平成 29 年度 ( 第 39 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 29 ~8 年月 73 月日開催 31 日 Riemann Riemann ( ). π(x) := #{p : p x} x log x (x ) Hadamard de

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1 Riemann Riemann ). π) : #{p : p } log ) Hadamard de la Vallée Poussin 896 )., f) g) ) lim f) g).. π) Chebychev. 4 3 Riemann. 6 4 Chebychev Riemann. 9 5 Riemann Res). A :. 5 B : Poisson Riemann-Lebesgue ). 5.,.,, 3, 5, 7,, 3, 7, 9, 3,....,. Gauss, π) Erdös Selberg 949 ), ) ).

2 Gauss ) π) log, π) log ) )., ) : π) π) / log Riemann 859, Riemann Hadamard de la Vallée Poussin ) Riemann ) 8 Euler, Riemann,.

3 ) 896. Li) : d log, π) Li) ). ) l Hôpial / log / log /log ) lim lim Li) lim / log, π) log ) log π) Li) ) 3. ). : von Mangold n Z >0 Λn) : { log p n p m p :, m ), 0, Chebyshev ψ Chebyshev ϑ > 0 ψ) : Λn), ϑ) : log p n p. Riemann ψ ) : ψ)d. π) log ϑ) ψ) ψ ) ) ), ψ ) ). ψ ) Riemann 3 ) ψ ) c+ i ) s ζ s) ds c > ) πi ci ss + ) 4 ). ζ s) s 3.3 ), ψ ) ) c+ i πi ci 4 ). hs) : ss+) ψ ) ) πi c+ i ci s ζ s) ss + ) ) ds c > ) s ζ s) s s hs)ds c π ), + hc + i)e i log d c > ) ) c lim π + hc + i)e i log d 0 3) 3, Li). 3

4 . Riemann-Lebesgue B), Lebesgue f) L R) Fourier fξ) : + f)e πiξ d lim ξ ± fξ) 0. c > + hc + i) d <, Riemann-Lebesgue lim π + hc + i)ei log d 0, c, 3). ) c c, h + i) Res) ζ s), Riemann-Lebesgue + h + i) d <., Res) 0, Res) ζ s)., Res) 0 Res), Res) ζ s) ), ) c, + h + i) d <. Riemann-Lebesgue 5 ),. Res) 0 Hadamard-de la Vallée Poussin ), cos θ + cos θ + 4 cos θ + cos θ + cos θ) 0 :, L, 4., Deligne Weil 5 ) 6, -Tae 7. π) Chebychev. π) ϑ). 4 Tauber,. 5 Weil I Weil II. 6, Laumon l Fourier Hadamard-de la Vallée Poussin. 7,, L. 4

5 . Abel ) Z >0 an) A) : n an). C 8 f) y<n fn)an) f)a) fy)ay) Riemann-Sieljes y<n fn)an) y A)df) f)a) fy)ay) A)f )d. y y. ϑ) π) log π) d, y A)f )d. f)da) f)a) fy)ay) π) ϑ) log + ϑ) log ) d. n Z >0 n an) :, n an) : 0 π) <n an), ϑ) <n an) log n. f) log, y Abel.) ϑ) <n fn)an) π) log π) log π) d π) log π) d < π) 0 )., n Z >0 bn) : an) log n ϑ) <n bn), π) 3/<n bn). f), y 3/ Abel.) log n log π) ϑ) 3/<n fn)bn) π3/) log3/) + ϑ) d ϑ) + ϑ) d log 3/ log ) log log ) < ϑ) 0 )..3 π) log ) ϑ) ).. ). lim π) O 9, π) ) d d O log log + log log lim π) d 0. log.. lim ϑ) O) log d + log ) log ) log ) ϑ) ψ)..4 > 0 ϑ) log ) d O log lim log 0 ψ) π)., ) d log ) ϑ) log ) log. d 0. d log d log + d log ϑ) d 0. log )., ϑ) log ) d 0. p m Λpm ) m log d log ) d + log ) p /m log p ψ) n Λn) m m log ϑ/m ), 0 ψ) ϑ) m log ϑ/m )., ϑ) p log log 0 ψ) ϑ) m log /m log /m ) log ) / log / ) / log ). log 8. 9 O ) Landau., f) Og)) lim f) <. 5 g)

