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✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ ❚❘❺◆ ❚❍➚ ❍❒◆ ❱➋ ❙➮ ✣❆ ❚❍Ù❈ ❇❻❚ ❑❍❷ ◗❯❨ ❚❘➊◆ ❚❘×❮◆● ❍Ú❯ ❍❸◆ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✷✵ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ ❚❘❺◆ ❚❍➚ ❍❒◆ ❱➋ ❙➮ ✣❆ ❚❍Ù❈ ❇❻❚ ❑❍❷ ◗❯❨ ❚❘➊◆ ❚❘×❮◆● ❍Ú❯ ❍❸◆ ❈❤✉②➯♥ ♥❣➔♥❤✿ P❤÷ì♥❣ ♣❤→♣ t♦→♥ ❝➜♣ ▼➣ sè✿ ✽ ✹✻ ✵✶ ✶✸ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈ ❚❙✳ ◆●➷ ❚❍➚ ◆●❖❆◆ ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✷✵ ✐ ▼ư❝ ❧ư❝ ▼ð ✤➛✉ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶ ✹ ✶✳✶ ▼ët sè ❦❤→✐ ♥✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ❚r÷í♥❣ ❤ú✉ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸ ❍➔♠ ▼♦❜✐✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ❈❤÷ì♥❣ ✷ ❙ü t÷ì♥❣ tü ❣✐ú❛ Fq [T ] ✈➔ Z ✶✵ ❈❤÷ì♥❣ ✸ ✣➳♠ sè ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② ✶✹ ✷✳✶ ▼ët sè t➼♥❤ ❝❤➜t ❝❤✉♥❣ ❝õ❛ Fq [T ] ✈➔ Z ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ t t tữỡ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✸✳✶ ❙è ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② ♠♦♥✐❝ ❜➟❝ n tr➯♥ Fq ✸✳✷ ❙è ❝→❝ ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② ✈ỵ✐ ❜➟❝ ≤ n ✳ ✳ ✸✳✸ ❚➼♥❤ ❧✐➯♥ tö❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✹ ✣✐➲✉ ❝❤➾♥❤ ❤➔♠ ✤➳♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✽ ✷✺ ✷✽ ✸✺ ✐✐ ▲í✐ ❝↔♠ ì♥ ▲✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ t❤ü❝ ❤✐➺♥ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✕ t ữợ sỹ ữợ ❞➝♥ ❝õ❛ ❚❙✳ ◆❣ỉ ❚❤à ◆❣♦❛♥✳ ❚→❝ ❣✐↔ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ s➙✉ s➢❝ tợ ữớ ữợ ữớ t ự tớ ữợ ✈➔ t➟♥ t➻♥❤ ❣✐↔✐ ✤→♣ ♥❤ú♥❣ t❤➢❝ ♠➢❝ ❝õ❛ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❧➔♠ ❧✉➟♥ ✈➠♥✳ ❚→❝ ❣✐↔ ❝ơ♥❣ ✤➣ ❤å❝ t➟♣ ✤÷đ❝ r➜t ♥❤✐➲✉ ❦✐➳♥ t❤ù❝ ❝❤✉②➯♥ ♥❣➔♥❤ ❜ê ➼❝❤ ❝❤♦ ❝æ♥❣ t→❝ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❜↔♥ t❤➙♥✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❝↔♠ ì♥ s➙✉ s➢❝ tỵ✐ ❝→❝ t❤➛② ❣✐→♦✱ ❝ỉ ❣✐→♦ ✤➣ t❤❛♠ ợ trữớ ✈➔ ❝→❝ ♣❤á♥❣ ❝❤ù❝ ♥➠♥❣ ❝õ❛ ❚r÷í♥❣❀ ❑❤♦❛ ❚♦→♥ ✕ ❚✐♥✱ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✕ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ q✉❛♥ t➙♠ ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ t↕✐ tr÷í♥❣✳ ❚→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ s➙✉ s➢❝ tỵ✐ ❚r✉♥❣ t➙♠ ◆❣❤✐➯♥ ❝ù✉ ✈➔ P❤→t tr✐➸♥ ❣✐→♦ ❞ö❝ ❍↔✐ P❤á♥❣ ✤➣ ❣✐ó♣ ✤ï✱ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ❣✐ó♣ tỉ✐ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚→❝ ❣✐↔ ❝ô♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ t➟♣ t❤➸ ❧ỵ♣ ❈❛♦ ❤å❝ ❚♦→♥ ❑✶✷❆✼ ✤➣ ❧✉ỉ♥ ✤ë♥❣ ✈✐➯♥ ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ r➜t ♥❤✐➲✉ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❧➔♠ ❧✉➟♥ ✈➠♥✳ ❈✉è✐ ❝ị♥❣✱ tỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣ ❣✐ó♣ ✤ï ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ❝❤♦ tæ✐ ❦❤✐ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✻ ♥➠♠ ✷✵✷✵ ❚→❝ ❣✐↔ ❚r➛♥ ❚❤à ❍ì♥ ✶ ▼ð ✤➛✉ ▼ët tr ỳ ữủ ự tr ỵ tt sè ✤â ❧➔ sü ♣❤➙♥ ❜è ❝→❝ sè ♥❣✉②➯♥ tè✳ ◆❣÷í✐ t❛ ♥❤➟♥ t❤➜② r➡♥❣ ❝→❝ sè ♥❣✉②➯♥ tè ♥❤ä ♥➡♠ t÷ì♥❣ ✤è✐ ❣➛♥ ♥❤❛✉✱ tr♦♥❣ ❦❤✐ ❝→❝ sè ♥❣✉②➯♥ tố ợ t õ ữợ ♥❤❛✉ ❤ì♥✳ ❚❛ ✤➦t ❝➙✉ ❤ä✐ ✈➲ sü ❧✐➯♥ q✉❛♥ ❣✐ú❛ ♠➟t ✤ë ❝õ❛ ❝→❝ sè ♥❣✉②➯♥ tè ✈ỵ✐ ✤ë ❧ỵ♥ ❝õ❛ ❝❤ó♥❣✳ ❇➡♥❣ ❝→❝❤ ❧➟♣ ❜↔♥❣ sè ♥❣✉②➯♥ tè ✈➔ ♥❣❤✐➯♥ ❝ù✉ ♠➟t ✤ë✱ ●❛✉ss t❤➜② r➡♥❣ “①✉♥❣ q✉❛♥❤ x ♠➟t ✤ë ❝õ❛ ❝→❝ sè ♥❣✉②➯♥ tè ❧➔ ①➜♣ ①➾ ❧♦❣1(x) ” t❤❡♦ ❬✾❪✳ P❤→t ❤✐➺♥ ♥➔② ❧➔ ❝❤➻❛ õ t ỵ số tố ❝❤ù♥❣ ♠✐♥❤ ♣❤→t ❤✐➺♥ ♥➔②✱ ●❛✉ss ✤➣ ♥❣❤✐➯♥ ❝ù✉ ❤➔♠ ✤➳♠ sè ♥❣✉②➯♥ tè✿ ●å✐ x ❧➔ sè t❤ü❝ ❞÷ì♥❣✱ π(x) ❜✐➸✉ t❤à sè ❝→❝ sè ♥❣✉②➯♥ tè ♥❤ä ❤ì♥ ❤♦➦❝ ❜➡♥❣ x✳ ❚ù❝ ❧➔ t❛ ❝â π(x) = 1✳ ❱➻ ♥❣÷í✐ t❛ ✤➣ ❞ü ✤♦→♥ ✈➲ ♠➟t ✤ë ❝→❝ p≤x ❧♦❣(x) ✱ ♥➯♥ ❤å ❝ô♥❣ ❞ü ✤♦→♥ r➡♥❣ π(x) ①➜♣ ①➾ sè ♥❣✉②➯♥ tè q✉❛♥❤ x ❧➔ ✈ỵ✐ ♠ët tê♥❣ ❧♦❣❛r✐t ❤♦➦❝ ♠ët t➼❝❤ ♣❤➙♥ ❧♦❣❛r✐t✳ ❈❤ó♥❣ t÷ì♥❣ ù♥❣ ✤÷đ❝ ❝❤♦ ❜ð✐✿ ❧s(x) := , ❧♦❣ (n) 2≤n≤x x ❧✐(x) := dt ❧♦❣(t) ❚❛ ♥â✐ ❤❛✐ ❤➔♠ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ t÷ì♥❣ ✤÷ì♥❣ tữỡ số ú f (x) t tợ x t tợ ổ ũ sỷ ỵ ❤✐➺✉ f (x) ∼ g(x) ❦❤✐ g(x) ❧♦❣(2) t❤❡♦ ❍➺ q✉↔ ✶✳✺✳✶ tr♦♥❣ ❬✹❪✳ ❉♦ ✤â✱ ❤❛✐ ❤➔♠ tê♥❣ ❧♦❣❛r✐t ✈➔ t➼❝❤ ♣❤➙♥ ❧♦❣❛r✐t ❧➔ t÷ì♥❣ ✤÷ì♥❣✳ ❍❛✐ ❤➔♠ ụ tữỡ ữỡ ợ x(x) q tr ỵ số tố ữủ ss ✈➔ ▲❡❣❡♥❞r❡ ✭✶✼✾✽✮ ♥➯✉ r❛ x → ∞✳ ❱ỵ✐ ♠é✐ x ≥ 2✱ ❤✐➺✉ sè ❣✐ú❛ ❧s(x) ✈➔ ❧✐(x) ❜à ❝❤➦♥ ❜ð✐ ✷ ❣✐↔ t❤✉②➳t r➡♥❣ ❤➔♠ ✤➳♠ sè ♥❣✉②➯♥ tố (x) tữỡ ữỡ ợ õ ữủ tr ữợ x (x) (x) (x → ∞) ▼ët tr➠♠ ♥➠♠ s❛✉ ✈➔♦ ♥➠♠ ✶✽✾✻ ✤à♥❤ ỵ ữủ ự r ❱❛❧❧➨❡ ▼✉❢❢s✐♥ ♠ët ❝→❝❤ ✤ë❝ ❧➟♣✳ ❈↔ ❤❛✐ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ❤å ✤➲✉ ❞ü❛ tr➯♥ ❤➔♠ ③❡t❛ ❘✐❡♠❛♥♥✱ ♠ët ♠ð rë♥❣ ❣✐↔✐ t➼❝❤ ❝õ❛ tê♥❣ ∞ ζ(s) = ✳ ❘✐❡♠❛♥♥ ✤➣ ❝❤➾ r❛ r➡♥❣ sü ♣❤➙♥ ❜ê ❝→❝ sè ♥❣✉②➯♥ tè ❝â s n=1 n ❧✐➯♥ q✉❛♥ trü❝ t✐➳♣ ✤➳♥ t➟♣ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ❤➔♠ ♥➔②✳ ❍❛❞❛♠❛r❞ ✈➔ ▲❛ ❱❛❧❧➨❡ ▼✉❢❢s✐♥ ✤➣ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ❤➔♠ ❘✐❡♠❛♥♥ ③❡t❛ ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ tr➯♥ ✤÷í♥❣ t❤➥♥❣ ❘❡(s) = 1✱ ❝❤ó♥❣ ✤➣ ữủ sỷ ự ỵ số tè✳ ❈→❝ ❣✐→ trà ①➜♣ ①➾ ❝õ❛ ❧s(x) ✈➔ ❧✐(x) tèt ❤ì♥ ❧♦❣x(x) ❞♦ ✤â ❝❤ó♥❣ t❤÷í♥❣ ✤÷đ❝ ÷✉ t✐➯♥ ❤ì♥ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♣❤➛♥ s❛✐ sè✳ ✣è✐ ✈ỵ✐ s số t sỷ ỵ O ❱ỵ✐ ❤❛✐ ❤➔♠ f ✈➔ g ❜➜t ❦ý✱ t❛ ❝â f (x) = O(g(x)) tỗ t số C s❛♦ ❝❤♦ ✈ỵ✐ x ✤õ ❧ỵ♥✱ ❣✐→ trà t✉②➺t ✤è✐ ❝õ❛ f (x) ❜à ❝❤➦♥ ❜ð✐ Cg(x)✳ ❱➻ ❧s(x) ✈➔ ❧✐(x) ❝❤➾ ❦❤→❝ ♥❤❛✉ ♠ët sè ❜à ❝❤➦♥✱ ♥➯♥ ♣❤➛♥ s❛✐ sè ❝ơ♥❣ ✤ó♥❣ ✈ỵ✐ ❧s(x)✳ ❇➡♥❣ ❝→❝❤ sû ❞ư♥❣ ❤➔♠ ζ ❦❤ỉ♥❣ ❝â ♥❣❤✐➺♠ tr➯♥ ✤÷í♥❣ t❤➥♥❣ ❘❡(s) = t ỵ tr ự ữủ tỗ t số c s c ❧♦❣(x) ✭✷✮ π(x) = ❧✐(x) + O xe P❤➛♥ s❛✐ sè ð ✤➙② ❝â t❤➸ ✤÷đ❝ ❦❤→✐ q✉→t ❤ì♥ ❜ð✐ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ζ ✳ ✣➦t Θ = s✉♣ζ(s)=0 ❘❡(s) ❧➔ ❝➟♥ tr➯♥ ✤ó♥❣ ❝õ❛ ❝→❝ ♣❤➛♥ t❤ü❝ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ζ ✳ ❑❤✐ ✤â t❤❡♦ ❬✺❪ t❛ ❝â✿ π(x) = ❧✐(x) + O xΘ ❧♦❣(x) ✭✸✮ ❘✐❡♠❛♥♥ ✤➣ ❝❤♦ r➡♥❣ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ ❝õ❛ ζ ♥➡♠ tr➯♥ ✤÷í♥❣ t❤➥♥❣ ❘❡(s) = 21 ✳ ●✐↔ t❤✐➳t ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ❣✐↔ t❤✉②➳t ❘✐❡♠❛♥♥✳ ●✐↔ t❤✉②➳t ❘✐❡♠❛♥♥ s✉② r❛ Θ = 12 ✱ ✤✐➲✉ ♥➔② ❝❤♦ t❛ ①➜♣ ①➾ π(x) = ❧✐(x) + O √ x❧♦❣(x) ✸ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ t❛ s➩ t➻♠ ❤✐➸✉ ✈➲ ởt sỹ tữỡ tỹ ỵ số tố ữ ✤÷đ❝ ♣❤→t ❜✐➸✉ tr♦♥❣ ✈➔♥❤ Fq [T ] ❧➔ ✈➔♥❤ ❝→❝ ✤❛ t❤ù❝ ♠ët ❜✐➳♥ ❚ ✈ỵ✐ ❝→❝ ❤➺ sè t❤✉ë❝ tr÷í♥❣ ❤ú✉ ❤↕♥ Fq ✳ ❚❛ s➩ ♥❣❤✐➯♥ ❝ù✉ tữỡ ữỡ ợ số t❤ù❝ ❜➜t ❦❤↔ q✉② ✈➔ ❝â sü s♦ s→♥❤ ❝→❝ ❦➳t q✉↔ ♥➔② ✈ỵ✐ ❤➔♠ ✤➳♠ sè ♥❣✉②➯♥ tè π(x)✳ ▼ët tr♦♥❣ ♥❤ú♥❣ ❧đ✐ t❤➳ ❦❤✐ ❧➔♠ ✈✐➺❝ ✈ỵ✐ Fq [T ] ❧➔ ❝æ♥❣ t❤ù❝ ❝õ❛ ●❛✉ss✱ ♠ët ❝æ♥❣ t❤ù❝ trü❝ t✐➳♣ ✈➲ sè ❧÷đ♥❣ ✤❛ t❤ù❝ ♠♦♥✐❝ ❜➜t ❦❤↔ q✉② ❜➟❝ n✳ ✣➙② ❧➔ ♠ët ❝ỉ♥❣ ❝ư r➜t ♠↕♥❤ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❤➔♠ t÷ì♥❣ ✤÷ì♥❣ ❤➔♠ ✤➳♠✳ ▲✉➟♥ ữủ t ữỡ ữỡ ỗ ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✈➲ tr÷í♥❣ ❤ú✉ ❤↕♥ ỵ s ỳ tự ♣❤ö❝ ✈ö ❝❤♦ ✈✐➺❝ tr➻♥❤ ❜➔② ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ tr♦♥❣ ♥❤ú♥❣ ❝❤÷ì♥❣ s❛✉✳ ❈❤÷ì♥❣ ✷ ♥➯✉ ❧➯♥ ♥❤ú♥❣ t➼♥❤ ❝❤➜t ❝ì ❜↔♥✱ ❝❤ó♥❣ ❝❤♦ t❛ t❤➜② sü t÷ì♥❣ tü ❣✐ú❛ ❤❛✐ ♠✐➲♥ ♥❣✉②➯♥ Z ✈➔ Fq [T ]✳ ❈❤÷ì♥❣ ✸ tr➻♥❤ ❜➔② ✈➲ ❤➔♠ ✤➳♠ sè ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥ q ♣❤➛♥ tỷ ỗ tớ ụ t t sỹ tữỡ tỹ ợ ỵ số tố tr ✈➔♥❤ ❝→❝ sè ♥❣✉②➯♥✳ ✹ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ♣❤➛♥ ♥➔②✱ t❛ s➩ tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ✈➲ ❝→❝ tr÷í♥❣ ❤ú✉ ❤↕♥ ✈➔ ❦❤→✐ ♥✐➺♠ ❤➔♠ ▼♦❜✐✉s✳ ◆❤ú♥❣ ❦➳t q✉↔ ♥➔② s➩ ✤÷đ❝ sû ❞ư♥❣ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❝æ♥❣ t❤ù❝ ❝õ❛ ●❛✉ss ✈➲ sè ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② ✤à♥❤ ❝❤✉➞♥ ❜➟❝ n ✈➔ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ❝→❝ ❝❤÷ì♥❣ s❛✉ ❝õ❛ ❧✉➟♥ ✈➠♥✳ ✶✳✶ ▼ët sè ❦❤→✐ ♥✐➺♠ ❚❛ ♥❤➢❝ ❧↕✐✱ ♠ët tr÷í♥❣ F ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❦❤→❝ ❦❤æ♥❣ ✈➔ ♠å✐ ♣❤➛♥ tû ❦❤→❝ ❦❤ỉ♥❣ ✤➲✉ ❦❤↔ ♥❣❤à❝❤✳ ▼ët tr÷í♥❣ ❝â ❤ú✉ ❤↕♥ ♣❤➛♥ tû ✤÷đ❝ ❣å✐ ❧➔ ♠ët tr÷í♥❣ ❤ú✉ ❤↕♥✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❚r÷í♥❣ F ✤÷đ❝ ❣å✐ ❧➔ ♠ët tr÷í♥❣ ♥❣✉②➯♥ tè ♥➳✉ ♥â ❦❤ỉ♥❣ ❝â tr÷í♥❣ ❝♦♥ ♥➔♦ ♥❣♦➔✐ ❜↔♥ t❤➙♥ ♥â✳ ◆❤➟♥ ①➨t ✶✳✶✳✷✳ (i) ❈❤♦ F ❧➔ tr÷í♥❣ ♥❣✉②➯♥ tè✳ ❑❤✐ ✤â ❝❤➾ ❝â t❤➸ ①↔② r❛ ♠ët tr♦♥❣ ❤❛✐ tr÷í♥❣ ❤đ♣✿ ♥➳✉ F ❝â ✤➦❝ sè ✵ t❤➻ F ∼ = Zp = Q❀ ♥➳✉ F ❝â ✤➦❝ sè p t❤➻ F ∼ ❚r÷í♥❣ ❤đ♣ F ∼ = Zp t❛ t❤÷í♥❣ ❦➼ ❤✐➺✉ Fp t❤❛② F (ii) E ởt trữớ tũ ỵ ❦❤✐ ✤â ♥➳✉ ❣å✐ F ❧➔ ❣✐❛♦ ❝õ❛ ♠å✐ tr÷í♥❣ ❝♦♥ ❝õ❛ E t❤➻ F ❝ơ♥❣ ❧➔ ♠ët tr÷í♥❣ ❝♦♥ ❝õ❛ E, rã r➔♥❣ F ❧➔ tr÷í♥❣ ❝♦♥ ♥❤ä ♥❤➜t ❝õ❛ E ✱ ❞♦ ✤â F ❧➔ tr÷í♥❣ ♥❣✉②➯♥ tè✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ t❛ ♥â✐ F ❧➔ tr÷í♥❣ ❝♦♥ ♥❣✉②➯♥ tè ❝õ❛ E ✳ ◆❤÷ ✈➟②✱ ♠å✐ tr÷í♥❣ ✤➲✉ ❝❤ù❛ ♠ët tr÷í♥❣ ❝♦♥ ♥❣✉②➯♥ tè✳ ✺ ✶✳✷ ❚r÷í♥❣ ❤ú✉ ❤↕♥ ●✐↔ sû p ❧➔ sè ♥❣✉②➯♥ tè✱ ✈➔♥❤ Z/pZ ❧➔ ♠ët tr÷í♥❣ ❝â ✤ó♥❣ p ♣❤➛♥ tû✳ ✣➙② ❧➔ tr÷í♥❣ ❤ú✉ ❤↕♥ ❞✉② ♥❤➜t ✭s❛✐ ❦❤→❝ ✤➥♥❣ ❝➜✉✮ ❝â ú p tỷ L ởt trữớ ợ p ♣❤➛♥ tû✱ ❣å✐ p′ ❧➔ ✤➦❝ sè ❝õ❛ L✳ ❑❤✐ ✤â Z/p′Z ❧➔ ✤➥♥❣ ❝➜✉ ❝õ❛ ♠ët tr÷í♥❣ ❝♦♥ ❝õ❛ L✱ ♥➯♥ p′ ❝❤✐❛ ❤➳t p✳ ✣✐➲✉ ♥➔② ❝❤➾ ✤ó♥❣ ♥➳✉ p′ = p ❞♦ ✤â L ∼ = Z/pZ ỵ Fp := Z/pZ qt ỡ ♥➳✉ q ❧➔ ❧ô② t❤ø❛ ❝õ❛ ♠ët ♥❣✉②➯♥ tè✱ t❤➻ tỗ t ởt trữớ t ợ q tỷ ỵ Fq trú trữớ ❤ú✉ ❤↕♥✮✳ (i) ❈❤♦ F ❧➔ tr÷í♥❣ ❤ú✉ ❤↕♥ ❝â q tỷ õ tỗ t số tố p s❛♦ ❝❤♦ q = pn ✈ỵ✐ sè tü ♥❤✐➯♥ n ♥➔♦ ✤â✳ (ii) ❱ỵ✐ ♠é✐ sè ♥❣✉②➯♥ tè p số tỹ n = tỗ t ♥❤➜t ♠ët tr÷í♥❣ ❤ú✉ ❤↕♥ ❝â pn ♣❤➛♥ tû ✭s❛✐ ❦❤→❝ ♠ët ✤➥♥❣ ❝➜✉ tr÷í♥❣✮✳ ❈❤ù♥❣ ♠✐♥❤✳ (i) ●å✐ p ❧➔ ✤➦❝ sè ❝õ❛ tr÷í♥❣ F ✱ ❦❤✐ ✤â p ❧➔ sè ♥❣✉②➯♥ tè✳ ●å✐ Fp ❧➔ tr÷í♥❣ ❝♦♥ ♥❣✉②➯♥ tè ❝õ❛ F ✱ ❦❤✐ ✤â Fp ∼ = Zp ✳ ❚❛ ❜✐➳t r➡♥❣ F ❧➔ Fp −❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì ❤ú✉ ❤↕♥ ❝❤✐➲✉✳ ●✐↔ sû dimF (F ) = n < ∞✱ ❦❤✐ ✤â F ❝â ♠ët ❝ì n sð ❧➔ {e1, , en} ✈➔ ✈➻ t❤➳ ♠é✐ ♣❤➛♥ tû ❝õ❛ F ❝â ❞↕♥❣ x = aiei ✈ỵ✐ i=1 a1 , , an ∈ Fp ❚ø ✤â s✉② r❛ sè ♣❤➛♥ tû ❝õ❛ F ❜➡♥❣ sè ❝→❝ ❜ë ♣❤➛♥ tû (a1 , , an ) ∈ Fp × × Fp ✭n ❧➛♥✮✳ ❉♦ ✤â q = pn (ii) ỹ tỗ t trữớ õ q = pn ♣❤➛♥ tû✳ ❳➨t ✤❛ t❤ù❝ f (x) = xq − x ∈ Fp [x] ✈ỵ✐ Fp ∼ = Zp ❧➔ tr÷í♥❣ ♥❣✉②➯♥ tè ❝â ✤➦❝ sè ♥❣✉②➯♥ tè p✳ ●å✐ E ❧➔ tr÷í♥❣ ♣❤➙♥ r➣ ❝õ❛ f (x) tr➯♥ Fp ✣➦t p K = {α ∈ E | f (α) = 0} ✤â ❝❤➼♥❤ ❧➔ t➟♣ ❤ñ♣ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ f (x)✳ ❑❤✐ ✤â K ❧➔ ♠ët tr÷í♥❣ ❝♦♥ ❝õ❛ E ✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠å✐ α, β ∈ K t❛ ❝â (α − β)q = αq − β q = α − β, (αβ)q = αq β q = αβ ❉♦ ✤â α − β, αβ ∈ K ✳ ◆➳✉ α ∈ K ∗ t❤➻ (α−1)q = (aq )−1 = α−1 s✉② r❛ α−1 ∈ K ◆❣♦➔✐ r❛✱ rã r➔♥❣ 1q = ♥➯♥ ∈ K ❈✉è✐ ❝ò♥❣✱ t❛ t❤➜② r➡♥❣ ♠å✐ a ∈ Fp ✤➲✉ t❤ä❛ ♠➣♥ ap = a ❞♦ ✤â aq = ap = a ❝❤ù♥❣ tä Fp ⊆ K n ✻ ◆❤÷ ✈➟② K ❝❤➼♥❤ ❧➔ tr÷í♥❣ ♣❤➙♥ r➣ ❝õ❛ f (x) tr➯♥ Fp✱ tr÷í♥❣ ♥➔② ❝â q = pn ♣❤➛♥ tû ✭❧÷✉ þ r➡♥❣ ✤❛ t❤ù❝ f (x) ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ ❜ë✐✮✳ ❚➼♥❤ ❞✉② ♥❤➜t ❝õ❛ tr÷í♥❣ ❝â q = pn ♣❤➛♥ tû✳ ●✐↔ sû Fq ❧➔ tr÷í♥❣ ❝â q = pn ♣❤➛♥ tû✳ ❑❤✐ ✤â Fq ❝â ✤➦❝ sè ❧➔ p ✭❣✐↔ sû p1 ❧➔ ✤➦❝ sè ❝õ❛ Fq t❤➻ t❤❡♦ (i) s✉② r❛ q = pn1 ❀ ❞♦ ✤â pn = pn1 ✈➻ t❤➳ p = p1) ❱➻ F∗q = Fq \ {0} ❧➔ ♥❤â♠ ✈ỵ✐ ♣❤➨♣ ♥❤➙♥ ♥➯♥ αq−1 = ✈ỵ✐ ♠å✐ α ∈ F∗q ❀ ❞♦ ✤â αq = α ✈ỵ✐ ♠å✐ α ∈ Fq ✳ ❈❤ù♥❣ tä ♠å✐ ♣❤➛♥ tû ❝õ❛ Fq ✤➲✉ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✤❛ t❤ù❝ f (x) = xq − x ∈ Fp [x] ợ Fp trữớ tố Fq ❙✉② r❛ tr÷í♥❣ Fq ❝❤➼♥❤ ❧➔ tr÷í♥❣ ♣❤➙♥ r➣ ❝õ❛ f (x) tr➯♥ Fp ✣✐➲✉ ✤â ❦❤➥♥❣ ✤à♥❤ t➼♥❤ ❞✉② ♥❤➜t ❝õ❛ Fq s❛✐ ❦❤→❝ ♠ët ✤➥♥❣ ❝➜✉ tr÷í♥❣✳ ❚❛ ♥❤➢❝ ❧↕✐✱ ♠ët ♠ð rë♥❣ tr÷í♥❣ E/F ✭F ⊂ E ✮ ❧➔ ♠ët ♠ð rë♥❣ ●❛❧♦✐s ♥➳✉ ♥â ❧➔ ♠ð rë♥❣ ❝❤✉➞♥ t➢❝ ✈➔ t→❝❤ ✤÷đ❝ ✭❈❤÷ì♥❣ ✷ t➔✐ ❧✐➺✉ ❬✶❪✮✳ ❚❛ ❝â ❦➳t q✉↔ s❛✉✿ ′ ′ ❇ê ✤➲ ✶✳✷✳✷✳ ❈❤♦ E/F ❧➔ ♠ët ♠ð rë♥❣ ❤ú✉ ❤↕♥ ❦❤✐ ✤â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ t÷ì♥❣ ✤÷ì♥❣✿ ✭✐✮ E/F ❧➔ ♠ð rë♥❣ ●❛❧♦✐s❀ ✭✐✐✮ ◆➳✉ p(x) ∈ F (x) ❧➔ ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② tr➯♥ F ❝â ♠ët ♥❣❤✐➺♠ tr♦♥❣ E t❤➻ ♥â t→❝❤ ✤÷đ❝ ✈➔ ❝â ♠å✐ ♥❣❤✐➺♠ tr♦♥❣ E ✭tù❝ ❧➔ p(x) t→❝❤ ✤÷đ❝ ✈➔ ♣❤➙♥ r➣ tr➯♥ E ✮❀ ✭✐✐✐✮ E ❧➔ tr÷í♥❣ ♣❤➙♥ r➣ ❝õ❛ ♠ët ✤❛ t❤ù❝ t→❝❤ ✤÷đ❝ f (x) ∈ F [x] ỵ q ụ tứ ♠ët sè ♥❣✉②➯♥ tè ✈➔ a, b ❧➔ sè ♥❣✉②➯♥ ữỡ a ữợ b t Fqa tr÷í♥❣ ❝♦♥ ❝õ❛ Fqb ✳ ❍ì♥ ♥ú❛✱ ♠ð rë♥❣ tr÷í♥❣ Fqb /Fqa ❧➔ ♠ð rë♥❣ ●❛❧♦✐s✳ ▼å✐ ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② tr➯♥ Fqa ✤➲✉ t→❝❤ ✤÷đ❝ ✈➔ ♥➳✉ ♥â ❝â ♥❣❤✐➺♠ tr♦♥❣ Fqb t❤➻ ♠å✐ ♥❣❤✐➺♠ ❝õ❛ ♥â ✤➲✉ t❤✉ë❝ Fqb ✳ ●✐↔ sû a, b ❧➔ ❝→❝ sè ữỡ s a ữợ b ❞ư♥❣ ❧➟♣ ❧✉➟♥ ♥❤÷ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ❇ê ✤➲ ✶✳✷✳✶✱ tr÷í♥❣ ♣❤➙♥ r➣ ❝õ❛ P (T ) = T q − T tr➯♥ Fq ❝â ✤ó♥❣ q b ♣❤➛♥ tỷ ợ Fq rữớ r ♥➔② ❝ô♥❣ ❝❤ù❛ Fq ✱ ✈➔ ❞♦ ✤â Fq ❧➔ ♠ët tr÷í♥❣ ❝♦♥ ❝õ❛ Fq ✳ ❍ì♥ ♥ú❛✱ ✈➻ P (T ) ❧➔ ✤❛ t❤ù❝ t→❝❤ ✤÷đ❝✱ Fq ❧➔ tr÷í♥❣ ♣❤➙♥ r➣ ❝õ❛ P (T ) tr➯♥ ❈❤ù♥❣ ♠✐♥❤✳ b a b a a b b ✷✶ 4k − − 4k + k = k − ≥ 0✱ ❜➙② ❣✐í t❛ sû ❞ư♥❣ 2k − > ✈➔ n ≥ 2k + s✉② r❛ t➜t ❝↔ ❝→❝ sè ❤↕♥❣ tr♦♥❣ tê♥❣ ❝✉è✐ ❝ò♥❣ ❧➔ ❦❤æ♥❣ ➙♠✳ ❉♦ ✈➟② qn qk qn − q − q−1 2Gq (n) − Fq (n) ≥ > = = Gq (n), n n q k≤n n q ❞♦ ✤â Fq (n) < − q+1 q−1 Gq (n) = Gq (n) q q ❱➟② t❛ ❤♦➔♥ t❤➔♥❤ ❝❤ù♥❣ ♠✐♥❤ ✭✐✮✳ ✭✐✐✮ ✣➸ ❝❤ù♥❣ ♠✐♥❤ Fq (n) ∼ Gq (n) ❦❤✐ n → ∞✱ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ lim n ữ ỵ qk qn Gq (n) = ≥ n k≤n n Fq (n) − Gq (n) = Gq (n) ✭✸✳✶✶✮ ✈➔ ❞♦ ✤â Fq (n) − Gq (n) Fq (n) − Gq (n) = ≤ qn Gq (n) n n − k k−n q k k≤n ❚❛ t→❝❤ tê♥❣ ❝✉è✐ ❝ò♥❣ ♥➔② t❤➔♥❤ ❤❛✐ ♣❤➛♥ ✈ỵ✐ k < n2 ✈➔ ✭❤↕♥❣ tû t❤ù n ❜➡♥❣ ố ợ trữớ ủ k < n2 t ❝â✿ n ≤ k < n n k< n2 n n − k k−n q − − q 1−n k−n nq ≤n q ≤ ≤ nq − k q1 k< n ố ợ trữớ ủ n2 ≤ k ≤ n t❛ ❧↕✐ ❝â✿ n − k k−n q ≤ k n ≤k 0✱ ♥➯♥ s✉② r❛ limsup{x→∞}A(x) ≥ 12 ✳ (x) > tự sỷ ỵ f (x) = Ω(g(x)) ♥➳✉ limsup{x→∞} fg(x) ð ✤➙② t❛ ❝â qx |πq (x) − f (x)| = Ω x ✳ ✣à♥❤ ỵ q ụ tứ ởt số tố õ ổ tỗ t ởt tư❝ f : R → R ✤➸ πq (x) t÷ì♥❣ ữỡ ợ f sỷ f x ∞✳ ❑❤✐ ✤â✿ ❈❤ù♥❣ ♠✐♥❤✳ :R→R ❧➔ ♠ët ❤➔♠ ❧✐➯♥ tö❝ s❛♦ ❝❤♦ πq (x) ∼ f |πq (x) − f (x)| =0 x q (x) lim ữ ỵ t ỵ ỵ t õ ✤â qk q+1 q + q [x] − q + qx qk πq (x) ≤ ≤ = ≤ k q [x] q − [x] q − x k≤x k≤x q − |πq (x) − f (x)| |πq (x) − f (x)| ≥ qx πq (x) q+1 x |πq (x) − f (x)| = 0✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✣à♥❤ ỵ s r x lim qx x |πq (x) − f (x)| ≥ ✳ ❚❛ ❦➳t ❧✉➟♥ lim sup x q x→∞ x f t÷ì♥❣ ✤÷ì♥❣ ợ q (x) x ổ tỗ t ♠ët ❤➔♠ ❧✐➯♥ tö❝ ✸✳✹ ✣✐➲✉ ❝❤➾♥❤ ❤➔♠ ✤➳♠ ✣➸ õ ữủ ởt tữỡ tỹ ợ ỵ số tè ✭♠ð rë♥❣ tr➯♥ t➟♣ sè t❤ü❝✮✳ ❚❛ s➩ sû ❞ö♥❣ ❤➔♠ ✤➳♠ ✈➔ ❝→❝ ❦➳t q✉↔ ✤➣ ❝â tr♦♥❣ ❬✼❪✳ ❚❛ ①➨t s♦♥❣ →♥❤ s❛✉ ✤➙② tø t➟♣ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣ ✤➳♥ t➟♣ ❝→❝ ✤❛ t❤ù❝ tr➯♥ ✷✾ Fq ợ ởt số ữỡ N trữợ t❛ ❜✐➸✉ ❞✐➵♥ N t❤❡♦ ❝ì sè q ✳ ❚❛ ❣å✐ n ❧➔ sè ♥❣✉②➯♥ ❧ỵ♥ ♥❤➜t s❛♦ ❝❤♦ q n N õ tỗ t t ❜ë ≤ a0 , , an ≤ q − s❛♦ ❝❤♦ N = an q n + + a1 q + a0 ❚❛ ❝❤♦ ù♥❣ N ✈ỵ✐ ✤❛ t❤ù❝ f (T ) = an T