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おきみちゆきしげ
5 years ago
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1 For Tutor MeBio ζ Eite by kamei

2 MeBio : Bernoulli Bernoulli 4 Bernoulli Bernoulli Γ 8 Γ Γ Γ ζ 4 ζ ζ Laurent ζ ζ ζ Chebyhev von Mangolt ψ ϑ B A Re Wiener ζ 49 Jenen Haamar ζ Π Möbiu Π ζ

3 MeBio :

4 MeBio : 4 BernoulliBernoulli k k +, k k + + 6, k k k 9 k n n + B n+ + B n+, J.Bernoulli B n Bernoulli k B n Bernoulli ζ k Bernoulli n Bernoulli B n n B n n! u n ueu e u ue u e u + u + u + 3 u u u + u 6 + u3 4 + u4 + + u + u + 3 u u u + + u u u + u4 7 + u u B, B, B + 6, B B n n +, B n Bernoulli B n n B n n! u n u e u

5 MeBio : Bernoulli Bernoulli 5 B n B n B, B, B 6, B 3, B 3 3 Bernoulli Bernoulli Bernoulli Bernoulli 3 B n n B n n! n nc k B n k k k u n u e u eu l B l l! u l k k k! u k n n k B n k k n k!k! un Σ Bernoulli Bernoulli Bernoulli ζ 4 B Bernoulli B 6, B 4 3, B 6 4 Bernoulli n n B n n! B n n! u n u e u n u n { n } u ueu e u B n n! u n u 3 ζm m π m m! u e u ueu e u ueu e u B m ζm > 5 n, k n n B n + B n n! k u n ue+u e u n + B n+ + B n+ ueu e u ueu e u e u ue u n n u n+ n! u n+ B n+ + B n+ n +! n n! k n k k B n+ k + B n+ k n + B n+ + B n+ n + B n+ + B n+ n + k 3 B 4 B 4 4 k

6 MeBio : Bernoulli Bernoulli 6 Bernoulli n k 9 B Bernoulli k Bernoulli n B n+ n + B n B n n! u n ueu e u n B n n! u n u e u e u n B n n! u n+ u n+ B n+ n + B n + C n > B n B n k n k n + B n+ B n+ B n B n+ B n+, B n+ B n+ B n+ B n+, B n+ B n 3 a B B + B, B B + B 3 3 b B 3 B B B 6 B + 6 a B + 6 B B 33, B B 3 + B 4 4 b B 4 B B 3 B 3 B B n B B n B n+ B n+ b B n B n [, ] Euler Maclaurin B n a []f n

7 MeBio : Bernoulli Bernoulli 7 B B + 6 B B B B B k 9 k

8 MeBio : 8 Γ Γ Γ > Γ e > Γ + Γ Γ n Γn + n! Γ Re > Γ Γ + Γ Γ n n n! lim Γ Euler, Gau Im ± Γz 5! ! z z! lim n n zz + z + n nz lim n Γz n zz + z + n nz Euler, Gau Γz lim n n zz + z + n nz 3 Weiertra z Γz zeγz n + z n e z n

9 MeBio : Γ 9 Γz z,,, Γz 3 Γ Γ π B B, y t t y t B, y ΓΓy Γ + y y 3 3 Γ + y Γ3 B, 3 3 π 3 Γ Γ Γ π π! fz ΓzΓ z fz + Γz + Γ z π 8 fz in πz z,,,,,, zγz Γ z z f π f lim z fz z in πz z π co πz z π lim z zγzγ z fz lim z π Γ Γ + zγ z Γ ΓzΓ z in πz π { ΓzΓ z Γz zγ z ze γz z n z z n n + z n in πz π } { e z n ze γz n z n } e z n Γz z,,, 3, Γz z,,, Γ z + z, 3, Γz ΓzΓ z +

10 MeBio : Γ fz ΓzΓ z + f z + Γ z + Γz + Γ z + zγz zfz Γ z + Γz + zγz fz Γz z ΓzΓ z + k z Γz z π k k π 3 Gau, Legenre ΓzΓ z + lim n lim n lim n ΓzΓ z + Γz n! n z zz + z + n n!n! n! n!n! n! n n z n n z π z Γz z + z + 3 n! n z n! n z zz + z + z + n n z n z n n z n z + n Stirling n! n πn e ΓzΓ z + Γz lim n n!n! n! n n! n!n! n πn n z n π z 3 Γ Γz z z ΓzΓ z + n Γ z + n n n π n n nz Γnz

MeBio 7.8.3 : Γ {z C ΓZ } {z C ΓZ } lim Γz +, lim Γz + Γz R + z + z + Γ.