6 :.5 ϑ) ) ψ) )., ψ) ψ )..6 A), A ) : A)d. a > 0 C A ) C a ), A) ac a ). A) a β β > A β) A ) β ) A β) β a A ) β) a a A) β + lim sup ac. a 0 < α < A ) A α), A) a α A ) a A α) α) a α a ) A)d β A)d A)β ), C βa. a β. lim sup A) A)d A)d A) α) α α A). lim inf C αa. a α α lim inf A) a ac. A) ψ) :.7 ψ ) ) ψ) )..3,.5,.7 ψ ) ). 3 Riemann. 3. Riemann ) σ : Res) > n < n d s n σ n σ n n d <, Res) > s s σ σ : n Res) >. Riemann. Riemann : 3. Euler ) Res) > p : n s p s. Res) >, Res) > 0., Euler. 6

7 Euler, Euclid : lim s p : <, ) ) + 8 p s lim s., Euler s, Euler 737 p : p n< log) n,, p : < log log) p ). n + n n nn ) + n n n) < 0, ). 000 )., Res) > ) p : p s p s p 3s p :. p s 3.3 ) Riemann s, ), π s s ) ) Γ π s s Γ ζ s). Γs) A )., ɛ > 0 Res) > ɛ ),., 5. Res) > 0 ),.,. 3.3 s, s ζ0) 3., ζ) π 6 s ζ ) π4, ζ4) 90 s 4 ζ 3) 4 ). 0 0 ζ) n n Basel, Euler π ),. Riemann. Dirichle., A B B.. 3, Z {±} ). 4. 7

8 n s n Res) >, s 0,, 3,..., + + +, , Res) > 0, Γs) s 0,,,... A ), 3.3 Res) < 0 0 s C) s, 4, 6,.... Riemann,, s, 4, 6,... Res), 5., 6 θ) : n Z π e n. f) : e π Fourier fξ) : e π e πiξ d e π +i ξ ) πξ d e πξ, Poisson B. ) θ) ) θ., Riemann. Res) >. π s Γ s 0 ) 0 e π n ) s n e π d e πξ π π e πξ d 0 e πn s n θ) ) s d θ) ) s d 0 + θ) s d ) + θ) ) s d θ) ) s d + s s d ) + θ) ) s d s s + d θ) ) s θ) ) s d θ) ) s d. d 5, π) Riemann. 6 Chebyshev ϑ. 8

9 5 θ) θ )., θ), s 0,., s s, π s Γ s s ) π Γ s )ζ s). s. s 0,, Γ s ), s 0. 4 Chebychev Riemann. ψ ).. 4. ψ ) n n)λn). an) Λn), A) ψ), f), y Abel.), n nλn) f)ψ) f)λ) ψ)d ψ) ψ )., ψ ) ψ) n nλn) n n)λn). 4. c > 0 u > 0 k Z >0, c+ i πi ci u z zz + ) z + k) dz { u) k k! 0 < u, 0 u >. u πi C z R dz Cauchy. zz+) z+k) C R. 0 < u, R> k + c) Rez) c Imz) < 0 Imz) > 0, R Rez) c. u >, R> k + c) Rez) c Imz) > 0 Imz) < 0, R Rez) c., z + iy C R R, 0 < u u >, u z zz+) z+k) u z z+ z+k, n k z + n z n R n R k R/ R > k ), C R R πr u c OR k ). k R 0. RR/) k u c R z+ z+k, C R R. u >, C R C R 0. 0 < u, C R z 0,,..., k, C R u z dz k πi zz+) z+k) n0 Res u z z n k zz+) z+k) n0 k ) n u n n0 k k ) n!k n)! k! n0 n u) n u)k.. k! 4.3 c > ψ ) πi c+ i ci ) s ζ s) ds. ss + ) 9 u n n) n+) ) n+k)

