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Mët trong nhúng v§n · ÷ñc nghi¶n cùu trong lþ thuy¸t sè â l sü ph¥n bè c¡c sè nguy¶n tè. Ng÷íi ta nhªn th§y r¬ng c¡c sè nguy¶n tè nhä n¬m t÷ìng èi g¦n nhau, trong khi c¡c sè nguy¶n tè c ng lîn th¼ c ng câ xu h÷îng c¡ch xa nhau hìn. Ta °t c¥u häi v· sü li¶n quan giúa mªt ë cõa c¡c sè nguy¶n tè vîi ë lîn cõa chóng. B¬ng c¡ch lªp b£ng sè nguy¶n tè v nghi¶n cùu mªt ë, Gauss th§y r¬ng “xung quanh x mªt ë cõa c¡c sè nguy¶n tè l x§p x¿ 1 log(x) ” theo 9. Ph¡t hi»n n y l ch¼a khâa º h¼nh th nh ành lþ sè nguy¶n tè. º chùng minh ph¡t hi»n n y, Gauss ¢ nghi¶n cùu h m ¸m sè nguy¶n tè: Gåi x l sè thüc d÷ìng, π(x) biºu thà sè c¡c sè nguy¶n tè nhä hìn ho°c b¬ng x. Tùc l ta câ π(x) = P p≤x 1. V¼ ng÷íi ta ¢ dü o¡n v· mªt ë c¡c sè nguy¶n tè quanh x l 1 log(x) , n¶n hå công dü o¡n r¬ng π(x) x§p x¿ vîi mët têng logarit ho°c mët t½ch ph¥n logarit. Chóng t÷ìng ùng ÷ñc cho bði: ls(x) := X 2≤n≤x 1 log(n) , li(x) := Z x 2 dt log(t) . Ta nâi hai h m f v g l hai h m t÷ìng ÷ìng n¸u th÷ìng sè cõa chóng f(x) g(x) ti¸n tîi 1 khi x ti¸n tîi væ còng. Ta sû döng kþ hi»u f(x) ∼ g(x) khi x → ∞. Vîi méi x ≥ 2, hi»u sè giúa ls(x) v li(x) bà ch°n bði 1 log(2) theo H» qu£ 1.5.1 trong 4. Do â, hai h m têng logarit v t½ch ph¥n logarit l t÷ìng ÷ìng. Hai h m n y công t÷ìng ÷ìng vîi x log(x) (H» qu£ 1.5.3 trong 4). ành lþ sè nguy¶n tè ÷ñc c£ Gauss (1792) v Legendre (1798) n¶u ra
✣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 ✶✵ ✷✳✶ ▼ët sè t➼♥❤ ❝❤➜t ❝❤✉♥❣ ❝õ❛ Fq [T ] ✈➔ Z ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ t t tữỡ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ❈❤÷ì♥❣ ✸ ✣➳♠ sè ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② ✶✹ ✸✳✶ ❙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❛ ❝→❝ ” t❤❡♦ ❬✾❪✳ P❤→t ❤✐➺♥ ♥➔② ❧➔ ❝❤➻❛ ❦❤â❛ ✤➸ ❧♦❣(x) ❤➻♥❤ t❤➔♥❤ ✤à♥❤ ỵ số tố ự t ●❛✉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 ❝→❝ sè ♥❣✉②➯♥ tè ❧➔ ①➜♣ ①➾ p≤x sè ♥❣✉②➯♥ tè q✉❛♥❤ x ❧➔ ✱ ♥➯♥ ❤å ❝ô♥❣ ❞ü ✤♦→♥ r➡♥❣ π(x) ①➜♣ ①➾ ❧♦❣(x) ✈ỵ✐ ♠ët tê♥❣ ❧♦❣❛r✐t ❤♦➦❝ ♠ët t➼❝❤ ♣❤➙♥ ❧♦❣❛r✐t✳ ❈❤ó♥❣ t÷ì♥❣ ù♥❣ ✤÷đ❝ ❝❤♦ ❜ð✐✿ x ❧s(x) := , ❧♦❣ (n) 2≤n≤x ❧✐(x) := dt ❧♦❣(t) ❚❛ ♥â✐ ❤❛✐ ❤➔♠ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ t÷ì♥❣ ữỡ tữỡ số ú f (x) t tợ ✶ ❦❤✐ x t✐➳♥ tỵ✐ ✈ỉ ❝ị♥❣✳ ❚❛ sû ❞ư♥❣ ỵ f (x) g(x) g(x) x → ∞✳ ❱ỵ✐ ♠é✐ x ≥ 2✱ ❤✐➺✉ sè ❣✐ú❛ ❧s(x) ✈➔ ❧✐(x) ❜à ❝❤➦♥ ❜ð✐ t❤❡♦ ❧♦❣(2) ❍➺ q✉↔ ✶✳✺✳✶ tr♦♥❣ ❬✹❪✳ ❉♦ ✤â✱ ❤❛✐ ❤➔♠ tê♥❣ ❧♦❣❛r✐t ✈➔ t➼❝❤ ♣❤➙♥ ❧♦❣❛r✐t x ❧➔ t÷ì♥❣ ✤÷ì♥❣✳ ❍❛✐ ❤➔♠ ♥➔② ụ tữỡ ữỡ ợ q (x) tr ỵ số tố ữủ ss ▲❡❣❡♥❞r❡ ✭✶✼✾✽✮ ♥➯✉ r❛ ✷ ❣✐↔ t❤✉②➳t r➡♥❣ ❤➔♠ ✤➳♠ số tố (x) tữỡ ữỡ ợ õ ữủ tr ữợ (x) x (x) (x → ∞) ✭✶✮ ▼ët tr➠♠ ♥➠♠ s❛✉ ✈➔♦ ♥➠♠ ỵ ữủ ự r ✈➔ ▲❛ ❱❛❧❧➨❡ ▼✉❢❢s✐♥ ♠ët ❝→❝❤ ✤ë❝ ❧➟♣✳ ❈↔ ❤❛✐ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ❤å ✤➲✉ ❞ü❛ tr➯♥ ❤➔♠ ③❡t❛ ❘✐❡♠❛♥♥✱ ♠ët ♠ð rë♥❣ ❣✐↔✐ t➼❝❤ ❝õ❛ tê♥❣ ∞ ✳ ❘✐❡♠❛♥♥ ✤➣ ❝❤➾ 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✳ x ❈→❝ ❣✐→ trà ①➜♣ ①➾ ❝õ❛ ❧s(x) ✈➔ ❧✐(x) tèt ❤ì♥ ❞♦ ✤â ❝❤ó♥❣ t❤÷í♥❣ ❧♦❣(x) ✤÷đ❝ ÷✉ t✐➯♥ ❤ì♥ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♣❤➛♥ s❛✐ sè✳ ✣è✐ ợ s số t sỷ ỵ O✿ ❱ỵ✐ ❤❛✐ ❤➔♠ f ✈➔ g ❜➜t ❦ý✱ t❛ õ 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 ✭✷✮ ζ(s) = 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 ❝↔ ❝→❝ ♥❣❤✐➺♠ ❦❤æ♥❣ t➛♠ t❤÷í♥❣ ❝õ❛ ζ ♥➡♠ tr➯♥ ✳ ●✐↔ t❤✐➳t ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ❣✐↔ t❤✉②➳t ❘✐❡♠❛♥♥✳ ●✐↔ t❤✉②➳t ❘✐❡♠❛♥♥ s✉② r❛ Θ = ✱ ✤✐➲✉ ♥➔② ❝❤♦ t❛ ①➜♣ ①➾ √ π(x) = ❧✐(x) + O x❧♦❣(x) ✤÷í♥❣ t❤➥♥❣ ❘❡(s) = ✸ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ t❛ s➩ t➻♠ ❤✐➸✉ ✈➲ ♠ët sü t÷ì♥❣ tü ✤à♥❤ ỵ số tố ữ ữủ t tr Fq [T ] ❧➔ ✈➔♥❤ ❝→❝ ✤❛ t❤ù❝ ♠ët ❜✐➳♥ ợ số tở trữớ ỳ Fq s ự tữỡ ữỡ ợ ❤➔♠ ✤➳♠ sè ❝→❝ ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② ✈➔ ❝â sü s♦ s→♥❤ ❝→❝ ❦➳t q✉↔ ♥➔② ✈ỵ✐ ❤➔♠ ✤➳♠ sè ♥❣✉②➯♥ tè π(x)✳ ▼ët tr♦♥❣ ♥❤ú♥❣ ❧ñ✐ t❤➳ ❦❤✐ ❧➔♠ ✈✐➺❝ ✈ỵ✐ Fq [T ] ❧➔ ❝ỉ♥❣ t❤ù❝ ❝õ❛ ●❛✉ss✱ ♠ët ❝ỉ♥❣ t❤ù❝ trü❝ t✐➳♣ ✈➲ sè ❧÷đ♥❣ ✤❛ t❤ù❝ ♠♦♥✐❝ ❜➜t ❦❤↔ q✉② ❜➟❝ n✳ ✣➙② ❧➔ ♠ët ❝ỉ♥❣ ❝ư r➜t ♠↕♥❤ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❤➔♠ t÷ì♥❣ ✤÷ì♥❣ ❤➔♠ ✤➳♠✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ t❤➔♥❤ ❜❛ ữỡ ữỡ ỗ ởt số tự trữớ ỳ ỵ ▼♦❜✐✉s✳ ◆❤ú♥❣ ❦✐➳♥ t❤ù❝ ♥➔② ♣❤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➯♥ trữớ ỳ q tỷ ỗ tớ ụ t t sỹ tữỡ tỹ ợ ỵ ✤➳♠ sè ♥❣✉②➯♥ tè 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♥❣ ♠✐♥❤ ❝ỉ♥❣ 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 ❦❤→❝ ❦❤æ♥❣ ✤➲✉ ❦❤↔ ♥❣❤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 ∼ = Q❀ ♥➳✉ F ❝â ✤➦❝ sè p t❤➻ F ∼ = Zp ❚r÷í♥❣ ❤đ♣ F ∼ = Zp t❛ t❤÷í♥❣ ❦➼ ❤✐➺✉ Fp t❤❛② ❝❤♦ F (ii) ❈❤♦ E ởt trữớ tũ ỵ õ 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 ✤ó♥❣ p ♣❤➛♥ tû✳ ◆➳✉ L ởt trữớ ợ p tỷ p ❧➔ ✤➦❝ sè ❝õ❛ L✳ ❑❤✐ ✤â Z/p Z ❧➔ ✤➥♥❣ ❝➜✉ ❝õ❛ ♠ët tr÷í♥❣ ❝♦♥ ❝õ❛ L✱ ♥➯♥ p ❝❤✐❛ ❤➳t p✳ ✣✐➲✉ ♥➔② ❝❤➾ ✤ó♥❣ ♥➳✉ p = p õ L = Z/pZ ỵ Fp := Z/pZ✳ ❚ê♥❣ q✉→t ❤ì♥✱ ♥➳✉ q ❧➔ ❧ơ② tứ ởt tố t tỗ t ởt trữớ t ợ q tỷ ỵ Fq ❇ê ✤➲ ✶✳✷✳✶ ✭❈➜✉ tró❝ tr÷í♥❣ ❤ú✉ ❤↕♥✮✳ ❈❤♦ F ❧➔ tr÷í♥❣ ❤ú✉ ❤↕♥ ❝â q ♣❤➛♥ tû✳ ❑❤✐ ✤â tỗ t số tố p s q = pn ✈ỵ✐ sè tü ♥❤✐➯♥ n ♥➔♦ ✤â✳ (ii) ❱ỵ✐ ♠é✐ sè ♥❣✉②➯♥ tè p ✈➔ sè tü ♥❤✐➯♥ n = tỗ t t ởt trữớ ỳ ❝â pn ♣❤➛♥ tû ✭s❛✐ ❦❤→❝ ♠ët ✤➥♥❣ ❝➜✉ tr÷í♥❣✮✳ ❈❤ù♥❣ ♠✐♥❤✳ (i) (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 