n + + a1 T + a0 ❑❤✐ ✤â t÷ì♥❣ ù♥❣ ❧➔ ♠ët s♦♥❣ →♥❤✳ ❚❛ ❦➼ ❤✐➺✉ ||f || = N ✳ ❱ỵ✐ ♠é✐ ❦❤♦↔♥❣ I ⊂ R t❛ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ ✤➳♠ π ˆq (I) = #{f ∈ Fq [T ], f ❜➜t ❦❤↔ q✉②✱ ||f || ∈ I ⑥ ✈➔ π ˆq (X) := π ˆq ([0; X) ✳ ◆➳✉ X = q n ✱ t❤➻ πˆq (q n ) = (q − 1)πq (n − 1)✱ ✈➻ ❝→❝ ✤❛ t❤ù❝ f ∈ Fq [T ] ♠➔ ||f || < q n ❝❤➼♥❤ ❧➔ ❝→❝ ✤❛ t❤ù❝ ❝â ❞❡❣(f ) < n✳ ❍ì♥ ♥ú❛✱ ♥➳✉ X ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ❧ơ② t❤ø❛ ❝õ❛ q ✱ t❛ ❦❤ỉ♥❣ t❤➸ ❝❤➾ ❞ü❛ ✈➔♦ ❝æ♥❣ t❤ù❝ ❝õ❛ ●❛✉ss✳ ❈❤♦ l ❧➔ sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠✳ ●✐↔ sû A = T n + an−1 T n−1 + + a0 ✈➔ B = T m + bm−1 T m−1 + + b0 ❧➔ ✤❛ t❤ù❝ ♠♦♥✐❝✳ ❚❛ ♥â✐ A ✈➔ B ❝â ❝ò♥❣ l ❤➺ sè s❛✉ ❤➺ sè ✤➛✉ ❜➡♥❣ ♥❤❛✉ ♥➳✉ an−i = bm−i ✈ỵ✐ i = 1, , l✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ ♠ët q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ Rl tr➯♥ t➟♣ ❝→❝ ✤❛ t❤ù❝ ♠♦♥✐❝ tr♦♥❣ Fq [T ] ♥❤÷ s❛✉✿ A ≡ B ♠♦❞ Rl ⇐⇒ A ✈➔ B ❝â ❝ò♥❣ l ❤➺ sè s❛✉ ❤➺ sè ✤➛✉ ❜➡♥❣ ♥❤❛✉✳ ❚❛ ❝➛♥ sû ❞ö♥❣ ❜ê ✤➲ s❛✉ ✤➙② ❝õ❛ P♦❧❧❛❝❦ ❬✼✱ ❇ê ✤➲ ✷❪✳ ❇ê ✤➲ ✸✳✹✳✶✳ ❈❤♦ l ❧➔ sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠✳ ❑❤✐ ✤â sè ✤❛ t❤ù❝ ♠♦♥✐❝ ❜➜t ❦❤↔ q✉② n tr ởt ợ t ữ t Rl ❧➔ n q n−1 q2 + O (l + 1) n n ✭✸✳✶✼✮ ◆➳✉ t❛ ①➨t ❦ÿ ❤ì♥ ❝ỉ♥❣ t❤ù❝ (3.17)✱ t❛ ❝â t❤➸ t❤➜② ♠➟t ✤ë ❝õ❛ ❝→❝ ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② ❜➟❝ n tr♦♥❣ ♠ët ❧ỵ♣ t❤➦♥❣ ❞÷ t❤❡♦ ♠♦❞✉❧♦ Rl ①➜♣ ①➾ ❧➔ ✱ tr ợ t ữ õ õ ú q nl t❤ù❝ ❜➟❝ n✳ ●✐↔ sû ❝→❝ ✤❛ t❤ù❝ n ❜➜t ❦❤↔ q✉② ❜➟❝ ♥ ✤÷đ❝ ❞➔♥ ✤➲✉✱ t❤➻ t❛ ❝â ÷ỵ❝ ❧÷đ♥❣ ❝❤♦ πˆq (X) ♥❤÷ s❛✉✿ ||f ||0 = ❞❡❣(f ) ||f ||0 + ❞❡❣(f ) q n ≤||f ||0 ❞❡❣(f ) ✸✵ q k [X] − q n + = (q − 1) k n k≤n−1 ▲÷✉ þ r➡♥❣ n ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣ s❛♦ ❝❤♦ qn ≤ X < qn+1✳ ❚r♦♥❣ ❝æ♥❣ t❤ù❝ tr➯♥ ❜✐➸✉ t❤ù❝ ❝✉è✐ ❧➔ ♠ët ❤➔♠ ❜➟❝ t❤❛♥❣✱ ♥❤÷♥❣ t❛ ❝â t❤➸ t❤❛② ✤ê✐ ✤➸ ♥â trð t❤➔♥❤ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ tr♦♥❣ tø♥❣ ❦❤♦↔♥❣ ❜➡♥❣ ❝→❝❤ t❤❛② [X] ❜➡♥❣ X ✳ ❚❛ ✤➦t✿ ❧sˆ q (X) := (q − 1) qk X − qn + k n k≤n−1 ✣à♥❤ ỵ q ụ tứ ởt số ♥❣✉②➯♥ tè ✈➔ X ≥ q✳ ●å✐ n ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣ t❤ä❛ ♠➣♥ q n ≤ X < q n+1 ✳ ❑❤✐ ✤â t❛ ❝â ✭✸✳✶✽✮ n ˆ q (X) + O nq π ˆq (X) = s rữợ t t t [X] t ỡ số q tù❝ ❧➔ [X] = anqn + + a1 q + a0 ú ỵ r ự ♠✐♥❤✳ π ˆq (X) = π ˆq ([0, q n )) + π ˆq ([q n , an q n )) + π ˆq ([an q n , X)) ❍↕♥❣ tû ✤➛✉ πˆq ([0, qn)) ❧➔ sè ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② ❝â ❜➟❝ ≤ n − 1✱ t❤❡♦ ✣à♥❤ ỵ ợ qk q ([0, q )) = (q − 1) +O k k≤n−1 n n−1 q n−1 an −1 ✣è✐ ✈ỵ✐ ❤↕♥❣ tû t❤ù ❤❛✐ t❛ ✈✐➳t πˆq ([q , anq )) = πˆq ([kqn, (k + 1)qn))✳ k=1 n n ❑❤✐ ✤â ♠é✐ sè ❤↕♥❣ πˆq ([kq , (k + 1)q )) ❝❤➼♥❤ ❧➔ sè ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② ❜➟❝ n ✈ỵ✐ ❤➺ sè ✤➛✉ k✱ ❝ơ♥❣ ❝❤➼♥❤ ❧➔ sè ❝→❝ ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② ♠♦♥✐❝ n✳ ❚❤❡♦ ✣à♥❤ ỵ t õ n n n