10 , Res) > n Λn) n s ζ s). Euler p: ζ s) p s log p p s p: p s log p p: m p ms Λn) n., c > n s n., 4. k, u n/, n c > 0 n /n) s ds., ss+) 4. ψ ) n n )Λn) πi c+ i ci Λn) n c < ψ ) n c+ i πi ci Λn)/n) s ds ss + ) n c+ i πi ci Λn)/n) s ds ss + ) 4. n > 0 ). c >, c+ i n Λn)/n)s ds Λn) c+ i c ci ss+) n ds n c ci s s+ C Λn) n < C, c > ), n c ψ ) c+ i πi n c+ i πi ci ci Λn)/n) s ds ss + ) πi ) ζ s) ds s ss + ) c+ i ci s ss + ) n Λn) ds n s,. ζ s) 4.4 c > s,. ψ ) ) c+ i πi ci ), hs) : ζ s), ss+) s ψ ) ) πi c+ i ci s ζ s) ss + ) ) ds. s s hs)ds c π hc + i)e i log d. 4. k, u /, c > 0 ) c+ i s ds. πi ci ss+)s+) s s, c > ) c+ i s ds πi ci ss+) s. 0

11 5 Riemann Res)., Riemann Res). 5. Euler ) C f) M <n M fn) M []. f)d + M [])f )d. [, M] n n []f )d n n )f )d n )fn) n n fn )) nfn) n )fn )) fn). n + n M M []f )d MfM) f) <n M fn)., <n M fn) f)d MfM) f) M f )d M []f )d + MfM) f). M., Hadamard-de la Vallée Poussin ) Res), Ims) e, ζ s). 5. A > 0, s σ + i, σ, σ > A log, e M log, ζ s) Mlog ) A ) M., σ ζ), ζ s) ζ ), σ <, e. Euler 5.) f) / s Z >0 <n M M d s M [] d. Res) > M, n s s s+ n> n s d s [] d s s [] d, Res) > s s+ s s+ n n s s s [] d. Res) δ > 0 [] d s s+ d < s+ δ+ [] d Res) δ Res) δ. s+, Res) > 0 Z >0 n n s + s s s [] d s+, s lim s s Res) > ). σ <, e s σ + + <, s, n n + σ σ + d σ+ n n + σ σ + σ σ.

12 < +, n log n log log, A > 0 s σ + i σ < A/ log, n n ) σ. n, σ n e σ) log n < n ea log n/ log n ea O ) ) n O O), O + σ σ O)., O n n) + O) Olog ) + O) Olog ),.. n + s s [] d ζ s) n s s s+ log n n s s log [] d + s []) log d. s <, n s s ) s s+ s+ s, log σ σ + ζ s) n d σ+ σ σ σ σ, ζ s) n log n n σ + σ + σ log +, log n n σ + σ + σ log d + σ+ log d log σ+ σ σ log d. σ+ + + log + + σ σ σ σ σ σ σ d σ σ+. < +, A > 0 n O n n) σ, ζ s) O ) ) ) ) log n n n + O log + O + O + Olog ) + O) Olog ) ) Olog ) ). 5.3 Hadamard-de la Vallée Poussin). σ ζσ) 3 ζσ + i) 4 ζσ + i).. Res) 0. ): Res) > log p: log p s ) p: s σ+i, σ > ep p:, ep p: m m cosm log p) mp mσ ), ζσ) 3 ζσ + i) 4 ζσ + i) ep p: m mp ms ) ep p: m mp ms, e im log p m ). mp mσ cosm log p) + cosm log p). mp mσ cos θ + cos θ + 4 cos θ + cos θ + cos θ) 0 ) σ >. σ + 0 σ ).