dimFp (F ) = n < ∞✱ ❦❤✐ ✤â F ❝â ♠ët ❝ì sð ❧➔ {e1 , , en } ✈➔ ✈➻ t❤➳ ♠é✐ ♣❤➛♥ tû ❝õ❛ F ❝â ❞↕♥❣ x = n ei ✈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 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➡♥❣ n ♠å✐ a ∈ Fp ✤➲✉ t❤ä❛ ♠➣♥ ap = a ❞♦ ✤â aq = ap = a ❝❤ù♥❣ tä Fp ⊆ K ✻ ◆❤÷ ✈➟② K ❝❤➼♥❤ ❧➔ tr÷í♥❣ ♣❤➙♥ r➣ ❝õ❛ f (x) tr➯♥ Fp ✱ tr÷í♥❣ ♥➔② ❝â q = pn ♣❤➛♥ tỷ ữ ỵ r tự f (x) ổ õ ♥❣❤✐➺♠ ❜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 ❝õ❛ 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 ❝õ❛ ♠ët sè ♥❣✉②➯♥ tè ✈➔ a, b ❧➔ sè ữỡ a ữợ b t Fq ❧➔ tr÷í♥❣ ❝♦♥ ❝õ❛ Fq ✳ ❍ì♥ ♥ú❛✱ ♠ð rë♥❣ tr÷í♥❣ Fq /Fq ❧➔ ♠ð rë♥❣ ●❛❧♦✐s✳ ▼å✐ ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② tr➯♥ Fq ✤➲✉ t→❝❤ ✤÷đ❝ ✈➔ ♥➳✉ ♥â ❝â ♥❣❤✐➺♠ tr♦♥❣ Fq t❤➻ ♠å✐ ♥❣❤✐➺♠ ❝õ❛ ♥â ✤➲✉ t❤✉ë❝ Fq ✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû a, b số ữỡ s a ữợ ❝õ❛ b✳ ⑩♣ a b b a a b b ❞ư♥❣ ❧➟♣ ❧✉➟♥ ♥❤÷ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ❇ê ✤➲ ✶✳✷✳✶✱ tr÷í♥❣ ♣❤➙♥ r➣ ❝õ❛ b P (T ) = T q − T tr➯♥ Fqa ❝â ✤ó♥❣ q b tỷ ợ Fqb rữớ r➣ ♥➔② ❝ô♥❣ ❝❤ù❛ Fqa ✱ ✈➔ ❞♦ ✤â Fqa ❧➔ ♠ët tr÷í♥❣ ❝♦♥ ❝õ❛ Fqb ✳ ❍ì♥ ♥ú❛✱ ✈➻ P (T ) ❧➔ ✤❛ t❤ù❝ t→❝❤ ✤÷đ❝✱ Fqb ❧➔ tr÷í♥❣ ♣❤➙♥ r➣ ❝õ❛ P (T ) tr➯♥ ✷✶ 4k − − 4k + k = k − ≥ 0✱ ❜➙② ❣✐í t❛ sû ❞ư♥❣ 2k − > ✈➔ n ≥ 2k + s✉② r❛ t➜t ❝↔ ❝→❝ sè ❤↕♥❣ tr♦♥❣ tê♥❣ ❝✉è✐ ❝ị♥❣ ❧➔ ❦❤ỉ♥❣ ➙♠✳ ❉♦ ✈➟② qn qn − q − qk 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→∞ Fq (n) − Gq (n) = Gq (n) ✭✸✳✶✶✮ qn qk ≥ ✈➔ ❞♦ õ ữ ỵ Gq (n) = n kn n Fq (n) − Gq (n) Fq (n) − Gq (n) ≤ = qn Gq (n) n n − k k−n q k k≤n ✭✯✮ n n ✈➔ ≤ k < n 2 n ✭❤↕♥❣ tû t❤ù n ❜➡♥❣ ✵✮✳ ố ợ trữớ ủ k < t õ ❚❛ t→❝❤ tê♥❣ ❝✉è✐ ❝ị♥❣ ♥➔② t❤➔♥❤ ❤❛✐ ♣❤➛♥ ✈ỵ✐ k < n k< n2 n n − k k−n q − − q 1−n k−n nq ≤n q ≤ ≤ nq − k q−1 k< n n ≤ k ≤ n