11 MeBio : Γ {z C ΓZ } {z C ΓZ } lim Γz +, lim Γz + Γz R + z + z + Γ Riemann z.4663 ± + σ lim Γσ + τi Γz τ ± i i, + i +i R +,,

12 MeBio : Γ Γz Γz Tailor z Γz fz Γz ze γz n + z n z + γz + γ z n e z n + γ3 z γ4 z γ5 z z z n + z3 3n 3 z4 8n 4 + z5 3n 5 z6 44n 6 + z + γz + γ z ζ γ z + γz + γ γ ζ 4 + γ3 z 3 6 z + ζ3 3 ζ + γζ3 3 + γ4 z γ5 z 5 + z 3 + ζ ζ4 z 4 6ζ5 5ζζ3 + z γ z γζ + ζ3 z 4 3 z 5 + ζ ζ4 8 γ 5 + γ3 ζ + γ ζ3 + γζ ζ γ z + γ z π z 3 γ 3 π γ + 4 ζ 3 z γ 5 π γ ζ 3 γ + π 4 γ 4 ζ 3 π + 88 ζ 5 z ζ5 5ζζ3 + z γ 4 6 π γ + 48 ζ 3 γ + π 4 z z z z 3.463z z z 6 + fbi bi.57756b b 3 i.463b b 5 i b b.463b b b b b 5 + i b fbi b b 6 ± lim fz c c a + ai z

13 MeBio : Γ 3

14 MeBio : 4 3 ζ ζ Euler in πz in πz πz z Taylor z n n ζn ζ π 6 n, ζ4 π4 9, ζ6 π6 945 m ζm m π m m! B m ζ in πz πz z n n π cot πz z + n z n z n { z + z n n z n m m } z ζm + z m+ m z πz cot πz ζm + z m+ ζmz m m m πz cot πz πz co πz in πz πz e πiz + e πiz e πiz e πiz i πiz + e πiz πiz + πiz e πiz Bernoulli u e u k B k k! u k πz cot πz πiz + πiz e πiz πiz + k B k k! πiz k ζmz m πiz+ m ζm k B k k! πiz k B, B, B 3 B 5 B 7 B m m! πim ζm m π m m! B m

15 MeBio : 3 ζ 5 ζ 3 partial ζ ζ an n a mo N n Hurwitz ζ ζ, n n + a ζ a ζ, ζ, ζ L, χ 3 ζ, a N N χaζ an a N ζ an 3 Re σ > Γζ, e t t t e t n n t + n + n t t e +nu u u u u n t + n e u e u u u u,,,, σ > Bernoulli ue u e u n B n n! u n ue u e u n B n n! u n σ >

16 MeBio : 3 ζ 6 e u e u u u u ue u e u u u ue u e u u u n B n n! u n u u n n B n n! B n n! n u n+ u n [ u n+ n + ] n B n n! n n + ue u e u π lim up n n Bn n! π B n n! n O π + ε Γζ, Γ ζ, e u u e u u u n B n n! n n +,,, n B n n n! Γ n n n! 3 3 ζ n, n B n ζ, ζ, B n B n 3 4 ζ ζ m ζ Bernoulli ζ m m B m ζ, ζ, ζ 3, ζ 5 5, ζ 7 4, ζ 9 3, ζ , ζ 3 ζ m n m.. lim m {B m B m }.. m B m

17 MeBio : 3 ζ Riemann Γζ u e u u u + + i +δ + i +δ i + i C C u u + i e log u u i e log u+πi e πi u Im nπi 4πi O πi C C n Re πi 4πi nπi u σ > C u e u u e πi u e u u + u e u u e πi u e u u σ > ζ e πi Γζ C u e u u e πi e πi e πi e πi ie πi in π ΓΓ ζ πi e πi Γ C u e u u in π π