qn q2 ˆq ([q , an q )) = (an − 1) + O n n−1 n n ✣è✐ ✈ỵ✐ ❤↕♥❣ tû ❝✉è✐ t❛ ❝â πˆq ([anq , X)) = πˆq anq ❦❤♦↔♥❣ tr➯♥ r❛ t❤➔♥❤ ♥ ❦❤♦↔♥❣ ✈➔ ❝â t❤➸ ✈✐➳t ❧↕✐✿ n n π ˆq n q i an q n , i=0 = j=1  π ˆ q  n n n i=0 n q i , i=j q i , i=j−1 ✳ ❚❛ t→❝❤  q i  , ✸✶ tr♦♥❣ ✤â ♠é✐ sè ❤↕♥❣ πˆq ❝→❝ ❦❤♦↔♥❣ ♥❤ä ✈➔ t❛ ❝â✿ n aj−1 −1 j=1 k=0  π ˆq  n n q i , i=j ❧↕✐ ✤÷đ❝ t✐➳♣ tư❝ t→❝❤ r❛ q i i=j−1  n n q i + kq j−1 , i=j i=j n q i + (k + 1)q j−1  n ✣➳♥ ✤➙②✱ ♠é✐ sè ❤↕♥❣ πˆq q + kq , q i + (k + 1)q j−1 ❜➡♥❣ i=j i=j sè ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② ❜➟❝ n ❝â ❤➺ sè ✤➛✉ ✈➔ n − j + ❤➺ sè t✐➳♣ t❤❡♦ ❧➔ ❝è ✤à♥❤✳ ◆❤÷ ✈➟② ♥â ✤ó♥❣ ❜➡♥❣ sè ❝→❝ ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② ♠♦♥✐❝ tr♦♥❣ ởt ợ t ữ Rnj+1 õ t ✤➲ ✸✳✹✳✶ t❛ ❝â n aj−1 −1 n π ˆq ([an q , X)) = j=1 n = j=1 k=0 i j−1 n q2 q j−1 + O (n − j + 2) n n n aj−1 q j−1 q2 + O (n − j + 2) n n [X] − an q n = + n n n q2 O (n − j + 2) n j=1 X − an q n = +O n n n n q2 O (n − j + 2) + n j=1 ❑➳t ❤ñ♣ ❝→❝ ❦➳t q✉↔ ♥➔② ✈➔ ♥❤â♠ O ✈➔♦ O n nq t❛ ♥❤➟♥ ✤÷đ❝ n q k (an − 1)q n X − an q n π ˆq (X) = (q − 1) + + + O nq k n n k≤n−1 = (q − 1) n qk X − qn + + O nq , k n k≤n−1 ❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ỏ ú ỵ t r qn ≤ X ✱ ♥➯♥ nq ≤ X ❧♦❣q (X)✳ ❉♦ ✤â✱ t❛ ❝â t❤➸ ✈✐➳t ❧↕✐ ❝æ♥❣ t❤ù❝ ✭✸✳✶✽✮ t❤➔♥❤✿ √ ˆ q (X) + O( X ❧♦❣q (X)) π ˆq (X) = ❧s ✭✸✳✶✾✮ ❈æ♥❣ t❤ù❝ ♥➔② t❤➸ ❤✐➺♥ sỹ tữỡ tỹ ợ ổ tự số tố √ π(x) = ❧s(x) + O( x❧♦❣(x)) n ✸✷ ❙❛✉ ✤➙② t❛ s➩ ❝❤➾ r❛ πˆq (X) ✈➔ ❧sˆ q (X) tữỡ ữỡ ỵ q ❧➔ ❧ô② t❤ø❛ ❝õ❛ ♠ët sè ♥❣✉②➯♥ tè✳ ❑❤✐ ✤â t❛ ❝â✿ ✭✸✳✷✵✮ ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠é✐ X ✱ t❛ ✤➦t n = [❧♦❣q (X)]✱ ✈ỵ✐ q n ≤ X < q n+1 rữợ t t ự q (X) ❦❤✐ X → ∞ π ˆq (X) ∼ s n t t õ ữợ ữủ nq =0 lim ˆ q (X) X→∞ ❧s ❧sˆ q (X) ≥ (q − 1) ✈➻ ✈➟② t❛ ❝â✿ qk q n−1 ≥ , k n − k≤n−1 n n(n − 1).q nq ≤ lim ≤ lim = 0, n ˆ q (X) X→∞ X→∞ ❧s q2 s✉② r❛ n nq = lim ˆ q (X) X s ứ õ t ủ ợ ỵ t❛ ✤÷đ❝✿ ˆ q (X) + O(nq ) π ˆq (X) ❧s lim = lim = ˆ q (X) X→∞ X→∞ ❧s ❧sˆ q (X) ❱➟② t❛ s✉② r❛ πˆq (X) ∼ ❧sˆ q (X)✳ ✣✐➲✉ ♥➔② ❝❤♦ ♣❤➨♣ t❛ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ s❛✉ ✤➙② t÷ì♥❣ tü ♥❤÷ ỵ số tố ợ số tỹ n ỵ q ụ tứ ❝õ❛ sè ♥❣✉②➯♥ tè✳ ❑❤✐ ✤â t❛ ❝â✿ π ˆq (X) ∼ X , ❦❤✐ X → ∞ ❧♦❣q (X) ỵ t õ q (X) ❧sˆ q (X)✱ ♥➯♥ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ❧sˆ q (X) ∼ ❧♦❣X(X) ✳ ❱ỵ✐ ♠é✐ X ✱ t❛ ✤➦t n = [❧♦❣q (X)] ✈➔ q y = {❧♦❣q (X)} = ❧♦❣q (X) − n✳ ❑❤✐ ✤â X = q n+y ✱ ✈➔ t❛ ❝â✿ ❈❤ù♥❣ ♠✐♥❤✳ lim X→∞ ❧sˆ q (X) X ❧♦❣q (X) = lim X→∞ qk (q − 1) k k≤n−1 q n+y n+y q n+y − q n n + n+y q n+y ✸✸ q n−1 − qn qk ∼ q ∼ ỵ n t ❝â✿ (q −1) k n − n k≤n−1 tø ✤â t❛ s✉② r❛✿ lim X→∞ qk (q − 1) k k≤n−1 q n+y n+y qk (q − 1) k n+y k≤n−1 = lim = lim q −y qn X→∞ nq y X→∞ n ◆â✐ ❝→❝❤ ❦❤→❝✱ t❛ ❝â q n+y − q n (n + y)(1 − q −y ) n = lim − q −y = lim lim n+y X→∞ X→∞ X→∞ q n n+y ❑➳t ❤đ♣ ♥❤ú♥❣ ✤✐➲✉ tr➯♥ t❛ ♥❤➟♥ ✤÷đ❝ ❧sˆ q (X) = lim q−y + − q−y = lim X→∞ X ❧♦❣q (X) X→∞ ❱➟② t❛ ❝â ❧sˆ q (X) ∼ ❧♦❣X(X) ✳ ❙✉② r❛✱ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ q ✸✹ ❑➳t ❧✉➟♥ ❝õ❛ ❧✉➟♥ ✈➠♥ ❚r♦♥❣ ❧✉➟♥ ✈➠♥✱ tæ✐ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ♥❤ú♥❣ ❦➳t q✉↔ t❤ó ✈à ✈➲ ❜➔✐ t♦→♥ ✤➳♠ sè ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥✳ ❈→❝ ❦➳t q✉↔ ữủ tr ró r ợ ự tt ỗ ỳ t t t÷ì♥❣ tü Z ✈➔ Fq [T ]✳ ✭✷✮ ❍➔♠ ✤➳♠ sè ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② ❜➟❝ n tr➯♥ tr÷í♥❣ Fq ✳ ✭✸✮ ❍➔♠ ✤➳♠ sè ✤❛ t❤ù❝ ❜➜t ❦❤↔ q tr trữớ Fq ợ n sỹ tữỡ tỹ ợ ỵ số tố t q t q ỵ ✤➳♠ sè ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② ✈ỵ✐ ❜➟❝ ≤ x ợ x số tỹ sỹ tữỡ tỹ ợ ỵ số tố ợ số tỹ t q t q ỵ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ◆❣æ ❚❤à ◆❣♦❛♥✱ tỷ ổ ỵ tt t ữợ t ỹ õ tự ữớ rữớ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ❚✐➳♥❣ ❆♥❤ ❬✷❪ ❆❧❡① ❇r❛❛t✱ ✭✷✵✶✽✮✱“❈♦✉♥t✐♥❣ ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s ♦✈❡r ❢✐♥✐t❡ ❢✐❡❧❞s”✱ ❯♥✐✈❡rs✐t❡✐t ❯tr❡t❝❤✳ ❬✸❪ ❉✳ ❙✳ ❉✉♠♠✐t ❛♥❞ ❘✳▼✳❋♦♦t❡✱ ✭✷✵✵✹✮✱ ❆❜str❛❝t ❛❧❣❡❜r❛ ✭❱♦❧ ✸✮✳ ❍♦❜♦✲ ❦❡♥✿ ❲✐❧❡②✳ ❬✹❪ ●✳ ❏✳ ❖✳ ❏❛♠❡s♦♥✱ ✭✷✵✵✸✮✱ ❚❤❡ ♣r✐♠❡ ♥✉♠❜❡r t❤❡♦r❡♠✱ ✭❱♦❧ ✺✸✮✱ ❈❛♠✲ ❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❬✺❪ ●✳ ❚❡♥❡♥❜❛✉♠ ❛♥❞ ▼✳ ▼ ❋r❛♥❝❡✱ ✭✷✵✵✵✮✱ ❚❤❡ ♣r✐♠❡ ♥✉♠❜❡rs ❛♥❞ t❤❡✐r ❞✐str✐❜✉✲t✐♦♥ ✭❱♦❧✳✻✮✳ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✳ ❬✻❪ ◆✳ ❙♥②❞❡r✱✭✷✵✵✵✮✱” ❆♥ ❛❧t❡r♥❛t❡ ♣r♦♦❢ ♦❢ ▼❛s♦♥✬s t❤❡♦r❡♠”✳ ■♥✿ ❊❧❡✲ ♠❡♥t❡ ❞❡r ▼❛t❤❡♠❛t✐❦ ✺ ✺✳✸ ✱ ♣♣✳ ✾✸✲✾✹✳ ❬✼❪ P✳P♦❧❧❛❝❦✱✭✷✵✶✵✮✱”❘❡✈✐s✐♥❣ ●❛✉ss✬s ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ♣r✐♠❡ ♥✉♠❜❡r t❤❡✲ ♦r❡♠ ❢♦r ♣♦❧②♥♦♠✐❛❧s ♦✈❡r ❛ ❢✐♥✐t❡ ❢✐❡❧❞ ”✳ ■♥✿ ❋✐♥✐t❡ ❋✐❡❧❞s ❛♥❞ ❚❤❡✐r ❆♣♣❧✐❝❛t✐♦♥s ✶✻✳✹ ✱ ♣♣✳ ✷✾✵✲✷✾✾✳ ❬✽❪ P✳ P♦❧❧❛❝❦✱ ✭✷✵✶✸✮✱ ❆♥❛❧♦❣✐❡s ❜❡t✇❡❡♥ ✐♥t❡❣❡rs ❛♥❞ ♣♦❧②♥♦♠✐❛❧s✱ ✉r❧✿ ❤tt♣✿✴✴♣♦❧❧❛❝❦✳✉❣❛✳❡❞✉✴❈■▼P❆✷✵✶✸✴P❛rt✶✳♣❞❢✳ ❬✾❪ ❨✳ ❚s❝❤✐♥❦❡❧✱✭✷✵✵✻✮✱ ✧❆❜♦✉t t❤❡ ❝♦✈❡r✿ ♦♥ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ♣r✐♠❡s✲ ●❛✉ss✬ t❛❜❧❡s✧✳ ■♥✿ ❇✉❧❧❡t✐♥ ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t② ✹✸✳✶ ... ✤â t❛ ❝â✿ q qn − πq (n) ∼ ✭✸✳✶✸✮ q−1 n ❚❛ s s t q ợ ỵ số tố ỵ số tố ✤÷đ❝ ♣❤→t ❜✐➸✉ r➡♥❣ sè ❝→❝ sè ♥❣✉②➯♥ tè ❦❤ỉ♥❣ ữủt q ởt số tỹ x tữỡ ữỡ ợ x(x) ✳ ◆➳✉ x ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣✱ t❛ ❝â ✷✹... ✤➳♠ sè ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② tr➯♥ tr÷í♥❣ Fq ợ n sỹ tữỡ tỹ ợ ỵ số tố t q t q ỵ số tự ❜➜t ❦❤↔ q✉② ✈ỵ✐ ❜➟❝ ≤ x ✈ỵ✐ x ❧➔ số tỹ sỹ tữỡ tỹ ợ ỵ sè ♥❣✉②➯♥ tè ✈ỵ✐ ❜✐➳♥ sè t❤ü❝ ✭❦➳t q✉↔ t❤➸... x❧♦❣(x) ✸ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ t❛ s➩ t➻♠ ❤✐➸✉ ✈➲ ♠ët sü t÷ì♥❣ tü ỵ số tố ữ ữủ t tr ✈➔♥❤ Fq [T ] ❧➔ ✈➔♥❤ ❝→❝ ✤❛ t❤ù❝ ♠ët ợ số tở trữớ ỳ Fq ✳ ❚❛ s➩ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❤➔♠ t÷ì♥❣ ✤÷ì♥❣ ✈ỵ✐ ❤➔♠ ✤➳♠

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