t❛ ❧↕✐ ❝â✿ n − k k−n n − k k−n q ≤ q = n k n n n ≤k 0✱ ♥➯♥ s✉② r❛ limsup{x→∞} A(x) ≥ ✳ f (x) ❚❛ sỷ ỵ f (x) = (g(x)) limsup{x} > 0✱ tù❝ ❧➔ g(x) qx ð ✤➙② t❛ ❝â |πq (x) − f (x)| = Ω ✳ x ❝❤ó♥❣ t õ limsup{x} A(x) ỵ q ❧➔ ❧ô② t❤ø❛ ❝õ❛ ♠ët sè ♥❣✉②➯♥ tè✳ ❑❤✐ ✤â ổ tỗ t ởt tử f : R R q (x) tữỡ ữỡ ợ f ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû f : R → R ❧➔ ♠ët ❤➔♠ ❧✐➯♥ tö❝ s❛♦ ❝❤♦ πq (x) ∼ f ❦❤✐ x → ∞✳ ❑❤✐ ✤â✿ |πq (x) − f (x)| =0 x q (x) lim ữ ỵ t ỵ ỵ t õ qk q+1 qk q + q [x] − q + qx πq (x) ≤ ≤ = ≤ k q [x] q − [x] q − x k≤x k≤x ❉♦ ✤â |πq (x) − f (x)| q − |πq (x) − f (x)| ≥ qx πq (x) q+1 x |πq (x) − f (x)| = t ợ ỵ ✸✳✸✳✶ ❧➔ qx x→∞ x |πq (x) − f (x)| t ổ tỗ t ởt ❤➔♠ ❧✐➯♥ tö❝ lim sup x q x→∞ x f tữỡ ữỡ ợ q (x) x s✉② r❛ lim ✸✳✹ ✣✐➲✉ ❝❤➾♥❤ ❤➔♠ ✤➳♠ ✣➸ ❝â ữủ ởt tữỡ tỹ ợ ỵ số tố ✭♠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 ≤ 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❤➻ π ˆq (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♥❣ ❝❤♦ π ˆq (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 ữ ỵ r n số ữỡ s q n ≤ X < q n+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✿ ˆ q (X) := (q − 1) ❧s qk X − qn + k n kn1 ỵ q ❧ô② t❤ø❛ ❝õ❛ ♠ët sè ♥❣✉②➯♥ tè ✈➔ X ≥ q✳ ●å✐ n ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣ t❤ä❛ ♠➣♥ qn ≤ X < qn+1✳ ❑❤✐ ✤â t❛ ❝â ˆ q (X) + O nq π ˆq (X) = ❧s ự rữợ t t t [X] t ỡ sè q✱ tù❝ ❧➔ [X] = anqn + + n a1 q + a0 ✳ ❚❛ ú ỵ r q (X) = q ([0, q n )) + π ˆq ([q n , an q n )) + π ˆq ([an q n , X)) ❍↕♥❣ tû ✤➛✉ π ˆq ([0, q n )) ❧➔ sè ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② ❝â ❜➟❝ ≤ n t ỵ ✈ỵ✐ n−1 qk +O π ˆq ([0, q )) = (q − 1) k k≤n−1 q n−1 n ✣è✐ ✈ỵ✐ ❤↕♥❣ tû t❤ù ❤❛✐ t❛ ✈✐➳t π ˆq ([q , an q )) = n n an −1 π ˆq ([kq n , (k + 1)q n ))✳ k=1 ❑❤✐ ✤â ♠é✐ sè ❤↕♥❣ π ˆq ([kq , (k + 1)q )) ❝❤➼♥❤ ❧➔ sè ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② n n ❜➟❝ n ✈ỵ✐ ❤➺ sè ✤➛✉ k ✱ ❝ô♥❣ ❝❤➼♥❤ ❧➔ sè ❝→❝ ✤❛ t❤ù❝ ❜➜t q n ỵ t õ n qn q2 π ˆq ([q , an q )) = (an − 1) + O n n−1 n n ✣è✐ ✈ỵ✐ ❤↕♥❣ tû ❝✉è✐ t❛ ❝â π ˆq ([an q , X)) = π ˆq n π ˆq an q n , n q i i=0 = n j=1 i=j ✳ ❚❛ t→❝❤ i=0 n q i , π ˆq q i an q , ❦❤♦↔♥❣ tr➯♥ r❛ t❤➔♥❤ ♥ ❦❤♦↔♥❣ ✈➔ ❝â t❤➸ ✈✐➳t ❧↕✐✿ n n n q i , i=j−1 ✸✶ n tr♦♥❣ ✤â ♠é✐ sè ❤↕♥❣ π ˆq n q i , i=j−1 i=j ❝→❝ ❦❤♦↔♥❣ ♥❤ä ✈➔ t❛ ❝â✿ n aj−1 −1 n j=1 q i + (k + 1)q j−1 i=j k=0 n q i + kq j−1 , π ˆq ✣➳♥ ✤➙②✱ ♠é✐ sè ❤↕♥❣ π ˆq ❧↕✐ ✤÷đ❝ t✐➳♣ tư❝ t→❝❤ r❛ q i i=j n i q + kq j−1 n ❜➡♥❣ 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❤❡♦ ❇ê ✤➲ ✸✳✹✳✶ t❛ ❝â n aj−1 −1 n π ˆq ([an q , X)) = j=1 n = j=1 k=0 n q j−1 q2 + 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 n ❑➳t ❤ñ♣ ❝→❝ ❦➳t q✉↔ ♥➔② ✈➔ ♥❤â♠ O ✈➔♦ O 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♥❣ n ỏ ú ỵ t r q n ≤ 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❝ t sỹ tữỡ tỹ ợ ổ tự sè ♥❣✉②➯♥ tè✿ √ π(x) = ❧s(x) + O( x❧♦❣(x)) ✸✷ ˆ q (X) ❧➔ t÷ì♥❣ ✤÷ì♥❣✳ ❙❛✉ ✤➙② t❛ s➩ ❝❤➾ r❛ π ˆq (X) ✈➔ ❧s ✣à♥❤ ỵ q ụ tứ ởt số ♥❣✉②➯♥ tè✳ ❑❤✐ ✤â t❛ ❝â✿ ˆ q (X) π ˆq (X) ∼ ❧s ❦❤✐ X → ∞ ✭✸✳✷✵✮ ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠é✐ X ✱ t❛ ✤➦t n = [❧♦❣q (X)] ợ qn X < qn+1 rữợ t t ❝❤ù♥❣ ♠✐♥❤✿ n nq lim =0 ˆ q (X) X s t t õ ữợ ữủ q (X) ≥ (q − 1) ❧s qk q n−1 ≥ , k n − k≤n−1 ✈➻ ✈➟② t❛ ❝â✿ n n n(n − 1).q nq nq ≤ lim = 0 ≤ lim = 0, s✉② r❛ lim n ˆ q (X) X→∞ ˆ q (X) X→∞ ❧s X→∞ ❧s q2 ❚ø ✤â ❦➳t ❤đ♣ ✈ỵ✐ ✣à♥❤ ỵ t ữủ q (X) + O(nq n2 ) π ˆq (X) ❧s lim = lim = ˆ q (X) X→∞ ˆ q (X) X→∞ ❧s ❧s ˆ q (X)✳ ❱➟② t❛ s✉② r❛ π ˆq (X) ∼ ❧s ✣✐➲✉ ♥➔② ❝❤♦ ♣❤➨♣ t❛ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q s tữỡ tỹ ữ ỵ số tố ợ số tỹ ỵ q ❧➔ ❧ô② t❤ø❛ ❝õ❛ sè ♥❣✉②➯♥ tè✳ ❑❤✐ ✤â t❛ ❝â✿ π ˆq (X) ∼ X ❧♦❣q (X) , X ự ỵ t❛ ❝â πˆq (X) ✭✸✳✷✶✮ ˆ q (X)✱ ♥➯♥ t❛ ❝❤➾ ❝➛♥ ∼ ❧s X ✳ ❱ỵ✐ ♠é✐ X ✱ t❛ ✤➦t n = [❧♦❣q (X)] ✈➔ ❧♦❣q (X) y = {❧♦❣q (X)} = ❧♦❣q (X) − n✳ ❑❤✐ ✤â X = q n+y ✱ ✈➔ t❛ ❝â✿ ˆ q (X) ∼ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ❧s lim X→∞ ˆ q (X) ❧s 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 − q n qk ∼ q ∼ ✱ ỵ n t õ (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 = lim = lim − q −y n+y X→∞ X→∞ X→∞ q n n+y ❑➳t ❤đ♣ ♥❤ú♥❣ ✤✐➲✉ tr➯♥ t❛ ♥❤➟♥ ✤÷đ❝ lim X→∞ ˆ q (X) ∼ ❱➟② t❛ ❝â ❧s ˆ q (X) ❧s X ❧♦❣q (X) = lim q −y + − q −y = X→∞ X ✳ ❙✉② r❛✱ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❧♦❣q (X) ✸✹ ❑➳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 ✭✶✮ ◆❤ú♥❣ t➼♥❤ ❝❤➜t t÷ì♥❣ tü Z ✈➔ Fq [T ]✳ ✭✷✮ ❍➔♠ ✤➳♠ sè ✤❛ t❤ù❝ ❜➜t ❦❤↔ q✉② ❜➟❝ n tr➯♥ tr÷í♥❣ Fq ✳ ✭✸✮ ❍➔♠ ✤➳♠ sè tự 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 ✈ỵ✐ x ❧➔ sè t❤ü❝ sỹ tữỡ tỹ ợ ỵ số tè ✈ỵ✐ ❜✐➳♥ sè t❤ü❝ ✭❦➳t q✉↔ t❤➸ ❤✐➺♥ q✉❛ ỵ t t ❬✶❪ ◆❣æ ❚❤à ◆❣♦❛♥✱ ✭✷✵✶✶✮✱ “❇➔✐ ❣✐↔♥❣ ✤✐➺♥ tû ♠æ♥ ỵ tt t ữợ t ỹ õ tự ♥❣÷í✐ ❤å❝ ”✱ ❚r÷í♥❣ ✣↕✐ ❤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 ✹✸✳✶ ... t❤ù❝ ❜➜t ❦❤↔ q✉② 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... (n) ❚ø ✤â t❛ ❝â✿ πq (n) ∼ q qn − q−1 n ✭✸✳✶✸✮ s s t q ợ ỵ số tố ỵ số tè ✤÷đ❝ ♣❤→t ❜✐➸✉ r➡♥❣ sè ❝→❝ sè ♥❣✉②➯♥ tè ổ ữủt q ởt số tỹ x tữỡ ữỡ ợ x ✳ ◆➳✉ x ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣✱ t❛ ❝â ❧♦❣(x)... tr÷í♥❣ ❤ú✉ ❤↕♥✮✳ ❈❤♦ F ❧➔ tr÷í♥❣ ❤ú✉ ❤↕♥ õ q tỷ õ tỗ t số tè p s❛♦ ❝❤♦ q = pn ✈ỵ✐ sè tü ♥❤✐➯♥ n ♥➔♦ ✤â✳ (ii) ❱ỵ✐ ♠é✐ sè ♥❣✉②➯♥ tè p số tỹ n = tỗ t ❞✉② ♥❤➜t ♠ët tr÷í♥❣ ❤ú✉ ❤↕♥ ❝â pn ♣❤➛♥ tû