18 MeBio : 3 ζ z Riemann z fz e z z R 3 7 C u e u u u log fu u e u u, 3, Γ C,, 3, > ζ > πi Γ ζ 3 8 ζ ζ Γ + n z n n ζ n πi e z z n log C z n C πi Γ + n ζ n πi C n! πi C n n! πi n C k z n e z B k k! z z e z z n z z k n z k n k n + n n! πi n B n+ n + πi B n+ n +! 3 9 ζ ζ m ζ Bernoulli ζ m m B m 3 ζ πi e πi Γ C u e u u

19 MeBio : 3 ζ 9 C u e u ζ, 4, 6, 8, Γ,, 3, 4, 5, u e u u, 6, 4,,, 3, 4, 5, C u C u e u u C u e u u u C k B k u k k! u u C k B k k! u k+ u log u B, 6, 4,,, 3, 4, Bernoulli B k k ζ n + n + + n + + n n n n n n + n k + ζ + k + ζ ζ + + k k k n k k k k n ζ + k k lim ζ ζ ζ n + + n n + n n n n k + + n + + n k k n k n k k n + + ζ + k + + ζ ζ + + ζ + k k k k k ζ k k ζ + k k k ζ + k k Re > σ > ζ Re > Γ k

20 MeBio : 3 ζ 3 ζ ζ 3 ζ 4 ζ 3 Laurent γ lim N N n n log N Euler lim ζ Euler Maclaurin γ 3 3 a<n< b f C fn b a f + b a [] f + a [a] fa b [b] fb f ρ B [] [] Riemann Stieltje fn a<n< b b a b a b a b a f[] f ρ f b a fρ... f [ fρ ] b b + f ρ a a ζ σ Re > ζ n n [] + + [] [ [] ] [ ] [] [] +

21 MeBio : 3 ζ [] Re > ζ Re > lim ζ lim N lim N lim N lim N lim N lim N { { { { { + [] N N log N + N log N + log N log N log N + + [] + k+ k k k+ k N k N k N k k [ k ] k+ k [] } [ ] N } k N + N k k + k k k + N lim + N + + N log N N γ } N } } ζ + γ + O lim ζ γ 3 3 Euler Maclaurin a<n< b fn b a + f fa + fb + K k +! b a k B k k! B k+ [] f K+ f k b f k a ζ Re > K K 4 ζ ζ ζ Laurent ζ + γ + γ m m γ m Euler m

22 MeBio : 3 ζ { C ζ } { C ζ } Re Re > lim ζσ + iτ σ Riemann

23 MeBio : 3 ζ 3 ζ , ζ σ

24 MeBio : 4 4 ζ ζ ζ π Γ co π ζ 4 Riemann ζ πi e πi Γ C u e u u Im nπi 4πi O πi C C n Re πi 4πi nπi + +i n+π +i n+π n+π i + i C n u πki k ±kπi k,,, n kπi πi Cn u e u u πi C u n e u u + { kπi + kπi } k Re σ < n Σ ζ

25 MeBio : 4 ζ 5 Cn u e u u eu n u + n + πi e u.. e +n+πi e +.. n e u O u n + σ u πn + u Cn e u u On + σ Σ C { kπi + kπi } k k k } {kπ e πi + kπ e 3πi {kπ } e πi e πi + e πi π e πi e πi + e πi π e πi π co ζ π e πi in π ζ k k ζ πi e πi Γ C πi e πi Γ πi π in π u e u u { π e πi in π } ζ Γ ζ ζ π Γ co π ζ ζ 3, 5, 7, 9, Γ,,, 3, 4, 5, co π, 5, 3,,, 3, 5, ζ, 4, 6,

