤❡♥ ❛♥❞ ❙r✐✈❛st❛✈❛ ❬✼✱ ✺✱ ♣✳ ✶✾✶✱ ❊q✉❛t✐♦♥ ✭✻✵✮❪✮ ζ (3) = 2π ∞ log + k=0 ζ (2k) (k + 1) · 22k ✭✻✶✮ ■♥ ❢❛❝t✱ ✐♥ ✈✐❡✇ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛♠✐❧✐❛r s✉♠✿ ∞ k=0 ζ (2k) = − log 2, (2k + 1) · 22k ✭✻✷✮ ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✸ ❊✉❧❡r✬s ❢♦r♠✉❧❛ ✭✾✮✱ t❤❛t ✐s✱ ζ (3) = − 4π ∞ k=0 ✭✺✽✮❪✮✿ ζ (2k) , (2k + 1) (2k + 2) · 22k n j−1 (−1) j=1 ∞ ✐s ✐♥❞❡❡❞ ❛ r❛t❤❡r s✐♠♣❧❡ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ✭✻✶✮✳ ■♥ ♣❛ss✐♥❣✱ ✇❡ ✜♥❞ ✐t ✇♦rt❤✇❤✐❧❡ t♦ r❡♠❛r❦ t❤❛t ❛♥ ✐♥t❡❣r❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r ζ (2n + 1)✱ ✇❤✐❝❤ ✐s ❡❛s✐❧② s❡❡♥ t♦ ❜❡ ❡q✉✐✈❛❧❡♥t t♦ t❤❡ s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥ ✭✺✽✮✱ ✇❛s ❣✐✈❡♥ ❜② ❉☛ ❛❜r♦✇s❦✐ ❬✶✻✱ ♣✳ ✷✵✸✱ ❊q✉❛t✐♦♥ ✭✶✻✮❪✱ ✇❤♦ ❬✶✻✱ ♣✳ ✷✵✻❪ ♠❡♥✲ t✐♦♥❡❞ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ✭❜✉t ❞✐❞ ♥♦t ❢✉❧❧② st❛t❡✮ t❤❡ s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥ ✭✺✾✮ ❛s ✇❡❧❧✳ ❚❤❡ s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥ ✭✺✽✮ ✇❛s ❞❡r✐✈❡❞ ❛❧s♦ ✐♥ ❛ ♣❛♣❡r ❜② ❇♦r✇❡✐♥ ❡t ❛❧✳ ✭❝❢✳ ❬✻✱ ♣✳ ✷✻✾✱ ❊q✉❛t✐♦♥ ✭✺✼✮❪✮✳ ■❢ ✇❡ s✉✐t❛❜❧② ❝♦♠❜✐♥❡ t❤❡ s❡r✐❡s ♦❝❝✉rr✐♥❣ ✐♥ ✭✺✸✮✱ ✭✻✵✮✱ ❛♥❞ ✭✻✷✮✱ ✐t ✐s ♥♦t ❞✐✣❝✉❧t t♦ ❞❡❞✉❝❡ s❡✈❡r❛❧ ♦t❤❡r s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r ζ (3)✱ ✇❤✐❝❤ ❛r❡ ❛♥❛❧♦❣♦✉s t♦ ❊✉❧❡r✬s ❢♦r♠✉❧❛ ✭✾✮✱ t❤❛t ✐s✱ ζ (3) = − 4π ∞ k=0 ζ (2k) (2k + 1) (2k + 2) · 22k ▼♦r❡ ❣❡♥❡r❛❧❧②✱ s✐♥❝❡ λk + µk + ν (2k + 2n − 1) (2k + 2n) (2k + 2n + 1) A B C = + + , 2k + 2n − 2k + 2n 2k + 2n + ✭✻✸✮ = log + k=0 ζ (2n + 1) n−1 B = Bn (λ, µ, ν) := − λn2 − µn + ν , ✭✻✺✮ 2n (−1) (2π) 2n+1 (2n)! {(2 − 1) B + (2n + 1) (22n − 1) C} n−1 2n − 1 j (−1) λ log + 2j − j=1 2j (2j − 1) A · + [λ (4n − 1) − 2µ] nj + λn n + · (2j − 2)! 22j − ζ (2j + 1) · 2j (2π) ∞ + E(k) = k=0 ✭✻✽✮ ✇❤❡r❡ A = An (λ, µ, ν) ✭✻✹✮ ζ (2k) , (n ∈ N0 ) , k + n + 12 · 22k ✇✐t❤ n r❡♣❧❛❝❡❞ ❜② n − 1, ❙r✐✈❛st❛✈❛ ❬✺✷❪ ❞❡✲ r✐✈❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ✉♥✐✜❝❛t✐♦♥ ♦❢ ❛ ❧❛r❣❡ ♥✉♠✲ ❜❡r ♦❢ ❦♥♦✇♥ ✭♦r ♥❡✇✮ s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r ζ (2n + 1) (n ∈ N)✱ ✐♥❝❧✉❞✐♥❣ ✭❢♦r ❡①❛♠♣❧❡✮ ❊✉✲ ❧❡r✬s ❢♦r♠✉❧❛ ✭✾✮✿ E(k) := 1 λn2 − (λ + µ) n + (λ + 2µ + 4ν) , 2n + (2j)! 22j − ζ (2j + 1) 2j 2j (2π) ✭✻✼✮ ✇❤❡r❡✱ ❢♦r ❝♦♥✈❡♥✐❡♥❝❡✱ := | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ λk + µk + ν ζ (2k) (2k + 2n − 1) (k + n) (2k + 2n + 1) · 22k ❛♥❞ n ∈ N; λ, µ, ν ∈ C ❛♥❞ A, B, ❛♥❞ C ❛r❡ ❣✐✈❡♥ ❜② ✭✻✹✮✱ ✭✻✺✮✱ ❛♥❞ ✭✻✻✮✱ r❡s♣❡❝t✐✈❡❧②✳ ◆✉♠❡r♦✉s ♦t❤❡r ✐♥t❡r❡st✐♥❣ s❡r✐❡s r❡♣r❡s❡♥t❛✲ t✐♦♥s ❢♦r ζ (2n + 1)✱ ✇❤✐❝❤ ❛r❡ ❛♥❛❧♦❣♦✉s t♦ ✭✺✽✮ ❛♥❞ ✭✺✾✮✱ ✇❡r❡ ❛❧s♦ ❣✐✈❡♥ ❜② ❙r✐✈❛st❛✈❛ ❡t ❛❧✳ ❬✻✷❪✳ ❛♥❞ ✹✳ ✭✻✻✮ ❈♦♠♣✉t❛t✐♦♥❛❧❧② ❯s❡❢✉❧ ❉❡❞✉❝t✐♦♥s ❛♥❞ ❈♦♥s❡q✉❡♥❝❡s ❜② ❛♣♣❧②✐♥❣ ✭✺✽✮✱ ✭✺✾✮✱ ❛♥❞ ❛♥♦t❤❡r r❡s✉❧t ✭♣r♦✈❡♥ ❜② ❙r✐✈❛st❛✈❛ ❬✺✷✱ ♣✳ ✸✹✶✱ ❊q✉❛t✐♦♥ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ s✉✐t❛❜❧② s♣❡❝✐❛❧✐③❡ t❤❡ ♣❛✲ r❛♠❡t❡r λ, µ, ❛♥❞ ν ✐♥ ✭✻✽✮ ❛♥❞ t❤❡♥ ❛♣♣❧② ❛ C = Cn (λ, µ, ν) := 1 λn2 + (λ − µ) n + (λ − 2µ + 4ν) , ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✸✹✶ ❱❖▲❯▼❊✿ ✸ r❛t❤❡r ❡❧❛❜♦r❛t❡ s❝❤❡♠❡✳ ❲❡ t❤✉s ❡✈❡♥t✉❛❧❧② ❛r✲ r✐✈❡ ❛t t❤❡ ❢♦❧❧♦✇✐♥❣ r❡♠❛r❦❛❜❧② r❛♣✐❞❧② ❝♦♥✈❡r✲ ❣❡♥t s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r ζ (2n + 1) (n ∈ N)✱ ✇❤✐❝❤ ✇❛s ❞❡r✐✈❡❞ ❜② ❙r✐✈❛st❛✈❛ ❬✺✷✱ ♣♣✳ ✸✹✽✕ ✸✹✾✱ ❊q✉❛t✐♦♥ ✭✸✳✺✵✮❪✮✿ | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ ■♥ ✐ts s♣❡❝✐❛❧ ❝❛s❡ ✇❤❡♥ n = 1, ✭✻✾✮ ②✐❡❧❞s t❤❡ ❢♦❧❧♦✇✐♥❣ ✭r❛t❤❡r ❝✉r✐♦✉s ✮ s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥✿ ζ (3) = − 6π 23 ∞ k=0 (98k + 121) ζ (2k) n−1 (2π) ζ (2n + 1) = (−1) (2k + 1) (2k + 2) (2k + 3) (2k + 4) (2k + 5) · 22k (2n)!∆n ✭✼✸✮ n−1 2j (2j)! − j (−1) · ζ (2j + 1) 2j (2π) O k −4 · 2−2k (k → ∞) j=1 2n+2 (2n − 3) − 2n ✇❤❡r❡ t❤❡ s❡r✐❡s ♦❜✈✐♦✉s❧② ❝♦♥✈❡r❣❡s ♠✉❝❤ ♠♦r❡ 2n − r❛♣✐❞❧② t❤❛♥ t❤❛t ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❝❡❧❡❜r❛t❡❞ r❡s✉❧ts 2n + − ✭✾✮ ❛♥❞ ✭✶✵✮✱ t❤❛t ✐s✱ 2j 2j · ∞ 2n − ζ (2k) 4π +6n ζ (3) = − 2j − (2k + 1) (2k + 2) · 22k k=0 ✭✻✾✮ · · − 22n+3 − 2n O k −2 · 2−2k (k → ∞) 2n + − 2j 2j ❛♥❞ · ∞ k−1 2n + (−1) ζ (3) = +3 2k 2j − k=1 k k ∞ (ξn k + ηn ) ζ (2k) +12 E (k) k=0 (k → ∞) O k − · 2−2k 2n ✇❤❡r❡ n ∈ N✱ ❆♥ ✐♥t❡r❡st✐♥❣ ❝♦♠♣❛♥✐♦♥ ♦❢ ✭✼✸✮ ✐♥ t❤❡ ❢♦❧❧♦✇✲ ✐♥❣ ❢♦r♠✿ E (k) := (2k + 2n − 1) (2k + 2n) · (2k + 2n + 1) (2k + 2n + 2) ζ (3) = − · (2k + 2n + 3) · 22k ∆n := 22n+3 − − (2n − 3) · 2n+2 k=0 O k −5 · 2−2k (2n + 1) 2n2 − 4n + 3 2n+2 ∞ 8576k + 24286k + 17283 ζ (2k) E(k) 22k ❛♥❞✱ ❢♦r ❝♦♥✈❡♥✐❡♥❝❡✱ · 120 π 1573 2n 2n+1 −1 −2 (k → ∞) , +1 − 2n + n (2n − 3) 22n − − , ✭✼✹✮ ✭✼✵✮ ✇❤❡r❡ E(k) := (2k + 1) (2k + 2) · (2k + 3) (2k + 4) · (2k + 5) (2k + 6) (2k + 7) , ξn := (2n − 5) 22n+2 − 2n + , ✭✼✶✮ ❛♥❞ ηn := 4n2 − 4n − 22n+2 − (2n + 1) ✭✼✷✮ ✸✹✷ ✇❛s ❞❡❞✉❝❡❞ ❜② ❙r✐✈❛st❛✈❛ ❛♥❞ ❚s✉♠✉r❛ ❬✼✶❪✱ ✇❤♦ ✐♥❞❡❡❞ ♣r❡s❡♥t❡❞ ❛♥ ✐♥❞✉❝t✐✈❡ ❝♦♥str✉❝✲ t✐♦♥ ♦❢ s❡✈❡r❛❧ ❣❡♥❡r❛❧ s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r ζ (2n + 1) (n ∈ N) ✭s❡❡ ❛❧s♦ ❬✼✵❪✮✳ ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✸ ✺✳ ◆✉♠❡r✐❝❛❧ ❱❡r✐✜❝❛t✐♦♥s ❛♥❞ ❙②♠❜♦❧✐❝ ❈♦♠♣✉t❛t✐♦♥s ❇❛s❡❞ ❯♣♦♥ ✭❱❡rs✐♦♥ ✹✳✵✮ ▼❛t❤❡♠❛t✐❝❛ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✜rst s✉♠♠❛r✐③❡ t❤❡ r❡s✉❧ts ♦❢ ♥✉♠❡r✐❝❛❧ ✈❡r✐✜❝❛t✐♦♥s ❛♥❞ s②♠❜♦❧✐❝ ❝♦♠♣✉t❛✲ t✐♦♥s ✇✐t❤ t❤❡ s❡r✐❡s ✐♥ ✭✼✸✮ ❜② ✉s✐♥❣ ▼❛t❤❡♠❛t✲ ✐❝❛ ✭❱❡rs✐♦♥ ✹✳✵✮ ❢♦r ▲✐♥✉①✿ | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ ❖✉t❬✷❪ ❡✈✐❞❡♥t❧② ✈❛❧✐❞❛t❡s t❤❡ s❡r✐❡s r❡♣r❡s❡♥t❛✲ t✐♦♥ ✭✼✸✮ s②♠❜♦❧✐❝❛❧❧②✳ ❋✉rt❤❡r♠♦r❡✱ ♦✉r ♥✉♠❡r✲ ✐❝❛❧ ❝♦♠♣✉t❛t✐♦♥s ✐♥ ❖✉t❬✸❪✱ ❖✉t❬✹❪✱ ❛♥❞ ❖✉t❬✺❪✱ t♦❣❡t❤❡r✱ ❡①❤✐❜✐t t❤❡ ❢❛❝t t❤❛t ♦♥❧② ✺✵ t❡r♠s (k = t♦ k = 50) ♦❢ t❤❡ s❡r✐❡s ✐♥ ✭✼✸✮ ❝❛♥ ♣r♦✲ ❞✉❝❡ ❛♥ ❛❝❝✉r❛❝② ♦❢ ❛s ♠❛♥② ❛s s❡✈❡♥ ❞❡❝✐♠❛❧ ♣❧❛❝❡s✳ ❖✉r s②♠❜♦❧✐❝ ❝♦♠♣✉t❛t✐♦♥s ❛♥❞ ♥✉♠❡r✐❝❛❧ ✈❡r✐✜❝❛t✐♦♥s ✇✐t❤ t❤❡ s❡r✐❡s ✐♥ ✭✼✹✮ ✉s✐♥❣ ▼❛t❤✲ ❡♠❛t✐❝❛ ✭❱❡rs✐♦♥ ✹✳✵✮ ❢♦r ▲✐♥✉① ❧❡❛❞ ✉s t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ t❛❜❧❡✿ ■♥❬✶❪:= (98k + 121)❩❡t❛[2k] /E1 (k), ✇❤❡r❡ E1 (k) := ❖✉t❬✶❪ = (2k + 1) (2k + 2) (2k + 3) · (2k + 4) (2k + 5) (2k) (121 + 98k) ❩❡t❛ [2k] , E2 (k) ✇❤❡r❡ ◆✉♠❜❡r ♦❢ ❚❡r♠s ✹ ✶✵ ✷✵ ✺✵ ✾✽ ■♥ ❢❛❝t✱ s✐♥❝❡ t❤❡ ❣❡♥❡r❛❧ t❡r♠ ♦❢ t❤❡ s❡r✐❡s ✐♥ ✭✼✹✮ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ♦r❞❡r ❡st✐♠❛t❡✿ O 2−2k · k −5 E2 (k) :=22k (1 + 2k) (2 + 2k) (3 + 2k) · (4 + 2k) (5 + 2k) ■♥❬✷❪ := ❙✉♠❬✪✱ {k, 1, ■♥✜♥✐t②}] ✴✴ ❙✐♠♣❧✐❢② ❖✉t❬✷❪ = 121 23 ❩❡t❛❬✸❪ − 240 6P✐2 ■♥❬✸❪ := ◆[%] ❖✉t❬✸❪ = 0.0372903 ■♥❬✹❪ := ❙✉♠[◆ [%1] // ❊✈❛❧✉❛t❡✱ {k, 1, 50}] ❖✉t❬✹❪ = 0.0372903 Pr❡❝✐s✐♦♥ ♦❢ ❈♦♠♣✉t❛t✐♦♥ ✻ ✶✶ ✶✽ ✸✽ ✻✾ (k −→ ∞) , ❢♦r ❣❡tt✐♥❣ p ❡①❛❝t ❞✐❣✐ts✱ ✇❡ ♠✉st ❤❛✈❡ 2−2k · k −5 < 10−p ❯♣♦♥ s♦❧✈✐♥❣ t❤✐s ✐♥❡q✉❛❧✐t② s②♠❜♦❧✐❝❛❧❧②✱ ✇❡ ✜♥❞ t❤❛t k∼ = Pr♦❞✉❝t▲♦❣ log 10p/5 log , ✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥ Pr♦❞✉❝t▲♦❣ ✭❛❧s♦ ❦♥♦✇♥ ❛s ▲❛♠❜❡rt✬s ❢✉♥❝t✐♦♥✮ ✐s t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛✲ t✐♦♥✿ xex = a ❙♦♠❡ r❡❧❡✈❛♥t ❞❡t❛✐❧s ❛❜♦✉t t❤❡ s②♠❜♦❧✐❝ ❝♦♠✲ ♣✉t❛t✐♦♥s ❛♥❞ ♥✉♠❡r✐❝❛❧ ✈❡r✐✜❝❛t✐♦♥s ✇✐t❤ t❤❡ s❡r✐❡s ✐♥ ✭✼✹✮ ✉s✐♥❣ ▼❛t❤❡♠❛t✐❝❛ ✭❱❡rs✐♦♥ ✹✳✵✮ ❢♦r ▲✐♥✉① ❛r❡ ❜❡✐♥❣ s✉♠♠❛r✐③❡❞ ❜❡❧♦✇✳ ■♥❬✺❪ := ◆ ❙✉♠[%1 ✴✴ ❊✈❛❧✉❛t❡✱ {k, 1, ■♥✜♥✐t②}] ■♥ ❬✶❪ := ❡①♣r = 8576k + 24286k + 17283 ❩❡t❛[2k]/E1 (k), ❖✉t❬✺❪ = 0.0372903 ❙✐♥❝❡ ζ (0) = − , ✇❤❡r❡ E1 (k) := ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ (2k + 1) (2k + 2) (2k + 3) (2k + 4) · (2k + 5) (2k + 6) (2k + 7) (2k) ✸✹✸ ❱❖▲❯▼❊✿ ✸ ❖✉t ❬✶❪ = 17283 + 24286k + 8576k ❩❡t❛[2k]/E2 (k), ✇❤❡r❡ E2 (k) :=22k (1 + 2k) (2 + 2k) (3 + 2k) · (4 + 2k) (5 + 2k) (6 + 2k) (7 + 2k) ■♥ ❬✷❪ := ❙✉♠[❡①♣r, {k, 0, ✐♥✜♥✐t②}] ✴✴ ❙✐♠♣❧✐❢② ❖✉t ❬✷❪ = − 1573 ❩❡t❛❬✸❪ 120P✐2 ■♥ ❬✸❪ :=◆ −1573/ 120P✐ | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ t♦ ♣r❡s❡♥t ❤❡r❡ ✐♥ ❛ r❛t❤❡r ❝♦♥❝✐s❡ ❢♦r♠✮ ❤❛✈❡ ❡ss❡♥t✐❛❧❧② ♠♦t✐✈❛t❡❞ ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ❢✉rt❤❡r ✐♥✈❡st✐❣❛t✐♦♥s ♦♥ t❤❡ s✉❜❥❡❝t✱ ♥♦t ♦♥❧② ✐♥✈♦❧✈✐♥❣ t❤❡ ❘✐❡♠❛♥♥ ❩❡t❛ ❢✉♥❝t✐♦♥ ζ(s) ❛♥❞ t❤❡ ❍✉r✲ ✇✐t③ ✭♦r ❣❡♥❡r❛❧✐③❡❞✮ ❩❡t❛ ❢✉♥❝t✐♦♥ ζ(s, a) ✭❛♥❞ t❤❡✐r s✉❝❤ r❡❧❛t✐✈❡s ❛s t❤❡ ♠✉❧t✐♣❧❡ ❩❡t❛ ❢✉♥❝✲ t✐♦♥s ❛♥❞ t❤❡ ♠✉❧t✐♣❧❡ ●❛♠♠❛ ❢✉♥❝t✐♦♥s✮✱ ❜✉t ✐♥❞❡❡❞ ❛❧s♦ t❤❡ s✉❜st❛♥t✐❛❧❧② ❣❡♥❡r❛❧ ❍✉r✇✐t③✲ ▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ Φ(z, s, a) ❞❡✜♥❡❞ ❜② ✭❝❢✳✱ ❡✳❣✳✱ ❬✶✼✱ ♣✳ ✷✼✳ ❊q✳ ✶✳✶✶ ✭✶✮❪❀ s❡❡ ❛❧s♦ ❬✺✼✱ ♣✳ ✶✷✶✱ ❡t s❡q✳❪✮ ∞ Φ(z, s, a) := zn (n + a)s n=0 ✭✼✺✮ ❩❡t❛❬3❪✱ 50 a ∈ C \ Z− ; s ∈ C ✇❤❡♥ |z| < 1; (s) > ✇❤❡♥ |z| = 1) −❙✉♠[❡①♣r, {k, 0, 50}] ❖✉t ❬✸❪ = 4.00751120011 · 10−38 ■♥ ❬✹❪ :=◆ −1573/ 120P✐ ❩❡t❛❬3❪✱ 100 −❙✉♠ [❡①♣r, {k, 0, 50}] ❖✉t ❬✹❪ = 4.0075112001 3481 · 10−38 ❚❤✉s✱ ❝❧❡❛r❧②✱ t❤❡ r❡s✉❧t ❞♦❡s ♥♦t ❝❤❛♥❣❡ ❛♣♣r❡❝✐❛❜❧② ✇❤❡♥ ✇❡ ✐♥❝r❡❛s❡ t❤❡ ♣r❡❝✐s✐♦♥ ♦❢ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ s②♠❜♦❧✐❝ r❡s✉❧t ❢r♦♠ ✺✵ t♦ ✶✵✵✳ ❚❤✐s ✐s ❡①♣❡❝t❡❞✱ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥✉♠❡r✐❝❛❧ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❧❛st t❡r♠ ❢♦r k = 50✿ ❏✉st ❛s ✐♥ t❤❡ ❝❛s❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ ❩❡t❛ ❢✉♥❝✲ t✐♦♥ ζ(s) ❛♥❞ t❤❡ ❍✉r✇✐t③ ✭♦r ❣❡♥❡r❛❧✐③❡❞✮ ❩❡t❛ ❢✉♥❝t✐♦♥ ζ(s, a)✱ t❤❡ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝✲ t✐♦♥ Φ(z, s, a) ❝❛♥ ❜❡ ❝♦♥t✐♥✉❡❞ ♠❡r♦♠♦r♣❤✐❝❛❧❧② t♦ t❤❡ ✇❤♦❧❡ ❝♦♠♣❧❡① s✲♣❧❛♥❡✱ ❡①❝❡♣t ❢♦r ❛ s✐♠✲ ♣❧❡ ♣♦❧❡ ❛t s = ✇✐t❤ ✐ts r❡s✐❞✉❡ ✶✳ ■t ✐s ❛❧s♦ ❦♥♦✇♥ t❤❛t ❬✶✼✱ ♣✳ ✷✼✱ ❊q✉❛t✐♦♥ ✶✳✶✶ ✭✸✮❪ Φ(z, s, a) = Γ(s) = Γ(s) ( (a) > 0; ∞ s−1 t e−at dt − ze−t ∞ s−1 t e−(a−1)t dt et − z (s) > ✇❤❡♥ |z| ✭✼✻✮ (z = 1); (s) > ✇❤❡♥ z = 1) ❚❤❡ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ Φ(z, s, a) ❞❡✲ ✜♥❡❞ ❜② ✭✼✺✮ ❝♦♥t❛✐♥s✱ ❛s ✐ts s♣❡❝✐❛❧ ❝❛s❡s✱ ♥♦t ❖✉t ❬✺❪ = 1.36085303749223768614438874545515 ♦♥❧② t❤❡ ❘✐❡♠❛♥♥ ❩❡t❛ ❢✉♥❝t✐♦♥ ζ(s) ❛♥❞ t❤❡ ❍✉r✇✐t③ ✭♦r ❣❡♥❡r❛❧✐③❡❞✮ ❩❡t❛ ❢✉♥❝t✐♦♥ ζ(s, a) 14233575702860179 · 10−37 ❬❝❢✳ ❊q✉❛t✐♦♥s ✭✶✮ ❛♥❞ ✭✷✮❪✿ ■♥ ❬✺❪ := ◆ [❡①♣r /.k → 50, 50] ✻✳ ❚❤❡ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❋✉♥❝t✐♦♥ Φ( , , ) : ❊①t❡♥s✐♦♥s ❛♥❞ ●❡♥❡r❛❧✐③❛t✐♦♥s ③s❛ ❚❤❡ ♣♦t❡♥t✐❛❧❧② ❛♥❞ ❝♦♠♣✉t❛t✐♦♥❛❧❧② ✉s❡❢✉❧ ❢♦r❡✲ ❣♦✐♥❣ ❞❡✈❡❧♦♣♠❡♥ts ✭✇❤✐❝❤ ✇❡ ❤❛✈❡ ❛tt❡♠♣t❡❞ ✸✹✹ ζ(s) = Φ(1, s, 1) ❛♥❞ ζ(s, a) = Φ(1, s, a) ✭✼✼✮ ❛♥❞ t❤❡ ▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ s (ξ) ❞❡✜♥❡❞ ❜② ✭s❡❡✱ ❢♦r ❞❡t❛✐❧s✱ ❬✶✼✱ ❈❤❛♣t❡r ■❪ ❛♥❞ ❬✺✼✱ ❈❤❛♣t❡r ✷❪✮ ∞ s (ξ) := e2nπiξ = e2πiξ Φ e2πiξ , s, s n n=1 (ξ ∈ R; ✭✼✽✮ (s) > 1) , ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✸ ❜✉t ❛❧s♦ s✉❝❤ ♦t❤❡r ✐♠♣♦rt❛♥t ❢✉♥❝t✐♦♥s ♦❢ ❆♥✲ ❛❧②t✐❝ ❋✉♥❝t✐♦♥ ❚❤❡♦r② ❛s t❤❡ P♦❧②❧♦❣❛r✐t❤♠✐❝ ❢✉♥❝t✐♦♥ ✭♦r ❞❡ ❏♦♥q✉✐èr❡✬s ❢✉♥❝t✐♦♥✮ ▲✐s (z)✿ ∞ zn ▲✐s (z) := = z Φ(z, s, 1) ns n=1 ✭✼✾✮ (s ∈ C ✇❤❡♥ |z| < 1; (s) > ✇❤❡♥ |z| = 1) (k ∈ N; | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ (s) > ✇❤❡♥ |z| (a) > 0; (z = 1); (s) > ✇❤❡♥ z = 1) ▼♦t✐✈❛t❡❞ ❡ss❡♥t✐❛❧❧② ❜② t❤❡ s✉♠✲✐♥t❡❣r❛❧ r❡♣✲ r❡s❡♥t❛t✐♦♥s ✭✽✶✮ ❛♥❞ ✭✽✷✮✱ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ Φ(z, s, a) ✇❛s ✐♥tr♦❞✉❝❡❞ ❛♥❞ ✐♥✈❡st✐❣❛t❡❞ ❜② ▲✐♥ ❛♥❞ ❙r✐✈❛s✲ t❛✈❛ ❬✸✺❪ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠ ❬✸✺✱ ♣✳ ✼✷✼✱ ❊q✳ ✭✽✮❪✿ ∞ ❛♥❞ t❤❡ ▲✐♣s❝❤✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ ✭❝❢✳ ❬✺✼✱ ♣✳ ✶✷✷✱ ❊q✳ ✷✳✺ ✭✶✶✮❪✮✿ (ρ,σ) Φµ,ν (z, s, a) := zn (µ)ρn (ν)σn (n + a)s n=0 ✭✽✸✮ ∞ φ(ξ, a, s) : = e2nπiξ (n + a)s n=0 + µ ∈ C; a, ν ∈ C \ Z− ; ρ, σ ∈ R ; ρ < σ ✇❤❡♥ s, z ∈ C; ρ = σ ❛♥❞ s ∈ C = Φ e2πiξ , s, a =: L (ξ, s, a) ✭✽✵✮ (s) > ✇❤❡♥ ξ ∈ Z) , ✇❤✐❝❤ ✇❛s ✜rst st✉❞✐❡❞ ❜② ❘✉❞♦❧❢ ▲✐♣s❝❤✐t③ ✭✶✽✸✷✲✶✾✵✸✮ ❛♥❞ ▼❛t②á➨ ▲❡r❝❤ ✭✶✽✻✵✲✶✾✷✷✮ ✐♥ ❝♦♥♥❡❝t✐♦♥ ✇✐t❤ ❉✐r✐❝❤❧❡t✬s ❢❛♠♦✉s t❤❡♦r❡♠ ♦♥ ♣r✐♠❡s ✐♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥s✳ ❋♦r ❞❡t❛✐❧s✱ t❤❡ ✐♥t❡r❡st❡❞ r❡❛❞❡r s❤♦✉❧❞ ❜❡ r❡❢❡rr❡❞✱ ✐♥ ❝♦♥✲ ♥❡❝t✐♦♥ ✇✐t❤ s♦♠❡ ♦❢ t❤❡s❡ ❞❡✈❡❧♦♣♠❡♥ts✱ t♦ t❤❡ r❡❝❡♥t ✇♦r❦s ✐♥❝❧✉❞✐♥❣ ✭❛♠♦♥❣ ♦t❤❡rs✮ ❬✷❪✱ ❬✽❪ t♦ ❬✶✸❪✱ ❬✷✷❪✱ ❬✸✵❪✱ ❬✸✶❪ ❛♥❞ ❬✸✻❪✳ ❨❡♥ ❡t ❛❧✳ ❬✼✽✱ ♣✳ ✶✵✵✱ ❚❤❡♦r❡♠❪ ❞❡r✐✈❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉♠✲✐♥t❡❣r❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r t❤❡ ❍✉r✇✐t③ ✭♦r ❣❡♥❡r❛❧✐③❡❞✮ ❩❡t❛ ❢✉♥❝t✐♦♥ ζ(s, a) ❞❡✜♥❡❞ ❜② ✭✷✮✿ k−1 j=0 k ∈ N; ∞ s−1 e−(a+j)t dt − e−kt t (s) > 1; ✭✽✶✮ (a) > , ✇❤✐❝❤✱ ❢♦r k = 2✱ ✇❛s ❣✐✈❡♥ ❡❛r❧✐❡r ❜② ◆✐s❤✐♠♦t♦ ❡t ❛❧✳ ❬✹✶✱ ♣✳ ✾✹✱ ❚❤❡♦r❡♠ ✹❪✳ ❆ str❛✐❣❤t❢♦r✇❛r❞ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ s✉♠✲✐♥t❡❣r❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ✭✽✶✮ ✇❛s ❣✐✈❡♥ s✉❜s❡q✉❡♥t❧② ❜② ▲✐♥ ❛♥❞ ❙r✐✈❛s✲ t❛✈❛ ❬✸✺✱ ♣✳ ✼✷✼✱ ❊q✳ ✭✼✮❪ ✐♥ t❤❡ ❢♦r♠✿ Φ(z, s, a) = Γ(s) k−1 ∞ s−1 t zj j=0 ❛♥❞ (s − µ + ν) > ✇❤❡♥ |z| = δ) , ✇❤❡r❡ (λ)ν ❞❡♥♦t❡s t❤❡ P♦❝❤❤❛♠♠❡r s②♠❜♦❧ ❞❡✲ ✜♥❡❞ ✐♥ ❝♦♥❥✉♥❝t✐♦♥ ✇✐t❤ ✭✶✹✮ ❛♥❞ ✭✶✺✮✳ ❈❧❡❛r❧②✱ ✇❡ ✜♥❞ ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ✭✽✸✮ t❤❛t a ∈ C \ Z− ; (s) > ✇❤❡♥ ξ ∈ R \ Z; ζ(s, a) = Γ(s) ✇❤❡♥ |z| < δ := ρ−ρ σ σ ; ρ = σ e−(a+j)t dt − z k e−kt ✭✽✷✮ (0,0) Φ(σ,σ) ν,ν (z, s, a) = Φµ,ν (z, s, a) = Φ(z, s, a) ✭✽✹✮ ❛♥❞ (1,1) Φµ,1 (z, s, a) = Φ∗µ (z, s, a) ∞ := n=0 zn (µ)n n! (n + a)s ✭✽✺✮ µ ∈ C; a ∈ C \ Z− ; s ∈ C ✇❤❡♥ |z| < 1; (s − µ) > ✇❤❡♥ |z| = 1) , ✇❤❡r❡✱ ❛s ❛❧r❡❛❞② ♥♦t❡❞ ❜② ▲✐♥ ❛♥❞ ❙r✐✈❛s✲ t❛✈❛ ❬✸✺❪✱ Φ∗µ (z, s, a) ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ ❝♦♥s✐❞❡r❡❞ ❜② ●♦②❛❧ ❛♥❞ ▲❛❞❞❤❛ ❬✷✺✱ ♣✳ ✶✵✵✱ ❊q✉❛t✐♦♥ ✭✹✮❪✳ ❋♦r ❢✉rt❤❡r r❡s✉❧ts ✐♥✈♦❧✈✐♥❣ t❤❡s❡ ❝❧❛ss❡s ♦❢ ❣❡♥✲ ❡r❛❧✐③❡❞ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥s✱ s❡❡ t❤❡ r❡❝❡♥t ✇♦r❦s ❜② ●❛r❣ ❡t ❛❧✳ ❬✷✷❪ ❛♥❞ ▲✐♥ ❡t ❛❧✳ ❬✸✻❪✳ ❆ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❛❜♦✈❡✲❞❡✜♥❡❞ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥s Φ(z, s, a) ❛♥❞ Φ∗µ (z, s, a) ✇❛s st✉❞✐❡❞ ❜② ●❛r❣ ❡t ❛❧✳ ❬✷✶❪ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠ ❬✷✶✱ ♣✳ ✸✶✸✱ ❊q✳ ✭✶✵✮❪✿ ∞ Φλ,µ;ν (z, s, a) := ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ zn (λ)n (µ)n (ν)n · n! (n + a)s n=0 ✭✽✻✮ ✸✹✺ ❱❖▲❯▼❊✿ ✸ λ, µ ∈ C; ν, a ∈ C \ Z− ; s ∈ C ✇❤❡♥ |z| < 1; (s + ν − λ − µ) > ✇❤❡♥ |z| = 1) ❇② ❝♦♠♣❛r✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥s ✭✽✸✮ ❛♥❞ ✭✽✺✮✱ ✐t ✐s ❡❛s✐❧② ♦❜s❡r✈❡❞ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ Φλ,µ;ν (z, s, a) st✉❞✐❡❞ ❜② ●❛r❣ ❡t ❛❧✳ ❬✷✶❪ ❞♦❡s ♥♦t ♣r♦✈✐❞❡ (ρ,σ) ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ Φµ,ν (z, s, a) ✇❤✐❝❤ ✇❛s ✐♥tr♦❞✉❝❡❞ ❡❛r❧✐❡r ❜② ▲✐♥ ❛♥❞ ❙r✐✲ ✈❛st❛✈❛ ❬✸✺❪✳ ■♥❞❡❡❞✱ ❢♦r λ = 1✱ t❤❡ ❢✉♥❝t✐♦♥ Φλ,µ;ν (z, s, a) ❝♦✐♥❝✐❞❡s ✇✐t❤ ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ (ρ,σ) Φµ,ν (z, s, a) ✭✽✼✮ z Γ (λ + 1) z λ−µ = Γ (λ − µ + 1) (λ) > −1 , ✭✽✽✮ ✇❤✐❝❤✱ ✐♥ ✈✐❡✇ ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥ ✭✽✸✮✱ ②✐❡❧❞s t❤❡ ❢♦❧❧♦✇✐♥❣ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ (ρ,σ) Φµ,ν (z, s, a) ✇✐t❤ ρ = σ ❬✸✺✱ ♣✳ ✼✸✵✱ ❊q✳ ✭✷✹✮❪✿ Dzµ−ν z µ−1 Φ (z σ , s, a) = Γ (µ) ν−1 (σ,σ) σ z Φµ,ν (z , s, a) Γ (ν) ✭✽✾✮ (µ) > 0; σ ∈ R+ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❤❡♥ ν = σ = 1, t❤❡ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ❢♦r♠✉❧❛ ✭✽✾✮ ✇♦✉❧❞ r❡❞✉❝❡ ❛t ♦♥❝❡ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✿ Φ∗µ (z, s, a) = Dµ−1 z µ−1 Φ (z, s, a) , Γ (µ) z ✭✾✵✮ (µ) > ✇❤✐❝❤ ✭❛s ❛❧r❡❛❞② r❡♠❛r❦❡❞ ❜② ▲✐♥ ❛♥❞ ❙r✐✲ ✈❛st❛✈❛ ❬✸✺✱ ♣✳ ✼✸✵❪✮ ❡①❤✐❜✐ts t❤❡ ✐♥t❡r❡st✲ ✐♥❣ ✭❛♥❞ ✉s❡❢✉❧✮ ❢❛❝t t❤❛t Φ∗µ (z, s, a) ✐s ❡s✲ s❡♥t✐❛❧❧② ❛ ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛✲ t✐✈❡ ♦❢ t❤❡ ❝❧❛ss✐❝❛❧ ❍✉r✇✐t③✲▲❡r❝❤ ❢✉♥❝t✐♦♥ ✸✹✻ t❤❡ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ❢♦r♠✉❧❛ ✭✽✽✮ t❤❛t Φλ,µ;ν (z, s, a) Γ(ν) 1−λ λ−ν λ−1 ∗ z Dz z Φµ (z, s, a) Γ(λ) Γ(ν) = z 1−λ Γ(λ)Γ(µ) = · Dzλ−ν z λ−1 Dzµ−1 z µ−1 Φµ (z, s, a) , ✭✾✶✮ ✐t ✐s ❦♥♦✇♥ t❤❛t λ Φ (z, s, a)✳ ▼♦r❡♦✈❡r✱ ✐t ✐s ❡❛s✐❧② ❞❡❞✉❝❡❞ ❢r♦♠ ✇❤❡♥ ρ = σ = 1✳ ❋♦r t❤❡ ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛✲ t✐✈❡ ♦♣❡r❛t♦r Dzµ ❞❡✜♥❡❞ ❜② ✭s❡❡✱ ❢♦r ❡①❛♠♣❧❡✱ ❬✶✽✱ ♣✳ ✶✽✶❪✱ ❬✹✺❪ ❛♥❞ ❬✸✸✱ ♣✳ ✼✵ ❡t s❡q✳❪✮ z −µ−1 ∫ (z − t) f (t) ❞t Γ (−µ) ( (µ) < 0) µ Dz {f (z)} := dm µ−m {D {f (z)}} dz m z (m − (µ) < m (m ∈ N)), Dzµ | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ ✇❤✐❝❤ ❡①❤✐❜✐ts t❤❡ ❤✐t❤❡rt♦ ✉♥♥♦t✐❝❡❞ ❢❛❝t t❤❛t t❤❡ ❢✉♥❝t✐♦♥ Φλ,µ;ν (z, s, a) st✉❞✐❡❞ ❜② ●❛r❣ ❡t ❛❧✳ ❬✷✶❪ ✐s ❡ss❡♥t✐❛❧❧② ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❝❧❛ss✐❝❛❧ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ Φ(z, s, a) ✇❤❡♥ ✇❡ ❛♣♣❧② t❤❡ ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛✲ t✐✈❡ ♦♣❡r❛t♦r Dzµ t✇♦ t✐♠❡s ❛s ✐♥❞✐❝❛t❡❞ ❛❜♦✈❡ ✭s❡❡ ❛❧s♦ ❬✻✼❪✮✳ ▼❛♥② ♦t❤❡r ❡①♣❧✐❝✐t r❡♣r❡s❡♥t❛✲ (ρ,σ) t✐♦♥s ❢♦r Φ∗µ (z, s, a) ❛♥❞ Φµ,ν (z, s, a)✱ ✐♥❝❧✉❞✐♥❣ ❛ ♣♦t❡♥t✐❛❧❧② ✉s❡❢✉❧ ❊✉❧❡r✐❛♥ ✐♥t❡❣r❛❧ r❡♣r❡s❡♥t❛✲ t✐♦♥ ♦❢ t❤❡ ✜rst ❦✐♥❞ ❬✸✺✱ ♣✳ ✼✸✶✱ ❊q✳ ✭✷✽✮❪✱ ✇❡r❡ ♣r♦✈❡♥ ❜② ▲✐♥ ❛♥❞ ❙r✐✈❛st❛✈❛ ❬✸✺❪✳ ❆ ♠✉❧t✐♣❧❡ ✭♦r✱ s✐♠♣❧②✱ n✲❞✐♠❡♥t✐♦♥❛❧✮ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ Φn (z, s, a) ✇❛s st✉❞✐❡❞ r❡❝❡♥t❧② ❜② ❈❤♦✐ ❡t ❛❧✳ ❬✾✱ ♣✳ ✻✻✱ ❊q✳ ✭✻✮❪✳ ❘➔❞✉❝❛♥✉ ❛♥❞ ❙r✐✈❛st❛✈❛ ✭s❡❡ ❬✹✸❪ ❛♥❞ t❤❡ r❡❢❡r❡♥❝❡s ❝✐t❡❞ t❤❡r❡✐♥✮✱ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ♠❛❞❡ ✉s❡ ♦❢ t❤❡ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ Φ(z, s, a) ✐♥ ❞❡✜♥✐♥❣ ❛ ❝❡rt❛✐♥ ❧✐♥❡❛r ❝♦♥✈♦❧✉✲ t✐♦♥ ♦♣❡r❛t♦r ✐♥ t❤❡✐r s②st❡♠❛t✐❝ ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ ✈❛r✐♦✉s ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥ ❝❧❛ss❡s ✐♥ ●❡♦♠❡tr✐❝ ❋✉♥❝t✐♦♥ ❚❤❡♦r② ✐♥ ❈♦♠♣❧❡① ❆♥❛❧②s✐s✳ ❋✉rt❤❡r✲ ♠♦r❡✱ ●✉♣t❛ ❡t ❛❧✳ ❬✷✻❪ r❡✈✐s✐t❡❞ t❤❡ st✉❞② ♦❢ t❤❡ ❢❛♠✐❧✐❛r ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❞✐str✐❜✉t✐♦♥ ❜② ✐♥✈❡st✐❣❛t✐♥❣ ✐ts str✉❝t✉r❛❧ ♣r♦♣❡rt✐❡s✱ r❡❧✐❛❜✐❧✲ ✐t② ♣r♦♣❡rt✐❡s ❛♥❞ st❛t✐st✐❝❛❧ ✐♥❢❡r❡♥❝❡✳ ❚❤❡s❡ ✐♥✈❡st✐❣❛t✐♦♥s ❜② ●✉♣t❛ ❡t ❛❧✳ ❬✷✻❪ ❛♥❞ ♦t❤✲ ❡rs ✭s❡❡✱ ❢♦r ❡①❛♠♣❧❡✱ ❬✺✸❪✱ ❬✺✼❪✱ ❬✻✵❪ ❛♥❞ ❬✻✶❪✮✱ ❢r✉✐t❢✉❧❧② ✉s✐♥❣ t❤❡ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ Φ(z, s, a) ❛♥❞ s♦♠❡ ♦❢ ✐ts ❛❜♦✈❡✲♠❡♥t✐♦♥❡❞ ❣❡♥✲ ❡r❛❧✐③❛t✐♦♥s✱ ♠♦t✐✈❛t❡❞ ❙r✐✈❛st❛✈❛ ❡t ❛❧✳ ❬✻✼❪ t♦ ♣r❡s❡♥t ❛ ❢✉rt❤❡r ❣❡♥❡r❛❧✐③❛t✐♦♥ ❛♥❞ ❛♥❛❧♦❣♦✉s ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ ❛ ♥❡✇ ❢❛♠✐❧② ♦❢ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠ ❬✻✼✱ ♣✳ ✹✾✶✱ ❊q✉❛t✐♦♥ ✭✶✳✷✵✮❪✿ ∞ (ρ,σ,κ) Φλ,µ;ν (z, s, a) := (λ)ρn (µ)σn zn (ν)κn · n! (n + a)s n=0 ✭✾✷✮ ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✸ + λ, µ ∈ C; a, ν ∈ C \ Z− ; ρ, σ, κ ∈ R ; | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ (p, q ∈ N0 ; λj ∈ C (j = 1, · · · , p); κ − ρ − σ > −1 ✇❤❡♥ s, z ∈ C; a, µj ∈ C \ Z0− (j = 1, · · · , q); κ − ρ − σ = −1 ❛♥❞ s ∈ C ρj , σk ∈ R+ (j = 1, · · · , p; k = 1, · · · , q); ∗ ✇❤❡♥|z| < δ := ρ −ρ σ −σ κ − ρ − σ = −1 ❛♥❞ κ ∆ > −1 ✇❤❡♥ s, z ∈ C; κ ; (s + ν − λ − µ) > ∆ = −1 ❛♥❞ s ∈ C ✇❤❡♥ |z| < ∇∗ ; ✇❤❡♥ |z| = δ ∗ ) ∆ = −1 ❛♥❞ ❋♦r t❤❡ ❛❜♦✈❡✲❞❡✜♥❡❞ ❢✉♥❝t✐♦♥ ✐♥ ✭✾✷✮✱ ❙r✐✈❛s✲ t❛✈❛ ❡t ❛❧✳ ❬✻✼❪ ❡st❛❜❧✐s❤❡❞ ✈❛r✐♦✉s ✐♥t❡❣r❛❧ r❡♣✲ r❡s❡♥t❛t✐♦♥s✱ r❡❧❛t✐♦♥s❤✐♣s ✇✐t❤ t❤❡ H ✲❢✉♥❝t✐♦♥ ✇❤✐❝❤ ✐s ❞❡✜♥❡❞ ❜② ♠❡❛♥s ♦❢ ❛ ▼❡❧❧✐♥✲❇❛r♥❡s t②♣❡ ❝♦♥t♦✉r ✐♥t❡❣r❛❧ ✭s❡❡✱ ❢♦r ❡①❛♠♣❧❡✱ ❬✻✺❪ ❛♥❞ ❬✻✼❪✮✱ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ❛♥❞ ❛♥❛❧②t✐❝ ❝♦♥t✐♥✉❛t✐♦♥ ❢♦r♠✉❧❛s✱ ❛s ✇❡❧❧ ❛s ❛♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ (ρ,σ,κ) Φλ,µ;ν (z, s, a) ✐♥ ✭✾✷✮✳ ❚❤✐s ♥❛t✉r❛❧ ❢✉rt❤❡r ❡①t❡♥s✐♦♥ ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ (Ξ) > ✇❤❡♥ |z| = ∇∗ ❚❤❡ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥ ✭✾✺✮ ✇❤❡♥ p − = q = ✇♦✉❧❞ ♦❜✈✐♦✉s❧② ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❛❜♦✈❡✲✐♥✈❡st✐❣❛t❡❞ ❣❡♥❡r❛❧✐③❡❞ ❍✉r✇✐t③✲ (ρ,σ,κ) ▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ Φλ,µ;ν (z, s, a) ❞❡✜♥❡❞ ❜② ✭✾✷✮✳ ■❢ ✇❡ s❡t p → p+1 (ρ1 = · · · = ρp = 1; λp+1 = ρp+1 = 1) (ρ,σ,κ) Φλ,µ;ν (z, s, a) ✇❛s ✐♥❞❡❡❞ ❛❝❝♦♠♣❧✐s❤❡❞ ❜② ✐♥✲ ❛♥❞ tr♦❞✉❝✐♥❣ ❡ss❡♥t✐❛❧❧② ❛r❜✐r❛r② ♥✉♠❜❡rs ♦❢ ♥✉✲ ♠❡r❛t♦r ❛♥❞ ❞❡♥♦♠✐♥❛t♦r ♣❛r❛♠❡t❡rs ✐♥ t❤❡ ❞❡❢✲ ✐♥✐t✐♦♥ ✭✾✷✮✳ ❋♦r t❤✐s ♣✉r♣♦s❡✱ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ s②♠❜♦❧ ∇∗ ❞❡✜♥❡❞ ❜② p −ρ ρj j ∗ ∇ := · j=1 q σ σj j , (6.19) j=1 σj − j=1 = ρj k=0 ❛♥❞ = q p µj − j=1 λj + j=1 p−q ✭✾✸✮ (b1 )k · · · (bq )k , bq ; z) zk Γ(αk + β) Γ(β) ·p+1 Ψ∗q+1 (a1 , 1) , · · · , (ap , 1) , (1, 1); (ρ ,··· ,ρ ,σ ,··· ,σ ) Φλ11,··· ,λpp;µ11,··· ,µqq (z, s, a) ✐s ❞❡✜♥❡❞ ❜② ❬✻✼✱ ♣✳ ✺✵✸✱ ❊q✉❛t✐♦♥ ✭✼✻✮❪ (ρ ,··· ,ρ ,σ ,··· ,σ ) Φλ11,··· ,λpp;µ11,··· ,µqq (z, s, a) ✭✾✹✮ p (λj )nρj j=1 q := n=0 n! (µj )nσj z (b1 , 1) , · · · , (bq , 1) , (β, α); ❚❤❡♥ t❤❡ ❡①t❡♥❞❡❞ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ ∞ , t❤❡♥ ✭✾✺✮ r❡❞✉❝❡s t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❣❡♥❡r❛❧✲ ✐③❡❞ M ✲s❡r✐❡s ✇❤✐❝❤ ✇❛s r❡❝❡♥t❧② ✐♥tr♦❞✉❝❡❞ ❜② ❙❤❛r♠❛ ❛♥❞ ❏❛✐♥ ❬✹✻❪ ❛s ❢♦❧❧♦✇s✿ j=1 Ξ := s + µq+1 = β; σq+1 = α p Mq (a1 , · · · , ap ; b1 , · · · ∞ (a1 )k · · · (ap )k p ∆ := σ1 = · · · = σq = 1; α,β t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥s ✇✐❧❧ ❜❡ ❡♠♣❧♦②❡❞✿ q q →q+1 zn (n + a)s ✭✾✻✮ ❚❤✐s ❧❛st r❡❧❛t✐♦♥s❤✐♣ ✭✾✻✮ ❡①❤✐❜✐ts t❤❡ ❢❛❝t t❤❛t t❤❡ s♦✲❝❛❧❧❡❞ ❣❡♥❡r❛❧✐③❡❞ M ✲s❡r✐❡s ✐s✱ ✐♥ ❢❛❝t✱ ❛♥ ♦❜✈✐♦✉s ✈❛r✐❛♥t ♦❢ t❤❡ ❋♦①✲❲r✐❣❤t ❢✉♥❝t✐♦♥ p Ψ∗q ♦r p Ψ∗q (p, q ∈ N0 )✱ ✇❤✐❝❤ ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❢❛♠✐❧✐❛r ❣❡♥❡r❛❧✐③❡❞ ❤②♣❡r❣❡♦♠❡tr✐❝ ❢✉♥❝✲ t✐♦♥ p Fq (p, q ∈ N0 )✱ ✇✐t❤ p ♥✉♠❡r❛t♦r ♣❛r❛♠✲ ❡t❡rs a1 , · · · , ap ❛♥❞ q ❞❡♥♦♠✐♥❛t♦r ♣❛r❛♠❡t❡rs b1 , · · · , bq s✉❝❤ t❤❛t ✭✾✺✮ j=1 ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ aj ∈ C (j = 1, · · · , p), bj ∈ C \ Z− (j = 1, · · · , q), ✸✹✼ ❱❖▲❯▼❊✿ ✸ ❞❡✜♥❡❞ ❜② ✭s❡❡✱ ❢♦r ❞❡t❛✐❧s✱ ❬✶✼✱ ♣✳ ✶✽✸❪ ❛♥❞ ❬✻✹✱ ♣✳ ✷✶❪❀ s❡❡ ❛❧s♦ ❬✸✸✱ ♣✳ ✺✻❪✱ ❬✸✽✱ ♣✳ ✸✵❪ ❛♥❞ ❬✻✸✱ ♣✳ ✶✾❪✮ ∗ p Ψq (ρ ,··· ,ρ ,σ ,··· ,σ ) Φλ11,··· ,λpp;µ11,··· ,µqq (z, s, a) ❝❛♥ ❜❡ ♣r♦✈❡♥ ❜② ❛♣♣❧②✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ ✭✾✺✮ ✐♥ ♣r❡❝✐s❡❧② t❤❡ s❛♠❡ ♠❛♥♥❡r ❛s ❢♦r t❤❡ ❝♦rr❡✲ s♣♦♥❞✐♥❣ r❡s✉❧t ✐♥✈♦❧✈✐♥❣ t❤❡ ❣❡♥❡r❛❧ ❍✉r✇✐t③✲ z (b1 , B1 ) , · · · , (bq , Bq ) ; ∞ := ❊❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧ts ✐♥✈♦❧✈✐♥❣ t❤❡ ❡①✲ t❡♥❞❡❞ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ (a1 , A1 ) , · · · , (ap , Ap ) ; | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ (a1 )A1 n · · · (ap )Ap n z n (b1 )B1 n · · · (bq )Bq n n! n=0 (ρ,σ,κ) ▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ Φλ,µ;ν (z, s, a) ✭s❡❡✱ ❢♦r ❞❡✲ t❛✐❧s✱ ❬✻✼✱ ❙❡❝t✐♦♥ ✻❪✮✳ Γ (b1 ) · · · Γ (bq ) Γ (a1 ) · · · Γ (ap ) (a1 , A1 ) , · · · , (ap , Ap ) ; · p Ψq z (b1 , B1 ) , · · · , (bq , Bq ) ; ∞ (ρ ,··· ,ρ ,σ ,··· ,σ ) ts−1 e−at Φλ11,··· ,λpp;µ11,··· ,µqq (z, s, a) = Γ(s) (λ1 , ρ1 ), · · · , (λp , ρp ); ✭✾✼✮ · p Ψ∗q ze−t dt ✭✾✾✮ (µ1 , σ1 ), · · · , (µq , σq ); (j = 1, · · · , q) ; min{ (a), (s)} > , , = Aj > (j = 1, · · · , p) ; Bj > q p 1 + Bj − Aj j=1 q j=1 Γ (µj ) ✇❤❡r❡ t❤❡ ❡q✉❛❧✐t② ✐♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ❝♦♥❞✐t✐♦♥ ❤♦❧❞s tr✉❡ ❢♦r s✉✐t❛❜❧② ❜♦✉♥❞❡❞ ✈❛❧✉❡s ♦❢ |z| ❣✐✈❡♥ ❜② p |z| < ∇ := −A Aj j j=1 (ρ ,··· ,ρ ,σ ,··· ,σ ) Φλ11,··· ,λpp;µ11,··· ,µqq (z, s, a) Γ (λj ) s L Γ (λj + ρj ξ) j=1 q · j=1 2πi p Γ(−ξ) {Γ(ξ + a)} B Bj j · · j=1 q = j=1 p s {Γ(ξ + a + 1)} (−z)ξ dξ Γ (µj + σj ξ) j=1 ✭✶✵✵✮ ■♥ t❤❡ ♣❛rt✐❝✉❧❛r ❝❛s❡ ✇❤❡♥ Aj = B k = (j = 1, · · · , p; k = 1, · · · , q), ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥s❤✐♣ ✭s❡❡✱ ❢♦r ❞❡✲ t❛✐❧s✱ ❬✻✹✱ ♣✳ ✷✶❪✮✿ ∗ p Ψq (a1 , 1) , · · · , (ap , 1) ; ♦r✱ ❡q✉✐✈❛❧❡♥t❧②✱ (ρ ,··· ,ρ ,σ ,··· ,σ ) Φλ11,··· ,λpp;µ11,··· ,µqq (z, s, a) q Γ (µj ) z = (b1 , 1) , · · · , (bq , 1) ; a1 , · · · , ap ; = p Fq z b1 , · · · , bq ; Γ (b1 ) · · · Γ (bq ) Γ (a1 ) · · · Γ (ap ) (a1 , 1) , · · · , (ap , 1) ; · p Ψq z , (b1 , 1) , · · · , (bq , 1) ; 1,p+1 j=1 p · H p+1,q+2 [z | E], ✭✶✵✶✮ Γ (λj ) j=1 ✇❤❡r❡ E := (1 − λ1 , ρ1 ; 1), · · · , (1 − λp , ρp ; 1), (1 − a, 1; s) (0, 1), (1 − µ1 , σ1 ; 1), · · · , (1 − µq , σq ; 1), (−a, 1; s), = ✭✾✽✮ ✐♥ t❡r♠s ♦❢ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❤②♣❡r❣❡♦♠❡tr✐❝ ❢✉♥❝t✐♦♥ p Fq (p, q ∈ N0 )✳ ✸✹✽ | arg(−z)| < π ♣r♦✈✐❞❡❞ t❤❛t ❜♦t❤ s✐❞❡s ♦❢ t❤❡ ❛ss❡rt✐♦♥s ✭✾✾✮✱ ✭✶✵✵✮ ❛♥❞ ✭✶✵✶✮ ❡①✐st✱ t❤❡ ♣❛t❤ ♦❢ ✐♥t❡❣r❛t✐♦♥ L ✐♥ ✭✶✵✶✮ ❜❡✐♥❣ ❛ ▼❡❧❧✐♥✲❇❛r♥❡s t②♣❡ ❝♦♥t♦✉r ✐♥ t❤❡ ❝♦♠♣❧❡① ξ ✲♣❧❛♥❡, ✇❤✐❝❤ st❛rts ❛t t❤❡ ♣♦✐♥t −i∞ ❛♥❞ t❡r♠✐♥❛t❡s ❛t t❤❡ ♣♦✐♥t i∞ ✇✐t❤ ✐♥❞❡♥t❛t✐♦♥s, ✐❢ ♥❡❝❡ss❛r②, ✐♥ s✉❝❤ ❛ ♠❛♥♥❡r ❛s ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✸ t♦ s❡♣❛r❛t❡ t❤❡ ♣♦❧❡s ♦❢ Γ(−ξ) ❢r♦♠ t❤❡ ♣♦❧❡s ♦❢ Γ (λj + ρj ξ) (j = 1, · · · , p) ❝♦♥✈❡r❣❡s ❢♦r ❛♥② t ∈ R+ ✳ ❚❤❡♥ (ρ ,··· ,ρ ,σ ,··· ,σ ) ❚❤❡ H ✲❢✉♥❝t✐♦♥ r❡♣r❡s❡♥t❛t✐♦♥ ❣✐✈❡♥ ❜② ✭✶✵✶✮ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ ✐♥ ♦r❞❡r t♦ ❞❡r✐✈❡ ✈❛r✐♦✉s ♣r♦♣❡r✲ t✐❡s ♦❢ t❤❡ ❡①t❡♥❞❡❞ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ Φλ11,··· ,λpp;µ11,··· ,µqq (z, s, a) ∞ Γ(s) n=0 = (ρ ,··· ,ρ ,σ ,··· ,σ ) Φλ11,··· ,λpp;µ11,··· ,µqq (z, s, a) =z min{ (a), (s)} > , m,n+1 H p+1,q+1 [ωz κ |E], ✭✶✵✷✮ (λ) > 0; κ > , ✇❤❡r❡ E := n p (1 − λ, κ; 1) , (aj , Aj ; αj )j=1 , (aj , Aj )j=n+1 m q (bj , Bj )j=1 , (bj , Bj ; βj )j=m+1 , (1 − λ + ν, κ; 1) , ✇❡ r❡❛❞✐❧② ♦❜t❛✐♥ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ s✉❝❤ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ❢♦r♠✉❧❛s ❛s ✭❢♦r ❡①❛♠♣❧❡✮ ✭✽✾✮ ❣✐✈❡♥ ❜② q Γ (µj ) = 1,p+2 z τ −1 · H p+2,q+3 [−z κ |E] Γ (λj ) j=1 = Γ(ν) τ −1 (ρ1 ,··· ,ρp ,κ,σ1 ,··· ,σq ,κ) κ z Φλ1 ,··· ,λp ,ν;µ1 ,··· ,µq ,τ (z , s, a), Γ(τ ) ✭✶✵✸✮ (ν) > 0; κ > ✇❤❡r❡ E := E1 , · · · , Ep , (1 − ν, κ; 1), (1 − a, 1; s) (0, 1), F1 , · · · , Fq , (1 − τ, κ; 1), (−a, 1; s) ✇✐t❤ Ei = (1 − λi , ρi ; 1)✱ Fj = (1 − µj , σj ; 1) (i = 1, p, j = 1, q)✳ ❋✐♥❛❧❧②✱ ✇❡ ♣r❡s❡♥t t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①t❡♥s✐♦♥ ♦❢ ❛ ❦♥♦✇♥ r❡s✉❧t ❬✻✼✱ ♣✳ ✹✾✻✱ ❚❤❡♦r❡♠ ✸❪ ✭s❡❡ ❛❧s♦ ❬✻✼✱ ♣✳ ✺✵✺✱ ❚❤❡♦r❡♠ ✾❪✳ ▲❡t αn n∈N0 ❜❡ ❛ ♣♦s✐t✐✈❡ s❡q✉❡♥❝❡ s✉❝❤ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥✜♥✐t❡ s❡r✐❡s: ❚❤❡♦r❡♠✳ ∞ e−αn t n=0 ♣r♦✈✐❞❡❞ t❤❛t ❡❛❝❤ ♠❡♠❜❡r ♦❢ ✭✶✵✹✮ ❡①✐sts✳ ■t ✇♦✉❧❞ ❜❡ ♥✐❝❡ ❛♥❞ ✇♦rt❤✇❤✐❧❡ t♦ ❜❡ ❛❜❧❡ t♦ ❡①t❡♥❞ t❤❡ r❡s✉❧ts ♣r❡s❡♥t❡❞ ✐♥ ❙❡❝t✐♦♥s ✷ t♦ ✺ ♦❢ t❤✐s ❧❡❝t✉r❡ t♦ ❤♦❧❞ tr✉❡ ❢♦r t❤❡ ❍✉r✇✐t③✲ ▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ Φ (z, s, a) ❛♥❞ ❢♦r s♦♠❡ ♦❢ ✐ts ❣❡♥❡r❛❧✐③❛t✐♦♥s ❣✐✈❡♥ ❜② t❤❡ ▲✐♥✲❙r✐✈❛st❛✈❛ (ρ,σ) ❩❡t❛ ❢✉♥❝t✐♦♥ Φµ,ν (z, s, a) ❛♥❞ t❤❡ ❡①t❡♥❞❡❞ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ (ρ ,··· ,ρ ,σ ,··· ,σ ) Φλ11,··· ,λpp;µ11,··· ,µqq (z, s, a) ❞❡✜♥❡❞ ❜② ✭✾✺✮ ❢♦r s♣❡❝✐❛❧ ✈❛❧✉❡s ♦❢ t❤❡ ✈❛r✐✲ ♦✉s ♣❛r❛♠❡t❡rs ✐♥✈♦❧✈❡❞ ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥s ✭✽✸✮ ❛♥❞ ✭✾✺✮✳ ❙❡✈❡r❛❧ ♠✉❝❤ ♠♦r❡ ❣❡♥❡r❛❧ ♦♣❡♥ ♣r♦❜✲ ❧❡♠s ✇♦✉❧❞ ✐♥✈♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ λ✲❣❡♥❡r❛❧✐③❡❞ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ ✇❤♦s❡ ✐♥✈❡st✐❣❛✲ t✐♦♥ ✇❛s ✐♥✐t✐❛t❡❞ ❜② ❙r✐✈❛st❛✈❛ ❬✺✻❪✿ (ρ ,··· ,ρ ,σ ,··· ,σ ) Dzν−τ z ν−1 Φλ11,··· ,λpp;µ11,··· ,µqq (z κ , s, a) j=1 p ts−1 e−(a−α0 +αn )t ✭✶✵✹✮ m,n z λ−1 H p,q (ωz κ ) λ−ν−1 ∞ · − e−(αn+1 −αn )t (λ1 , ρ1 ), · · · , (λp , ρp ); · p Ψ∗q ze−t dt (µ1 , σ1 ), · · · , (µq , σq ); ❢r♦♠ t❤♦s❡ ♦❢ t❤❡ H ✲❢✉♥❝t✐♦♥✳ ❚❤✉s✱ ❢♦r ❡①❛♠✲ ♣❧❡✱ ❜② ♠❛❦✐♥❣ ✉s❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢r❛❝t✐♦♥❛❧✲ ❝❛❧❝✉❧✉s r❡s✉❧t ❞✉❡ t♦ ❙r✐✈❛st❛✈❛ ❡t ❛❧✳ ❬✻✺✱ ♣✳ ✾✼✱ ❊q✳ ✭✶✼✮❪✿ Dzν | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ (ρ ,··· ,ρ ,σ ,··· ,σ ) Φλ11,··· ,λpp;µ11,··· ,µqq (z, s, a; b, λ) ∞ b ts−1 exp −at − λ Γ(s) t (λ1 , ρ1 ), · · · , (λp , ρp ); · p Ψ∗q ze−t dt, (µ1 , σ1 ), · · · , (µq , σq ); := ✭✶✵✺✮ min{ (a), (s)} > 0; (b) 0; λ , s♦ t❤❛t✱ ♦❜✈✐♦✉s❧②✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛✲ t✐♦♥s❤✐♣✿ (ρ ,··· ,ρ ,σ ,··· ,σ ) Φλ11,··· ,λpp;µ11,··· ,µqq (z, s, a; 0, λ) (ρ ,··· ,ρ ,σ ,··· ,σ ) = Φλ11,··· ,λpp;µ11,··· ,µqq (z, s, a) (ρ ,··· ,ρ ,σ ,··· ,σ ) = eb Φλ11,··· ,λpp;µ11,··· ,µqq (z, s, a; b, 0) ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✭✶✵✻✮ ✸✹✾ ❱❖▲❯▼❊✿ ✸ ■♥❞❡❡❞✱ ❢♦r t❤❡ s❛❦❡ ♦❢ t❤❡ ✐♥t❡r❡st❡❞ r❡❛❞❡r✱ ✇❡ r❡❝❛❧❧ ❢r♦♠ ❙r✐✈❛st❛✈❛✬s ✇♦r❦ ❬✺✻❪ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣❧✐❝✐t s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r♠✉❧❛s ❤♦❧❞ tr✉❡ ❢♦r t❤❡ λ✲❣❡♥❡r❛❧✐③❡❞ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝✲ t✐♦♥ (ρ ,··· ,ρ ,σ ,··· ,σ ) Φλ11,··· ,λpp;µ11,··· ,µqq (z, s, a; b, λ) (ρ ,··· ,ρ ,σ ,··· ,σ ) Φλ11,··· ,λpp;µ11,··· ,µqq (z, s, a; b, λ) ❬✹❪ ❆♣♦st♦❧✱ ❚✳ ▼✳ ✭✶✾✼✻✮✳ ■♥tr♦❞✉❝t✐♦♥ t♦ ❆♥❛✲ ❧②t✐❝ ◆✉♠❜❡r ❚❤❡♦r②✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✱ ❍❡✐❞❡❧❜❡r❣ ❛♥❞ ❇❡r❧✐♥✱ ✶✾✼✻✳ p = λΓ (s) n=0 (λj )nρj j=1 q s (a + n) · (µj )nσj j=1 2,0 · H0,2 (a + n)b λ (s, 1), 0, λ1 n z , n! ✭✶✵✼✮ m,n ✇❤❡r❡ λ > ❛♥❞ Hp,q [·] ❞❡♥♦t❡s ❋♦①✬s H ✲ ❢✉♥❝t✐♦♥ ✭s❡❡✱ ❢♦r ❞❡t❛✐❧s✱ ❬✻✸❪✳ ❚❤❡ t❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ ✈❛r✐♦✉s s♣❡❝✐❛❧ ❛s ✇❡❧❧ ❛s ❧✐♠✐t ❝❛s❡s ♦❢ t❤❡ λ✲❣❡♥❡r❛❧✐③❡❞ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ (ρ ,··· ,ρ ,σ ,··· ,σ ) Φλ11,··· ,λpp;µ11,··· ,µqq (z, s, a; b, λ), ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤♦s❡ ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡✱ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ✭❢♦r ❡①❛♠♣❧❡✮ t❤❡ r❡❝❡♥t ✇♦r❦s ❬✺✻❪ ❛♥❞ ❬✻✻❪✱ ❛♥❞ ✐♥❞❡❡❞ ❛❧s♦ ✐♥ ♠❛♥② ♦❢ t❤❡ ❡❛r❧✐❡r r❡❢❡r❡♥❝❡s ✇❤✐❝❤ ❛r❡ ❝✐t❡❞ ✐♥ ❡❛❝❤ ♦❢ t❤❡s❡ r❡❝❡♥t ✇♦r❦s✳ ❘❡♠❛r❦❛❜❧②✱ ❥✉st ❛s ✐ts s✉❝❤ ❛❢♦r❡♠❡♥✲ t✐♦♥❡❞ s♣❡❝✐❛❧ ❝❛s❡s ❛s t❤❡ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ Φ(z, s, a) ❛♥❞ r❡❧❛t❡❞ ❩❡t❛ ❢✉♥❝t✐♦♥s✱ t❤❡ λ✲❣❡♥❡r❛❧✐③❡❞ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ (ρ ,··· ,ρ ,σ ,··· ,σ ) Φλ11,··· ,λpp;µ11,··· ,µqq (z, s, a; b, λ) ❞❡✜♥❡❞ ❜② ✭✶✵✺✮ ✐s ♣♦t❡♥t✐❛❧❧② ✉s❡❢✉❧ ❛♥❞ ✐s ❝✉r✲ r❡♥t❧② ❜❡✐♥❣ ❛♣♣❧✐❡❞ ✐♥ ♠❛♥② ❛r❡❛s ♦❢ t❤❡ ♠❛t❤❡✲ ♠❛t✐❝❛❧✱ st❛t✐st✐❝❛❧✱ ♣❤②s✐❝❛❧ ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣ s❝✐✲ ❡♥❝❡s✳ ❚❤❡ r❡❧❡✈❛♥t ❞❡t❛✐❧s ♦❢ s✉❝❤ ❞❡✈❡❧♦♣♠❡♥ts ❛r❡ ❡❛s✐❧② ❛❝❝❡ss✐❜❧❡ ✐♥ t❤❡ ❝✉rr❡♥t ❧✐t❡r❛t✉r❡ ♦♥ t❤❡ s✉❜❥❡❝t✳ ❘❡❢❡r❡♥❝❡s ❬✶❪ ❆❜r❛♠♦✇✐t③✱ ▼✳✱ ✫ ❙t❡❣✉♥✱ ■✳ ❆✳ ✭✶✾✻✺✮✳ ❍❛♥❞❜♦♦❦ ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ❢✉♥❝t✐♦♥s✿ ✇✐t❤ ❢♦r♠✉❧❛s✱ ❣r❛♣❤s✱ ❛♥❞ ♠❛t❤❡♠❛t✐❝❛❧ t❛❜❧❡s ✭❱♦❧✳ ✺✺✮✳ ❈♦✉r✐❡r ❈♦r♣♦r❛t✐♦♥✳ ✸✺✵ ❬✷❪ ❆❧③❡r✱ ❍✳✱ ❑❛r❛②❛♥♥❛❦✐s✱ ❉✳✱ ✫ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳ ✭✷✵✵✻✮✳ ❙❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r s♦♠❡ ♠❛t❤❡♠❛t✐❝❛❧ ❝♦♥st❛♥ts✳ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✸✷✵✭✶✮✱ ✶✹✺✲✶✻✷✳ ❬✸❪ ❆♣ér②✱ ❘✳ ✭✶✾✼✾✮✳ ■rr❛t✐♦♥❛❧✐té ❞❡ ζ (2) ❡t ζ (3)✱ ✐♥ ❏♦✉r♥é❡s ❆r✐t❤♠ét✐q✉❡s ❞❡ ▲✉♠✐♥②✱ ✻✶✱ ✶✶✲✶✸✱ ❞❡✜♥❡❞ ❜② ✭✶✵✺✮✿ ∞ | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ ❬✺❪ ❆②♦✉❜✱ ❘✳ ✭✶✾✼✹✮✳ ❊✉❧❡r ❛♥❞ t❤❡ ③❡t❛ ❢✉♥❝✲ t✐♦♥✳ ❚❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ▼♦♥t❤❧②✱ ✽✶✭✶✵✮✱ ✶✵✻✼✲✶✵✽✻✳ ❬✻❪ ❇♦r✇❡✐♥✱ ❏✳ ▼✳✱ ❇r❛❞❧❡②✱ ❉✳ ▼✳✱ ✫ ❈r❛♥✲ ❞❛❧❧✱ ❘✳ ❊✳ ✭✷✵✵✵✮✳ ❈♦♠♣✉t❛t✐♦♥❛❧ str❛t❡❣✐❡s ❢♦r t❤❡ ❘✐❡♠❛♥♥ ③❡t❛ ❢✉♥❝t✐♦♥✳ ❏♦✉r♥❛❧ ♦❢ ❈♦♠♣✉t❛t✐♦♥❛❧ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ✶✷✶✭✶✲✷✮✱ ✷✹✼✲✷✾✻✳ ❬✼❪ ❈❤❡♥✱ ▼✳ P✳✱ ✫ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳ ✭✶✾✾✽✮✳ ❙♦♠❡ ❢❛♠✐❧✐❡s ♦❢ s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r t❤❡ ❘✐❡♠❛♥♥ ζ (3)✳ ❘❡s✉❧ts ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ✸✸✭✸✲✹✮✱ ✶✼✾✲✶✾✼✳ ❬✽❪ ❈❤♦✐✱ ❏✳✱ ❈❤♦✱ ❨✳ ❏✳✱ ✫ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳ ✭✷✵✵✹✮✳ ❙❡r✐❡s ✐♥✈♦❧✈✐♥❣ t❤❡ ❩❡t❛ ❢✉♥❝t✐♦♥ ❛♥❞ ♠✉❧t✐♣❧❡ ●❛♠♠❛ ❢✉♥❝t✐♦♥s✳ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❈♦♠♣✉t❛t✐♦♥✱ ✶✺✾✭✷✮✱ ✺✵✾✲✺✸✼✳ ❬✾❪ ❈❤♦✐✱ ❏✳✱ ❏❛♥❣✱ ❉✳ ❙✳✱ ✫ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳ ✭✷✵✵✽✮✳ ❆ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❍✉r✇✐t③✲ ▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥✳ ■♥t❡❣r❛❧ ❚r❛♥s❢♦r♠s ❛♥❞ ❙♣❡❝✐❛❧ ❋✉♥❝t✐♦♥s✱ ✶✾✭✶✮✱ ✻✺✲✼✾✳ ❬✶✵❪ ❈❤♦✐✱ ❏✳✱ ✫ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳ ✭✷✵✵✺✮✳ ❈❡rt❛✐♥ ❢❛♠✐❧✐❡s ♦❢ s❡r✐❡s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❍✉r✇✐t③✕▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥✳ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❈♦♠♣✉t❛t✐♦♥✱ ✶✼✵✭✶✮✱ ✸✾✾✲✹✵✾✳ ❬✶✶❪ ❈❤♦✐✱ ❏✳✱ ✫ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳ ✭✷✵✵✺✮✳ ❊①✲ ♣❧✐❝✐t ❡✈❛❧✉❛t✐♦♥ ♦❢ ❊✉❧❡r ❛♥❞ r❡❧❛t❡❞ s✉♠s✳ ❚❤❡ ❘❛♠❛♥✉❥❛♥ ❏♦✉r♥❛❧✱ ✶✵✭✶✮✱ ✺✶✲✼✵✳ ❬✶✷❪ ❈❤♦✐✱ ❏✳✱ ✫ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳ ✭✷✵✶✹✮✳ ❙❡r✐❡s ✐♥✈♦❧✈✐♥❣ t❤❡ ③❡t❛ ❢✉♥❝t✐♦♥s ❛♥❞ ❛ ❢❛♠✐❧② ♦❢ ❣❡♥❡r❛❧✐③❡❞ ●♦❧❞❜❛❝❤✕❊✉❧❡r s❡r✐❡s✳ ❚❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ▼♦♥t❤❧②✱ ✶✷✶✭✸✮✱ ✷✷✾✲✷✸✻✳ ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✸ ❬✶✸❪ ❈❤♦✐✱ ❏✳✱ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳✱ ✫ ❆❞❛♠❝❤✐❦✱ ❱✳ ❙✳ ✭✷✵✵✸✮✳ ▼✉❧t✐♣❧❡ ●❛♠♠❛ ❛♥❞ r❡❧❛t❡❞ ❢✉♥❝t✐♦♥s✳ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❈♦♠✲ ♣✉t❛t✐♦♥✱ ✶✸✹✭✷✲✸✮✱ ✺✶✺✲✺✸✸✳ ❬✶✹❪ ❈✈✐❥♦✈✐❝✱ ❉✳✱ ✫ ❑❧✐♥♦✇s❦✐✱ ❏✳ ✭✶✾✾✼✮✳ ◆❡✇ r❛♣✐❞❧② ❝♦♥✈❡r❣❡♥t s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r ζ (2n + 1)✳ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ✶✷✺✭✺✮✱ ✶✷✻✸✲✶✷✼✶✳ ❬✶✺❪ ❈✈✐❥♦✈✐❝✱ ❉✳✱ ✫ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳ ✭✷✵✶✷✮✳ ▲✐♠✐t ❘❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ❘✐❡♠❛♥♥✬s ❩❡t❛ ❋✉♥❝t✐♦♥✳ ❚❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ▼♦♥t❤❧②✱ ✶✶✾✭✹✮✱ ✸✷✹✲✸✸✵✳ ❬✶✻❪ ❉❛❜r♦✇s❦✐✱ ❆✳ ✭✶✾✾✻✮✳ ❆ ♥♦t❡ ♦♥ ✈❛❧✉❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ ③❡t❛ ❢✉♥❝t✐♦♥ ❛t ♣♦s✐t✐✈❡ ♦❞❞ ✐♥t❡❣❡rs✳ ◆■❊❯❲ ❆❘❈❍■❊❋ ❱❖❖❘ ❲■❙❑❯◆❉❊✱ ✶✹✱ ✶✾✾✲✷✵✽✳ ❬✶✼❪ ❊r❞é❧②✐✱ ❆✳✱ ▼❛❣♥✉s✱ ❲✳✱ ❖❜❡r❤❡tt✐♥❣❡r✱ ❋✳✱ ❚r✐❝♦♠✐✱ ❋✳ ●✳ ✭✶✾✺✸✮✳ ❍✐❣❤❡r ❚r❛♥s❝❡♥✲ ❞❡♥t❛❧ ❋✉♥❝t✐♦♥s✳ ▼❝●r❛✇✲❍✐❧❧ ❇♦♦❦ ❈♦♠✲ ♣❛♥②✱ ◆❡✇ ❨♦r❦✱ ❚♦r♦♥t♦ ❛♥❞ ▲♦♥❞♦♥✱ ❱♦❧✳ ■✳ ❬✶✽❪ ❊r❞é❧②✐✱ ❆✳✱ ▼❛❣♥✉s✱ ❲✳✱ ❖❜❡r❤❡tt✐♥❣❡r✱ ❋✳✱ ❚r✐❝♦♠✐✱ ❋✳ ●✳ ✭✶✾✺✹✮✳ ❚❛❜❧❡s ♦❢ ■♥t❡✲ ❣r❛❧ ❚r❛♥s❢♦r♠s✳ ▼❝●r❛✇✲❍✐❧❧ ❇♦♦❦ ❈♦♠✲ ♣❛♥②✱ ◆❡✇ ❨♦r❦✱ ❚♦r♦♥t♦ ❛♥❞ ▲♦♥❞♦♥✱ ❱♦❧✳ ■■✳ ❬✶✾❪ ❊✇❡❧❧✱ ❏✳ ❆✳ ✭✶✾✾✵✮✳ ❆ ♥❡✇ s❡r✐❡s r❡♣r❡s❡♥✲ t❛t✐♦♥ ❢♦r ζ (3)✳ ❆♠❡r✳ ▼❛t❤✳ ▼♦♥t❤❧②✱ ✾✼✱ ✷✶✾✕✷✷✵✳ ❬✷✵❪ ❊✇❡❧❧✱ ❏✳ ❆✳ ✭✶✾✾✺✮✳ ❖♥ t❤❡ ❩❡t❛ ❢✉♥❝✲ t✐♦♥ ✈❛❧✉❡s ζ(2k + 1), k = 1, 2, · · · ❘♦❝❦② ▼♦✉♥t❛✐♥ ❏✳ ▼❛t❤✳✱ ✷✺✱ ✶✵✵✸✕✶✵✶✷✳ ❬✷✶❪ ●❛r❣✱ ▼✳✱ ❏❛✐♥✱ ❑✳✱ ✫ ❑❛❧❧❛✱ ❙✳ ▲✳ ✭✷✵✵✽✮✳ ❆ ❢✉rt❤❡r st✉❞② ♦❢ ❣❡♥❡r❛❧ ❍✉r✇✐t③✲▲❡r❝❤ ③❡t❛ ❢✉♥❝t✐♦♥✳ ❆❧❣❡❜r❛s ●r♦✉♣s ●❡♦♠✱ ✷✺✱ ✸✶✶✲✸✶✾✳ ❬✷✷❪ ●❛r❣✱ ▼✳✱ ❏❛✐♥✱ ❑✳✱ ✫ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳ ✭✷✵✵✻✮✳ ❙♦♠❡ r❡❧❛t✐♦♥s❤✐♣s ❜❡t✇❡❡♥ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❆♣♦st♦❧✕❇❡r♥♦✉❧❧✐ ♣♦❧②♥♦♠✐❛❧s ❛♥❞ ❍✉r✇✐t③✕▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥s✳ ■♥t❡❣r❛❧ ❚r❛♥s❢♦r♠s ❛♥❞ ❙♣❡❝✐❛❧ ❋✉♥❝t✐♦♥s✱ ✶✼✭✶✶✮✱ ✽✵✸✲✽✶✺✳ ❬✷✸❪ ●❧❛ss❡r✱ ▼✳ ▲✳ ✭✶✾✻✽✮✳ ❙♦♠❡ ✐♥t❡❣r❛❧s ♦❢ t❤❡ ❛r❝t❛♥❣❡♥t ❢✉♥❝t✐♦♥✳ ▼❛t❤❡♠❛t✐❝s ♦❢ ❈♦♠✲ ♣✉t❛t✐♦♥✱ ✷✷✭✶✵✷✮✱ ✹✹✺✲✹✹✼✳ | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ ❬✷✹❪ ●♦s♣❡r ❏r✱ ❘✳ ❲✳ ✭✶✾✼✻✮✳ ❆ ❝❛❧❝✉❧✉s ♦❢ s❡r✐❡s r❡❛rr❛♥❣❡♠❡♥ts✱ ✐♥ ❆❧❣♦r✐t❤♠s ❛♥❞ ❈♦♠♣❧❡①✐t②✿ ◆❡✇ ❉✐r❡❝t✐♦♥s ❛♥❞ ❘❡❝❡♥t ❘❡s✉❧ts✳ ❆❝❛❞❡♠✐❝ Pr❡ss✱ ◆❡✇ ❨♦r❦✱ ▲♦♥✲ ❞♦♥ ❛♥❞ ❚♦r♦♥t♦✱ ✶✷✶✕✶✺✶✳ ❬✷✺❪ ●♦②❛❧✱ ❙✳ P✳✱ ▲❛❞❞❤❛✱ ❘✳ ❑✳ ✭✶✾✾✼✮✳ ❖♥ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❩❡t❛ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ❣❡♥❡r❛❧✲ ✐③❡❞ ▲❛♠❜❡rt ❢✉♥❝t✐♦♥✳ ●❛♥✳✐t❛ ❙❛♥❞❡s❤✱ ✶✶✱ ✾✾✕✶✵✽✳ ❬✷✻❪ ●✉♣t❛✱ P✳ ▲✳✱ ●✉♣t❛✱ ❘✳ ❈✳✱ ❖♥❣✱ ❙✳ ❍✳✱ ✫ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳ ✭✷✵✵✽✮✳ ❆ ❝❧❛ss ♦❢ ❍✉r✲ ✇✐t③✕▲❡r❝❤ ❩❡t❛ ❞✐str✐❜✉t✐♦♥s ❛♥❞ t❤❡✐r ❛♣✲ ♣❧✐❝❛t✐♦♥s ✐♥ r❡❧✐❛❜✐❧✐t②✳ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✲ ✐❝s ❛♥❞ ❈♦♠♣✉t❛t✐♦♥✱ ✶✾✻✭✷✮✱ ✺✷✶✲✺✸✶✳ ❬✷✼❪ ❍❛♥s❡♥✱ ❊✳ ❘✳ ✭✶✾✼✺✮✳ ❆ ❚❛❜❧❡ ♦❢ ❙❡r✐❡s ❛♥❞ Pr♦❞✉❝ts✳ Pr❡♥t✐❝❡✲❍❛❧❧✱ ❊♥❣❧❡✇♦♦❞ ❈❧✐✛s✱ ◆❡✇ ❏❡rs❡②✳ ❬✷✽❪ ❍❥♦rt♥❛❡s✱ ▼✳ ▼✳ ✭✶✾✺✸✮✳ ❖✈❡r❢Ør✐♥❣ ❛✈ ∞ r❡❦❦❡♥ t✐❧ ❡t ❜❡st❡♠t ✐♥t❡❣r❛❧✳ k=1 1/k Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❚✇❡❧❢t❤ ❙❝❛♥❞❛♥❛✈✐❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❈♦♥❣r❡ss✱ ✷✶✶✕✷✶✸✳ ❬✷✾❪ ❑❛♥❡♠✐ts✉✱ ❙✳✱ ❑❛ts✉r❛❞❛✱ ▼✳✱ ❨♦s❤✐♠♦t♦✱ ▼✳ ✭✷✵✵✵✮✳ ❖♥ t❤❡ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛✲ ❢✉♥❝t✐♦♥✳ ❆❡q✉❛t✐♦♥❡s ▼❛t❤❡♠❛t✐❝❛❡✱ ✺✾✱ ✶✕✶✾✳ ❬✸✵❪ ❑❛♥❡♠✐ts✉✱ ❙✳✱ ❑✉♠❛❣❛✐✱ ❍✳✱ ✫ ❨♦s❤✐♠♦t♦✱ ▼✳ ✭✷✵✵✶✮✳ ❙✉♠s ✐♥✈♦❧✈✐♥❣ t❤❡ ❍✉r✇✐t③ ③❡t❛ ❢✉♥❝t✐♦♥✳ ❚❤❡ ❘❛♠❛♥✉❥❛♥ ❏♦✉r♥❛❧✱ ✺✭✶✮✱ ✺✲ ✶✾✳ ❬✸✶❪ ❑❛♥❡♠✐ts✉✱ ❙✳✱ ❑✉♠❛❣❛✐✱ ❍✳✱ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳✱ ✫ ❨♦s❤✐♠♦t♦✱ ▼✳ ✭✷✵✵✹✮✳ ❙♦♠❡ ✐♥t❡❣r❛❧ ❛♥❞ ❛s②♠♣t♦t✐❝ ❢♦r♠✉❧❛s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❍✉r✇✐t③ ❩❡t❛ ❢✉♥❝t✐♦♥✳ ❆♣♣❧✐❡❞ ♠❛t❤❡✲ ♠❛t✐❝s ❛♥❞ ❝♦♠♣✉t❛t✐♦♥✱ ✶✺✹✭✸✮✱ ✻✹✶✲✻✻✹✳ ❬✸✷❪ ❑❛ts✉r❛❞❛✱ ▼✳ ✭✶✾✾✾✮✳ ❘❛♣✐❞❧② ❝♦♥✈❡r❣❡♥t s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r ζ(2n+1) ❛♥❞ t❤❡✐r χ✲❛♥❛❧♦❣✉❡✳ ❆❝t❛ ❆r✐t❤♠❡t✐❝❛✱ ✾✵✭✶✮✱ ✼✾✲✽✾✳ ❬✸✸❪ ❑✐❧❜❛s✱ ❆✳ ❆✳ ❆✳✱ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳✱ ✫ ❚r✉❥✐❧❧♦✱ ❏✳ ❏✳ ✭✷✵✵✻✮✳ ❚❤❡♦r② ❛♥❞ ❛♣♣❧✐✲ ❝❛t✐♦♥s ♦❢ ❢r❛❝t✐♦♥❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭❱♦❧✳ ✷✵✹✮✳ ❊❧s❡✈✐❡r ❙❝✐❡♥❝❡ ▲✐♠✐t❡❞✳ ❬✸✹❪ ❑♦❜❧✐t③✱ ◆✳ ✭✶✾✼✼✮✳ p✲❆❞✐❝ ◆✉♠❜❡rs✱ p✲ ❆❞✐❝ ❆♥❛❧②s✐s✱ ❛♥❞ ❩❡t❛✲❋✉♥❝t✐♦♥s✱ ●r❛❞✉✲ ❛t❡ ❚❡①ts ✐♥ ▼❛t❤❡♠❛t✐❝s✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✱ ❍❡✐❞❡❧❜❡r❣ ❛♥❞ ❇❡r❧✐♥✱ ✺✽✳ ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✸✺✶ ❱❖▲❯▼❊✿ ✸ ❬✸✺❪ ▲✐♥✱ ❙✳ ❉✳✱ ✫ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳ ✭✷✵✵✹✮✳ ❙♦♠❡ ❢❛♠✐❧✐❡s ♦❢ t❤❡ ❍✉r✇✐t③✕▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥s ❛♥❞ ❛ss♦❝✐❛t❡❞ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛✲ t✐✈❡ ❛♥❞ ♦t❤❡r ✐♥t❡❣r❛❧ r❡♣r❡s❡♥t❛t✐♦♥s✳ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❈♦♠♣✉t❛t✐♦♥✱ ✶✺✹✭✸✮✱ ✼✷✺✲✼✸✸✳ ❬✸✻❪ ▲✐♥✱ ❙✳ ❉✳✱ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳✱ ✫ ❲❛♥❣✱ P✳ ❨✳ ✭✷✵✵✻✮✳ ❙♦♠❡ ❡①♣❛♥s✐♦♥ ❢♦r♠✉❧❛s ❢♦r ❛ ❝❧❛ss ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❍✉r✇✐t③✕▲❡r❝❤ ③❡t❛ ❢✉♥❝t✐♦♥s✳ ■♥t❡❣r❛❧ ❚r❛♥s❢♦r♠s ❛♥❞ ❙♣❡❝✐❛❧ ❋✉♥❝t✐♦♥s✱ ✶✼✭✶✶✮✱ ✽✶✼✲✽✷✼✳ ❬✸✼❪ ▼❛❣♥✉s✱ ❲✳✱ ❖❜❡r❤❡tt✐♥❣❡r✱ ❋✳✱ ✫ ❙♦♥✐✱ ❘✳ P✳ ✭✷✵✶✸✮✳ ❋♦r♠✉❧❛s ❛♥❞ t❤❡♦r❡♠s ❢♦r t❤❡ s♣❡❝✐❛❧ ❢✉♥❝t✐♦♥s ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ♣❤②s✐❝s ✭❱♦❧✳ ✺✷✮✳ ❙♣r✐♥❣❡r ❙❝✐❡♥❝❡ ✫ ❇✉s✐♥❡ss ▼❡✲ ❞✐❛✳ ❬✸✽❪ ▼❛t❤❛✐✱ ❆✳ ▼✳✱ ❙❛①❡♥❛✱ ❘✳ ❑✳✱ ✫ ❍❛✉❜♦❧❞✱ ❍✳ ❏✳ ✭✷✵✵✾✮✳ ❚❤❡ ❍✲❢✉♥❝t✐♦♥✿ t❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ❙♣r✐♥❣❡r ❙❝✐❡♥❝❡ ✫ ❇✉s✐♥❡ss ▼❡❞✐❛✳ ❬✸✾❪ ◆❛s❤✱ ❈✳✱ ✫ ❖✬❈♦♥♥♦r✱ ❉✳ ❏✳ ✭✶✾✾✸✮✳ ❘❛②✲ ❙✐♥❣❡r t♦rs✐♦♥✱ t♦♣♦❧♦❣✐❝❛❧ ✜❡❧❞ t❤❡♦r✐❡s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ❩❡t❛ ❢✉♥❝t✐♦♥ ❛t s = 3✳ ■♥ ▲♦✇✲❉✐♠❡♥s✐♦♥❛❧ ❚♦♣♦❧♦❣② ❛♥❞ ◗✉❛♥✲ t✉♠ ❋✐❡❧❞ ❚❤❡♦r② ✭♣♣✳ ✷✼✾✲✷✽✽✮✳ ❙♣r✐♥❣❡r✱ ❇♦st♦♥✱ ▼❆✳ ❬✹✵❪ ◆❛s❤✱ ❈✳✱ ✫ ❖✬❈♦♥♥♦r✱ ❉✳ ✭✶✾✾✺✮✳ ❉❡t❡r✲ ♠✐♥❛♥ts ♦❢ ▲❛♣❧❛❝✐❛♥s✱ t❤❡ ❘❛②✕❙✐♥❣❡r t♦r✲ s✐♦♥ ♦♥ ❧❡♥s s♣❛❝❡s ❛♥❞ t❤❡ ❘✐❡♠❛♥♥ ③❡t❛ ❢✉♥❝t✐♦♥✳ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s✱ ✸✻✭✸✮✱ ✶✹✻✷✲✶✺✵✺✳ ❬✹✶❪ ◆✐s❤✐♠♦t♦✱ ❑✳✱ ❨❡♥✱ ❈✳✲❊✱ ▲✐♥✱ ▼✳✲▲✳ ✭✷✵✵✷✮✳ ❙♦♠❡ ✐♥t❡❣r❛❧ ❢♦r♠s ❢♦r ❛ ❣❡♥❡r❛❧✲ ✐③❡❞ ❩❡t❛ ❢✉♥❝t✐♦♥✳ ❏✳ ❋r❛❝t✳ ❈❛❧❝✳ ✷✷✱ ✾✶✕ ✾✼✳ ❬✹✷❪ ❖❧✈❡r✱ ❋✳ ❲✳✱ ▲♦③✐❡r✱ ❉✳ ❲✳✱ ❇♦✐s✈❡rt✱ ❘✳ ❋✳✱ ✫ ❈❧❛r❦✱ ❈✳ ❲✳ ✭❊❞s✳✮✳ ✭✷✵✶✵✮✳ ◆■❙❚ ❤❛♥❞❜♦♦❦ ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ❢✉♥❝t✐♦♥s ❤❛r❞✲ ❜❛❝❦ ❛♥❞ ❈❉✲❘❖▼✳ ❈❛♠❜r✐❞❣❡ ✉♥✐✈❡rs✐t② ♣r❡ss✳ ❬✹✸❪ ❘➔❞✉❝❛♥✉✱ ❉✳✱ ✫ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳ ✭✷✵✵✼✮✳ ❆ ♥❡✇ ❝❧❛ss ♦❢ ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ❜② ♠❡❛♥s ♦❢ ❛ ❝♦♥✈♦❧✉t✐♦♥ ♦♣❡r❛t♦r ✐♥✈♦❧✈✐♥❣ t❤❡ ❍✉r✇✐t③✕▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥✳ ■♥t❡❣r❛❧ ❚r❛♥s❢♦r♠s ❛♥❞ ❙♣❡❝✐❛❧ ❋✉♥❝t✐♦♥s✱ ✶✽✭✶✷✮✱ ✾✸✸✲✾✹✸✳ ✸✺✷ | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ ❬✹✹❪ ❘❛♠❛s✇❛♠✐✱ ❱✳ ✭✶✾✸✹✮✳ ◆♦t❡s ♦♥ ❘✐❡✲ ♠❛♥♥✬s ζ ✲❢✉♥❝t✐♦♥✳ ❏✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝✳✱ ✾✱ ✶✻✺✕✶✻✾✳ ❬✹✺❪ ❙❛♠❦♦✱ ❙✳ ●✳✱ ❑✐❧❜❛s✱ ❆✳ ❆✳✱ ✫ ▼❛r✐❝❤❡✈✱ ❖✳ ■✳ ✭✶✾✾✸✮✳ ❋r❛❝t✐♦♥❛❧ ✐♥t❡❣r❛❧s ❛♥❞ ❞❡r✐✈❛✲ t✐✈❡s✿ t❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ❬✹✻❪ ❙❤❛r♠❛✱ ▼✳✱ ✫ ❏❛✐♥✱ ❘✳ ✭✷✵✵✾✮✳ ❆ ♥♦t❡ ♦♥ ❛ ❣❡♥❡r❛❧✐③❡❞ ▼✲s❡r✐❡s ❛s ❛ s♣❡❝✐❛❧ ❢✉♥❝t✐♦♥ ♦❢ ❢r❛❝t✐♦♥❛❧ ❝❛❧❝✉❧✉s✳ ❋r❛❝t✳ ❈❛❧❝✳ ❆♣♣❧✳ ❆♥❛❧✱ ✶✷✭✹✮✱ ✹✹✾✲✹✺✷✳ ❬✹✼❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳ ✭✶✾✽✽✮✳ ❆ ✉♥✐✜❡❞ ♣r❡s❡♥✲ t❛t✐♦♥ ♦❢ ❝❡rt❛✐♥ ❝❧❛ss❡s ♦❢ s❡r✐❡s ♦❢ t❤❡ ❘✐❡✲ ♠❛♥♥ ❩❡t❛ ❢✉♥❝t✐♦♥✳ ❘✐✈✳ ▼❛t✳ ❯♥✐✈✳ P❛r♠❛ ✭ ❙❡r✳ ✹✮✱ ✶✹✱ ✶✕✷✸✳ ❬✹✽❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳ ✭✶✾✽✽✮✳ ❙✉♠s ♦❢ ❝❡rt❛✐♥ s❡r✐❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ ❩❡t❛ ❢✉♥❝t✐♦♥✳ ❏♦✉r✲ ♥❛❧ ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ❛♥❛❧②s✐s ❛♥❞ ❛♣♣❧✐❝❛✲ t✐♦♥s✱ ✶✸✹✭✶✮✱ ✶✷✾✲✶✹✵✳ ❬✹✾❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳ ✭✶✾✾✼✮✳ ❈❡rt❛✐♥ ❢❛♠✐❧✐❡s ♦❢ r❛♣✐❞❧② ❝♦♥✈❡r❣❡♥t s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r ζ (2n + 1)✳ ▼❛t❤✳ ❙❝✐✳ ❘❡s✳ ❍♦t✲▲✐♥❡✱ ✶ ✭✻✮✱ ✶✕✻ ✭❘❡s❡❛r❝❤ ❆♥♥♦✉♥❝❡♠❡♥t✮✳ ❬✺✵❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳ ✭✶✾✾✽✮✳ ❋✉rt❤❡r s❡r✐❡s r❡♣✲ r❡s❡♥t❛t✐♦♥s ❢♦r ζ (2n + 1)✳ ❆♣♣❧✐❡❞ ♠❛t❤❡✲ ♠❛t✐❝s ❛♥❞ ❝♦♠♣✉t❛t✐♦♥✱ ✾✼✭✶✮✱ ✶✲✶✺✳ ❬✺✶❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ✭✶✾✾✾✮✳ ❙♦♠❡ r❛♣✐❞❧② ❝♦♥✲ ✈❡r❣✐♥❣ s❡r✐❡s ❢♦r ζ (2n + 1)✳ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ✶✷✼✭✷✮✱ ✸✽✺✲✸✾✻✳ ❬✺✷❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳ ✭✷✵✵✵✮✳ ❙♦♠❡ s✐♠♣❧❡ ❛❧❣♦✲ r✐t❤♠s ❢♦r t❤❡ ❡✈❛❧✉❛t✐♦♥s ❛♥❞ r❡♣r❡s❡♥t❛✲ t✐♦♥s ♦❢ t❤❡ ❘✐❡♠❛♥♥ ❩❡t❛ ❢✉♥❝t✐♦♥ ❛t ♣♦s✲ ✐t✐✈❡ ✐♥t❡❣❡r ❛r❣✉♠❡♥ts✳ ❏♦✉r♥❛❧ ♦❢ ♠❛t❤✲ ❡♠❛t✐❝❛❧ ❛♥❛❧②s✐s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✱ ✷✹✻✭✷✮✱ ✸✸✶✲✸✺✶✳ ❬✺✸❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳ ✭✷✵✵✵✱ ❏✉❧②✮✳ ❙♦♠❡ ❢♦r♠✉✲ ❧❛s ❢♦r t❤❡ ❇❡r♥♦✉❧❧✐ ❛♥❞ ❊✉❧❡r ♣♦❧②♥♦♠✐✲ ❛❧s ❛t r❛t✐♦♥❛❧ ❛r❣✉♠❡♥ts✳ ■♥ ▼❛t❤❡♠❛t✐❝❛❧ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❈❛♠❜r✐❞❣❡ P❤✐❧♦s♦♣❤✲ ✐❝❛❧ ❙♦❝✐❡t② ✭❱♦❧✳ ✶✷✾✱ ◆♦✳ ✶✱ ♣♣✳ ✼✼✲✽✹✮✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❬✺✹❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳ ✭✷✵✶✶✮✳ ❙♦♠❡ ❣❡♥❡r❛❧✐③❛✲ t✐♦♥s ❛♥❞ ❜❛s✐❝ ✭♦r q−✮ ❡①t❡♥s✐♦♥s ♦❢ t❤❡ ❇❡r♥♦✉❧❧✐✱ ❊✉❧❡r ❛♥❞ ●❡♥♦❝❝❤✐ ♣♦❧②♥♦♠✐❛❧s✳ ❆♣♣❧✳ ▼❛t❤✳ ■♥❢♦r♠✳ ❙❝✐✱ ✺✭✸✮✱ ✸✾✵✲✹✹✹✳ ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✸ ❬✺✺❪ ❍❛r✐✱ ▼✳ ✭✷✵✶✸✮✳ ●❡♥❡r❛t✐♥❣ r❡❧❛t✐♦♥s ❛♥❞ ♦t❤❡r r❡s✉❧ts ❛ss♦❝✐❛t❡❞ ✇✐t❤ s♦♠❡ ❢❛♠✐❧✐❡s ♦❢ t❤❡ ❡①t❡♥❞❡❞ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝✲ t✐♦♥s✳ ❙♣r✐♥❣❡rP❧✉s✱ ✷✭✶✮✱ ✻✼✳ ❬✺✻❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳ ✭✷✵✶✹✮✳ ❆ ♥❡✇ ❢❛♠✐❧② ♦❢ t❤❡ λ✲❣❡♥❡r❛❧✐③❡❞ ❍✉r✇✐t③✲▲❡r❝❤ ③❡t❛ ❢✉♥❝✲ t✐♦♥s ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s✳ ❆♣♣❧✳ ▼❛t❤✳ ■♥❢✳ ❙❝✐✱ ✽✭✹✮✱ ✶✹✽✺✲✶✺✵✵✳ ❬✺✼❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳✱ ✫ ❈❤♦✐✱ ❏✳ ✭✷✵✵✶✮✳ ❙❡r✐❡s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ③❡t❛ ❛♥❞ r❡❧❛t❡❞ ❢✉♥❝✲ t✐♦♥s ✭❱♦❧✳ ✺✸✵✮✳ ❙♣r✐♥❣❡r ❙❝✐❡♥❝❡ ✫ ❇✉s✐✲ ♥❡ss ▼❡❞✐❛✳ ❬✺✽❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳✱ ✫ ❈❤♦✐✱ ❏✳ ✭✷✵✶✷✮✳ ❩❡t❛ ❛♥❞ q−❩❡t❛ ❢✉♥❝t✐♦♥s ❛♥❞ ❛ss♦❝✐❛t❡❞ s❡r✐❡s ❛♥❞ ✐♥t❡❣r❛❧s✳ ❊❧s❡✈✐❡r✳ ❬✺✾❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳✱ ●❛❜♦✉r②✱ ❙✳✱ ✫ ❚r❡♠✲ ❜❧❛②✱ ❘✳ ✭✷✵✶✹✮✳ ◆❡✇ r❡❧❛t✐♦♥s ✐♥✈♦❧✈✐♥❣ ❛♥ ❡①t❡♥❞❡❞ ♠✉❧t✐♣❛r❛♠❡t❡r ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥ ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s✳ ■♥t❡r♥❛✲ t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❆♥❛❧②s✐s✱ ✷✵✶✹✳ ❬✻✵❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳✱ ●❛r❣✱ ▼✳✱ ✫ ❈❤♦✉❞✲ ❤❛r②✱ ❙✳ ✭✷✵✶✵✮✳ ❆ ♥❡✇ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❇❡r♥♦✉❧❧✐ ❛♥❞ r❡❧❛t❡❞ ♣♦❧②♥♦♠✐❛❧s✳ ❘✉ss✐❛♥ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s✱ ✶✼✭✷✮✱ ✷✺✶✲✷✻✶✳ ❬✻✶❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳✱ ●❛r❣✱ ▼✳✱ ✫ ❈❤♦✉❞❤❛r②✱ ❙✳ ✭✷✵✶✶✮✳ ❙♦♠❡ ♥❡✇ ❢❛♠✐❧✐❡s ♦❢ ❣❡♥❡r❛❧✲ ✐③❡❞ ❊✉❧❡r ❛♥❞ ●❡♥♦❝❝❤✐ ♣♦❧②♥♦♠✐❛❧s✳ ❚❛✐✲ ✇❛♥❡s❡ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✶✺✭✶✮✱ ✷✽✸✲ ✸✵✺✳ ❬✻✷❪ ❙r✐✈❛s❛t❛✱ ❍✳ ▼✳✱ ●❧❛ss❡r✱ ▼✳ ▲✳✱ ✫ ❆❞❛♠✲ ❝❤✐❦✱ ❱✳ ❙✳ ✭✷✵✵✵✮✳ ❙♦♠❡ ❞❡✜♥✐t❡ ✐♥t❡❣r❛❧s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❘✐❡♠❛♥♥ ③❡t❛ ❢✉♥❝t✐♦♥✳ ❩✳ ❆♥❛❧✳ ❆♥✇❡♥❞✉♥❣❡♥✱ ✶✾✱ ✽✸✶✕✽✹✻✳ ❬✻✸❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳✱ ●✉♣t❛✱ ❑✳ ❈✳✱ ●♦②❛❧✱ ❙✳ P✳ ✭✶✾✽✷✮✳ ❚❤❡ H ✲❋✉♥❝t✐♦♥s ♦❢ ❖♥❡ ❛♥❞ ❚✇♦ ❱❛r✐❛❜❧❡s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✳ ❙♦✉t❤ ❆s✐❛♥ P✉❜❧✐s❤❡rs✱ ◆❡✇ ❉❡❧❤✐ ❛♥❞ ▼❛❞r❛s✳ ❬✻✹❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳✱ ❑❛r❧ss♦♥✱ P✳ ❲✳ ✭✶✾✽✺✮✳ ▼✉❧t✐♣❧❡ ●❛✉ss✐❛♥ ❍②♣❡r❣❡♦♠❡tr✐❝ ❙❡r✐❡s✳ ❍❛❧st❡❞ Pr❡ss ✭❊❧❧✐s ❍♦r✇♦♦❞ ▲✐♠✐t❡❞✱ ❈❤✐❝❤❡st❡r✮✱ ❏♦❤♥ ❲✐❧❡② ❛♥❞ ❙♦♥s✱ ◆❡✇ ❨♦r❦✱ ❈❤✐❝❤❡st❡r✱ ❇r✐s❜❛♥❡ ❛♥❞ ❚♦r♦♥t♦✳ | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ ❬✻✺❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳✱ ▲✐♥✱ ❙✳ ❉✳✱ ✫ ❲❛♥❣✱ P✳ ❨✳ ✭✷✵✵✻✮✳ ❙♦♠❡ ❢r❛❝t✐♦♥❛❧✲❝❛❧❝✉❧✉s r❡s✉❧ts ❢♦r t❤❡ H ✲❢✉♥❝t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ❝❧❛ss ♦❢ ❋❡②♥♠❛♥ ✐♥t❡❣r❛❧s✳ ❘✉ss✐❛♥ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s✱ ✶✸✭✶✮✱ ✾✹✲✶✵✵✳ ❬✻✻❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳✱ ▲✉♦✱ ▼✳ ❏✳✱ ✫ ❘❛✐♥❛✱ ❘✳ ❑✳ ✭✷✵✶✸✮✳ ◆❡✇ r❡s✉❧ts ✐♥✈♦❧✈✐♥❣ ❛ ❝❧❛ss ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❍✉r✇✐t③✲▲❡r❝❤ ③❡t❛ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥s✳ ❚✉r❦✐s❤ ❏✳ ❆♥❛❧✳ ◆✉♠❜❡r ❚❤❡♦r②✱ ✶✭✶✮✱ ✷✻✲✸✺✳ ❬✻✼❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳✱ ❙❛①❡♥❛✱ ❘✳ ❑✳✱ P♦❣á♥②✱ ❚✳ ❑✳✱ ✫ ❙❛①❡♥❛✱ ❘✳ ✭✷✵✶✶✮✳ ■♥t❡❣r❛❧ ❛♥❞ ❝♦♠♣✉t❛t✐♦♥❛❧ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ t❤❡ ❡①✲ t❡♥❞❡❞ ❍✉r✇✐t③✕▲❡r❝❤ ③❡t❛ ❢✉♥❝t✐♦♥✳ ■♥✲ t❡❣r❛❧ ❚r❛♥s❢♦r♠s ❛♥❞ ❙♣❡❝✐❛❧ ❋✉♥❝t✐♦♥s✱ ✷✷✭✼✮✱ ✹✽✼✲✺✵✻✳ ❬✻✽❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳✱ ❚♦♠♦✈s❦✐✱ ❩✳✱ ✫ ▲❡s❦♦✈s❦✐✱ ❉✳ ✭✷✵✶✺✮✳ ❙♦♠❡ ❢❛♠✐❧✐❡s ♦❢ ▼❛t❤✐❡✉✲t②♣❡ s❡r✐❡s ❛♥❞ ❍✉r✇✐t③✲▲❡r❝❤ ❩❡t❛ ❢✉♥❝t✐♦♥s ❛♥❞ ❛ss♦❝✐❛t❡❞ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s✳ ❆♣♣❧✳ ❈♦♠♣✉t✳ ▼❛t❤✱ ✶✹✭✸✮✱ ✸✹✾✲✸✽✵✳ ❬✻✾❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳✱ ✫ ❚s✉♠✉r❛✱ ❍✳ ✭✷✵✵✵✮✳ ❆ ❝❡rt❛✐♥ ❝❧❛ss ♦❢ r❛♣✐❞❧② ❝♦♥✈❡r❣❡♥t s❡✲ r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r ζ (2n + 1)✳ ❏♦✉r♥❛❧ ♦❢ ❈♦♠♣✉t❛t✐♦♥❛❧ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✲ ✐❝s✱ ✶✶✽✭✶✲✷✮✱ ✸✷✸✲✸✸✺✳ ❬✼✵❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳✱ ❚s✉♠✉r❛✱ ❍✳ ✭✷✵✵✵✮✳ ◆❡✇ r❛♣✐❞❧② ❝♦♥✈❡r❣❡♥t s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r ζ (2n + 1)✱ L (2n, χ) ❛♥❞ L (2n + 1, χ)✳ ▼❛t❤✳ ❙❝✐✳ ❘❡s✳ ❍♦t✲▲✐♥❡✱ ✹✭✼✮✱ ✶✼✕✷✹ ✭❘❡✲ s❡❛r❝❤ ❆♥♥♦✉♥❝❡♠❡♥t✮✳ ❬✼✶❪ ❙r✐✈❛st❛✈❛✱ ❍✳ ▼✳✱ ✫ ❚s✉♠✉r❛✱ ❍✳ ✭✷✵✵✸✮✳ ■♥❞✉❝t✐✈❡ ❝♦♥str✉❝t✐♦♥ ♦❢ r❛♣✐❞❧② ❝♦♥✈❡r✲ ❣❡♥t s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r ζ (2n + 1)✳ ■♥t❡r♥❛t✐♦♥❛❧ ❥♦✉r♥❛❧ ♦❢ ❝♦♠♣✉t❡r ♠❛t❤❡✲ ♠❛t✐❝s✱ ✽✵✭✾✮✱ ✶✶✻✶✲✶✶✼✸✳ ❬✼✷❪ ❚✐t❝❤♠❛rs❤✱ ❊✳ ❈✳✱ ❚✐t❝❤♠❛rs❤✱ ❊✳ ❈✳ ❚✳✱ ✫ ❍❡❛t❤✲❇r♦✇♥✱ ❉✳ ❘✳ ✭✶✾✽✻✮✳ ❚❤❡ t❤❡♦r② ♦❢ t❤❡ ❘✐❡♠❛♥♥ ③❡t❛✲❢✉♥❝t✐♦♥✳ ❖①❢♦r❞ ❯♥✐✲ ✈❡rs✐t② Pr❡ss✳ ❬✼✸❪ ❚r✐❝♦♠✐✱ ❋✳ ●✳ ✭✶✾✻✾✮✳ ❙✉❧❧❛ s♦♠♠❛ ❞❡❧❧❡ ✐♥✲ ✈❡rs❡ ❞❡❧❧❡ t❡r③❡ ❡ q✉✐♥t❡ ♣♦t❡♥③❡ ❞❡✐ ♥✉♠❡r✐ ♥❛t✉r❛❧✐✳ ❆tt✐ ❆❝❝❛❞✳ ◆❛③✳ ▲✐♥❝❡✐ ❘❡♥❞✳ ❈❧✳ ❙❝✐✳ ❋✐s✳ ▼❛t✳ ◆❛t✉r✳ ✭❙❡r✳ ✽✮✱ ✹✼✱ ✶✻✕✶✽✳ ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✸✺✸ ❱❖▲❯▼❊✿ ✸ ❬✼✹❪ ❚s✉♠✉r❛✱ ❍✳ ✭✶✾✾✹✮✳ ❖♥ ❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ❉✐r✐❝❤❧❡t s❡r✐❡s ❛t ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs ❜② q−❝❛❧❝✉❧❛t✐♦♥✳ ❏♦✉r♥❛❧ ♦❢ ◆✉♠❜❡r ❚❤❡♦r②✱ ✹✽✭✸✮✱ ✸✽✸✲✸✾✶✳ ❬✼✺❪ ❲❤✐tt❛❦❡r✱ ❊✳ ❚✳✱ ❲❛ts♦♥✱ ●✳ ◆✳ ✭✶✾✷✼✮✳ ❆ ❈♦✉rs❡ ♦❢ ▼♦❞❡r♥ ❆♥❛❧②s✐s✿ ❆♥ ■♥tr♦✲ ❞✉❝t✐♦♥ t♦ t❤❡ ●❡♥❡r❛❧ ❚❤❡♦r② ♦❢ ■♥✜♥✐t❡ Pr♦❝❡ss❡s ❛♥❞ ♦❢ ❆♥❛❧②t✐❝ ❋✉♥❝t✐♦♥s❀ ✇✐t❤ ❛♥ ❆❝❝♦✉♥t ♦❢ t❤❡ Pr✐♥❝✐♣❛❧ ❚r❛♥s❝❡♥❞❡♥t❛❧ ❋✉♥❝t✐♦♥s✳ ❋♦✉rt❤ ❊❞✐t✐♦♥✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✲ ✈❡rs✐t② Pr❡ss✱ ❈❛♠❜r✐❞❣❡✱ ▲♦♥❞♦♥ ❛♥❞ ◆❡✇ ❨♦r❦✳ ❬✼✻❪ ❲✐❧t♦♥✱ ❏✳ ❘✳ ✭✶✾✷✷✲✶✾✷✸✮✳ ❆ ♣r♦♦❢ ♦❢ ❇✉r♥✲ s✐❞❡✬s ❢♦r♠✉❧❛ ❢♦r log Γ (x + 1) ❛♥❞ ❝❡rt❛✐♥ ❛❧❧✐❡❞ ♣r♦♣❡rt✐❡s ♦❢ ❘✐❡♠❛♥♥✬s ζ ✲❢✉♥❝t✐♦♥✳ ▼❡ss❡♥❣❡r ▼❛t❤✳✱ ✺✷✱ ✾✵✕✾✸✳ ❬✼✼❪ ❲✐tt❡♥✱ ❊✳ ✭✶✾✾✶✮✳ ❖♥ q✉❛♥t✉♠ ❣❛✉❣❡ t❤❡♦✲ r✐❡s ✐♥ t✇♦ ❞✐♠❡♥s✐♦♥s✳ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s✱ ✶✹✶✭✶✮✱ ✶✺✸✲✷✵✾✳ ❬✼✽❪ ❨❡♥✱ ❈✳✲❊✱ ▲✐♥✱ ▼✳✲▲✳✱ ◆✐s❤✐♠♦t♦✱ ❑✳ ✭✷✵✵✷✮✳ ❆♥ ✐♥t❡❣r❛❧ ❢♦r♠ ❢♦r ❛ ❣❡♥❡r❛❧✐③❡❞ ❩❡t❛ ❢✉♥❝t✐♦♥✳ ❏✳ ❋r❛❝t✳ ❈❛❧❝✳✱ ✷✷✱ ✾✾✕✶✵✷✳ ❬✼✾❪ ❩❤❛♥❣✱ ◆✳✲❨✳✱ ❲✐❧❧✐❛♠s✱ ❑✳ ❙✳ ✭✶✾✾✸✮✳ ❙♦♠❡ s❡r✐❡s r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ζ (2n + 1)✳ ❘♦❝❦② ▼♦✉♥t❛✐♥ ❏✳ ▼❛t❤✳✱ ✷✸✱ ✶✺✽✶✕✶✺✾✷✳ ❬✽✵❪ ❩❤❛♥❣✱ ◆✳✲❨✳✱ ❲✐❧❧✐❛♠s✱ ❑✳ ❙✳ ✭✶✾✾✸✮✳ ❙♦♠❡ ✐♥✜♥✐t❡ s❡r✐❡s ✐♥✈♦❧✈✐♥❣ t❤❡ ❘✐❡♠❛♥♥ ❩❡t❛ ❢✉♥❝t✐♦♥✱ ✐♥ ❆♥❛❧②s✐s✱ ●❡♦♠❡tr② ❛♥❞ ●r♦✉♣s✿ ❆ ❘✐❡♠❛♥♥ ▲❡❣❛❝② ❱♦❧✉♠❡✱ P❛rts ■ ❛♥❞ ■■ ✭❍✳ ▼✳ ❙r✐✈❛st❛✈❛ ❛♥❞ ❚❤✳ ▼✳ ❘❛s✲ s✐❛s✱ ❊❞✐t♦rs✮✱ P❛rt ■■✱ ✻✾✶✕✼✶✷✱ ❍❛❞r♦♥✐❝ Pr❡ss✱ P❛❧♠ ❍❛r❜♦r✱ ❋❧♦r✐❞❛✳ ❬✽✶❪ ❩❤❛♥❣✱ ◆✳✲❨✳✱ ❲✐❧❧✐❛♠s✱ ❑✳ ❙✳ ✭✶✾✾✺✮✳ ❱❛❧✉❡s ♦❢ t❤❡ ❘✐❡♠❛♥♥ ❩❡t❛ ❢✉♥❝t✐♦♥ ❛♥❞ ✐♥t❡❣r❛❧s ✐♥✈♦❧✈✐♥❣ log(2 sinh θ2 ) ❛♥❞ log(2 sin θ2 )✳ P❛❝✐✜❝ ❏✳ ▼❛t❤✳✱ ✶✻✽✱ ✷✼✶✕✷✽✾✳ | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ ❛❣❡ ♦❢ ✶✾ ②❡❛rs✳ ❍❡ ❡❛r♥❡❞ ❤✐s P❤✳❉✳ ❞❡❣r❡❡ ✐♥ ✶✾✻✺ ✇❤✐❧❡ ❤❡ ✇❛s ❛ ❢✉❧❧✲t✐♠❡ ♠❡♠❜❡r ♦❢ t❤❡ t❡❛❝❤✐♥❣ ❢❛❝✉❧t② ❛t ❏✳ ◆✳ ❱✳ ❯♥✐✈❡rs✐t② ♦❢ ❏♦❞❤♣✉r ✭s✐♥❝❡ ✶✾✻✸✮✳ ❈✉rr❡♥t❧②✱ Pr♦❢❡ss♦r ❙r✐✈❛st❛✈❛ ❤♦❧❞s t❤❡ ♣♦s✐t✐♦♥ ♦❢ Pr♦❢❡ss♦r ❊♠❡r✐t✉s ✐♥ t❤❡ ❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❙t❛t✐st✐❝s ❛t t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❱✐❝t♦r✐❛ ✐♥ ❈❛♥❛❞❛✱ ❤❛✈✐♥❣ ❥♦✐♥❡❞ t❤❡ ❢❛❝✉❧t② t❤❡r❡ ✐♥ ✶✾✻✾✳ Pr♦❢❡ss♦r ❙r✐✈❛st❛✈❛ ❤❛s ❤❡❧❞ ♥✉♠❡r♦✉s ✈✐s✐t✐♥❣ ♣♦s✐t✐♦♥s ❛t ♠❛♥② ✉♥✐✈❡rs✐t✐❡s ❛♥❞ r❡s❡❛r❝❤ ✐♥st✐t✉t❡s ✐♥ ❞✐✛❡r❡♥t ♣❛rts ♦❢ t❤❡ ✇♦r❧❞✳ ❍❛✈✐♥❣ r❡❝❡✐✈❡❞ s❡✈❡r❛❧ ❉✳❙❝✳ ✭❤♦♥♦r✐s ❝❛✉s❛✮ ❞❡❣r❡❡s ❛s ✇❡❧❧ ❛s ❤♦♥♦r❛r② ♠❡♠❜❡rs❤✐♣s ❛♥❞ ❢❡❧❧♦✇s❤✐♣s ♦❢ ♠❛♥② s❝✐❡♥t✐✜❝ ❛❝❛❞❡♠✐❡s ❛r♦✉♥❞ t❤❡ ✇♦r❧❞✱ ❤❡ ✐s ❛❧s♦ ❛❝t✐✈❡❧② ❛ss♦❝✐❛t❡❞ ❡❞✐t♦r✐❛❧❧② ✇✐t❤ ♥✉✲ ♠❡r♦✉s ✐♥t❡r♥❛t✐♦♥❛❧ s❝✐❡♥t✐✜❝ r❡s❡❛r❝❤ ❥♦✉r♥❛❧s✳ Pr♦❢❡ss♦r ❙r✐✈❛st❛✈❛✬s r❡s❡❛r❝❤ ✐♥t❡r❡sts ✐♥❝❧✉❞❡ s❡✈❡r❛❧ ❛r❡❛s ♦❢ ♣✉r❡ ❛♥❞ ❛♣♣❧✐❡❞ ♠❛t❤❡♠❛t✲ ✐❝❛❧ s❝✐❡♥❝❡s s✉❝❤ ❛s ✭❢♦r ❡①❛♠♣❧❡✮ ❘❡❛❧ ❛♥❞ ❈♦♠♣❧❡① ❆♥❛❧②s✐s✱ ❋r❛❝t✐♦♥❛❧ ❈❛❧❝✉❧✉s ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✱ ■♥t❡❣r❛❧ ❊q✉❛t✐♦♥s ❛♥❞ ❚r❛♥s✲ ❢♦r♠s✱ ❍✐❣❤❡r ❚r❛♥s❝❡♥❞❡♥t❛❧ ❋✉♥❝t✐♦♥s ❛♥❞ ❚❤❡✐r ❆♣♣❧✐❝❛t✐♦♥s✱ q ✲❙❡r✐❡s ❛♥❞ q ✲P♦❧②♥♦♠✐❛❧s✱ ❆♥❛❧②t✐❝ ◆✉♠❜❡r ❚❤❡♦r②✱ ❆♥❛❧②t✐❝ ❛♥❞ ●❡♦✲ ♠❡tr✐❝ ■♥❡q✉❛❧✐t✐❡s✱ Pr♦❜❛❜✐❧✐t② ❛♥❞ ❙t❛t✐st✐❝s✱ ❛♥❞ ■♥✈❡♥t♦r② ▼♦❞❡❧❧✐♥❣ ❛♥❞ ❖♣t✐♠✐③❛t✐♦♥✳ ❍❡ ❤❛s ♣✉❜❧✐s❤❡❞ ✷✹ ❜♦♦❦s✱ ♠♦♥♦❣r❛♣❤s ❛♥❞ ❡❞✐t❡❞ ✈♦❧✉♠❡s✱ ✸✵ ❜♦♦❦ ✭❛♥❞ ❡♥❝②❝❧♦♣❡❞✐❛✮ ❝❤❛♣t❡rs✱ ✹✽ ♣❛♣❡rs ✐♥ ✐♥t❡r♥❛t✐♦♥❛❧ ❝♦♥❢❡r❡♥❝❡ ♣r♦❝❡❡❞✐♥❣s✱ ❛♥❞ ♦✈❡r ✶✱✷✵✵ s❝✐❡♥t✐✜❝ r❡s❡❛r❝❤ ❥♦✉r♥❛❧ ❛rt✐❝❧❡s✳ ❋✉rt❤❡r ❞❡t❛✐❧s ❛❜♦✉t Pr♦❢❡ss♦r ❙r✐✈❛st❛✈❛✬s ♣r♦❢❡ss✐♦♥❛❧ ❛❝❤✐❡✈❡♠❡♥ts ❛♥❞ ❛❝❝♦♠♣❧✐s❤♠❡♥ts✱ ❛♥❞ ❤♦♥♦rs✱ ❛✇❛r❞s ❛♥❞ ❞✐st✐♥❝t✐♦♥s✱ ❝❛♥ ❜❡ ❢♦✉♥❞ ❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❲❡❜ ❙✐t❡✿ ❯❘▲✿ ❤tt♣✿✴✴✇✇✇✳♠❛t❤✳✉✈✐❝✳❝❛✴∼❤❛r✐♠sr✐✴ ❆❜♦✉t ❆✉t❤♦rs ❍❛r✐ ▼♦❤❛♥ ❙❘■❱❆❙❚❆❱❆ ❱✐❝t♦r✐❛✱ ❈❛♥❛❞❛✮ ✭❯♥✐✈❡rs✐t② ♦❢ Pr♦❢❡ss♦r ❍❛r✐ ▼♦❤❛♥ ❙r✐✈❛st❛✈❛ ❜❡❣❛♥ ❤✐s ✉♥✐✈❡rs✐t②✲❧❡✈❡❧ t❡❛❝❤✐♥❣ ❝❛r❡❡r r✐❣❤t ❛❢t❡r ❤❛✈✐♥❣ r❡❝❡✐✈❡❞ ❤✐s ▼✳❙❝✳ ❞❡❣r❡❡ ✐♥ ✶✾✺✾ ❛t t❤❡ ✸✺✹ ✧❚❤✐s ✐s ❛♥ ❖♣❡♥ ❆❝❝❡ss ❛rt✐❝❧❡ ❞✐str✐❜✉t❡❞ ✉♥❞❡r t❤❡ t❡r♠s ♦❢ t❤❡ ❈r❡❛t✐✈❡ ❈♦♠♠♦♥s ❆ttr✐❜✉t✐♦♥ ▲✐❝❡♥s❡✱ ✇❤✐❝❤ ♣❡r♠✐ts ✉♥r❡str✐❝t❡❞ ✉s❡✱ ❞✐str✐❜✉t✐♦♥✱ ❛♥❞ r❡♣r♦❞✉❝t✐♦♥ ✐♥ ❛♥② ♠❡❞✐✉♠ ♣r♦✈✐❞❡❞ t❤❡ ♦r✐❣✐♥❛❧ ✇♦r❦ ✐s ♣r♦♣❡r❧② ❝✐t❡❞ ✭❈❈ ❇❨ ✹✳✵✮✳✧