13 ): 3.3, s, s + i, 0. ) σ > 4 σ )ζσ)) 3 ζσ + i) σ ζσ + i) σ. σ , σ )ζσ)) 3, ζσ + i) ζ + i) <. ζσ + i) 0 lim σ + ζσ+i) σ 4 ζ + i) 4 <.,. ζ + i) 0. Hadamard-de la Vallée Poussin Res), Ims) e. Res) ζ s) Res) 0 ζ s) ). 5.4 s σ + i σ, e C log ), ζ s) 7 C log ) 9 C, C > 0., σ µn) Möbius n p m p m k k n m m k µn) : ) k, µn) : 0), p: ) µn) p s n n s µn) n n s n ζ) <, 4.3 ζ s) Λn) n n n s, ζ s) Λn) n <,, σ <, n e. 5.3) ζσ + i). σ )ζσ) σ ζσ) 3/4 ζσ+i) /4 M ζσ) M < σ., 5. σ σ ζσ + i) Olog ), < σ, e ζσ + i) Bσ )3/4 log ) /4 B., σ. < α < α. σ α, e, 5. ζσ + i) ζα + i) α σ ζ u + i) du α σ)m log ) α )M log ) M., ζσ + i) ζα + i) ζσ + i) ζα + i) ζα + i) α )M log ) Bα ) 3/4 log ) /4 α )M log ) σ α., α σ, σ ) 3/4 α ) 3/4 ζσ + i) Bσ )3/4 log ) /4. σ, e Bα )3/4 log ) /4 α )M log ) ζσ + i) Bα )3/4 log ) /4 α )M log ) 3

14 . α < α <, α Bα )3/4 α )M log ) ) α. log ) /4 α + ) B 4 α. α >, M log ) α <., 0, σ ζσ + i) α )M log ) C log ) 7 C. e 0 ζσ + i) C C log ) 7, C : min{c, C } ζσ + i) C σ, e log ) , σ, e ζ σ+i) ζσ+i).. log )7 C ζ σ + i) C log ) 9 C 4.4 c. ) 5.5 hs) : ζ s) 4.4. ss+) s.. h + i) d <. 3. ψ ) ). ψ ) ) h + i)e i log d. π ): 4.4 c >, ψ ) ) πi c+ i ci s hs)ds. Res) c> ) Res) Cauchy. Res) c> ), Res), Ims) ±T. 3.3, ζ s) s s +, ζ s) + )/ + ) +, s s ) s s ) ), hs) : ζ s) s ss+) s,., s hs), Cauchy 0. T s c ± it s ± it ) 0. s c + it s + it.., 5.4 s σ + i, σ, e ss+) T, ss+)s ) T 3 T ζ s)/ Mlog ) 9 M., T e, Mlog T )9 s c + it s + it hs). T c+it c s hs)ds c Mlog T )9 c log T )9 dσ M c ). T T +it 4

15 T 0., c+ i ci s hs)ds + i i s hs)ds. ) + i πi i s hs)ds h + π i)ei log d. ): h + i) d e + + e., 5.4 e e e Mlog )9 e, h + i).. h + i) d <. 3) ) Riemann-Lebesgue B.) ), Hadamard-de la Vallée Poussin) π) log )..3,.5,.7, 5.53). A :.. Res) > 0 Γs) : 0 e s d., c > 0, e s, e s d c, > 0 e s < s σ : Res) > 0 c 0 e s d c 0 s d c 0 σ d cσ < σ c 0 e s d Γs) : e s d Res) > 0. 0, Γs + ) e s d [ e s ] s e s d sγs). 0 Γs + ) sγs), s C, s 0,,,... Γs+) + Res) > 0, Γs) Γs) ss+) s+) ), Γs) C s 0,,,...., Γ) e d, n 0 Γn + ) n!. B : Poisson Riemann-Lebesgue ).,. B. Poisson ) R 7 f), Fourier fξ) : f)e πiξ d, fm). n Z fn) m Z 7, f), m, n 0 sup R m dn f d n ) <. f) e π. 5

16 g) : n Z f + n) g). g) Fourier g) ) m Z 0 n Z f + n)e πim d e πim ) m Z f)e πim d e πim fm)e m Z πim. 0 n Z fn) fm) m Z. B. Riemann-Lebesgue ) Lebesgue f) L R) Fourier fξ) : f)e πiξ d lim ξ ± fξ) 0. L R), C 8, ɛ > 0 f g L < ɛ, C g)., lim sup ξ ± fξ) lim supξ ± f) g))e πiξ d + lim sup ξ ± g)e πiξ d < ɛ + lim sup ξ ± πiξ g )e πiξ d ɛ + lim sup ξ ± π ξ g ) d ɛ. ɛ >

Z: Q: R: C: sin 6 5 ζ a, b

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