26 MeBio : 4 ζ 6 4 ζ ˆζ ˆζ π Γ ζ ˆζ ˆζ ˆζ ˆζ ˆζ ˆζ ζ, 4, 6,,, 4, Γ, ζ Γ π Γ co π ζ ζ π ΓΓ Γ co π Γ ζ Γ + Γ Γ ζ Gau, Legenre π Γ Γ ζ π Γ Γ Γ π Γ π + co π Γ co π Γ ζ ζ + Γ Γ in π π π co π Γ π Γ ζ π Γ ζ π Γ ζ π ζ π Γ ζ ˆζ ζ < Re < Re ˆζ ρ ρ, ρ, ρ ζ ρ ˆζ ρ ρ ρ

27 MeBio : 4 ζ 7 4 Jacobi > ϕ e πn ϕ ϕ n Z e πn e πn n Z n Z e πt t ft n Z e πn+t Fourier ft n Z e πn+t m Z a m e mπit a m a m fte mπit t mi πt+ e πm n Z t e πm e πn+t e mπit t e πt t e πm e πt e mπit t ft n Z e πn+t m Z e πm e mπit t Poion Whittaker, Waton p.4 fz + N + e πz e πiz z n + i N + πi e πn fz C : N + + i N i N i + N + i ϕ + i i e πz + +i e πiz z e πz +i e πiz z Imz e πiz e π > + i i + i i + i n e πz + i e πiz z i e πz e πiz e πiz z e πz e πiz + e πiz + e 4πiz + e 6πiz + z i n e πz πniz z e πn n + i ni πz πn i e z n e πn + i ni πz i e z

28 MeBio : 4 ζ 8 Imz e πiz e π < + +i +i + +i +i + i n e πz + +i e πiz z +i e πz z eπiz e πz + e πiz + e 4πiz + e 6πiz + z i n e πz +πniz z e πn n + i ni πz+ πn i e z n e πn + +i ni πz+ +i e z θz θz e πinz ϕ n z i ϕ ϕ ϕ , ϕ , ϕ ϕ 4 ϕ ϕ ϕ y ϕ

29 MeBio : 4 ζ ζ ˆζ π Γ ζ ˆζ ˆζ ψ e πn Chebyhev ϕ ψ + n Jacobi ψ + ψ ˆζ π Γ π π π + ψ ζ e t t n t t e t t n n e πn π n ψ ψ + n t t ψ ψ + t πn

30 MeBio : 4 ζ 3, ψ ψ ψ + ψ + ψ ψ ψ + + [ ] [ + ] ˆζ ψ + ψ ˆζ ˆζ < < 4 ζ ζ Bernoulli m ζm m π m m! B m ζ m m B m ζ m πm Γm co π m ζm π m m! m ζm ζm πm m! m ζ m m π m m! B m

31 MeBio : 3 5 π π log Gau Haamar, e la Valée Pouin 896 {S n } n S n n S n a k a k >, lim S n S n n k { n a n δ[π]n n 5 {a n} Dirichlet f A g f S n< a n A a n n A a n n a n > Re > n Re > a n n ζ ζ n Re > S [] n a k Dirichlet k a n n δ[π]n π p + Re n δ[π]nn p n p p log ζ Re > n πn δ[π]k δ[π] π k

32 MeBio : p p log ζ log π log p log p log { log p n a n δ[π]n log n n λn log p Chebyhev ϑ λn p< λn Dirichlet Φ n log p p p n Re > ϑ Chebyhev ψ von Mangolt Λn ψ ϑ ψ 5 a n S S n< a n f a n n Sn Sn n Snn Snn + n n n Snn n + Sn S n n n+ n n n n+ n S S O S f A g f A S Re > Re > S A a n S a n f a n Dirichlet a n f n S n a n A B A f S f + + A S A + < < S [, f S Mellin

33 MeBio : 5 33 Re > A 5 3 A 3 g g g + lim + Re > g S O S Re > lim + A g S A + S A S A + Re > S A A Laplace e t g + ge t e t t e t t ge t e t t ge t It It fe t e t A 5 5 A It It LI Ite t t Re > Re > LI lim lim T T Ite t t lim T T It t lim lim T T Ite t t It t, lim LI It t 5 6 B S A S S A Chebyhev ϑ, λn, Chebyhev ψ, von Mangolt Λn ψ ψ log p mi i p i < ϑ.. π log 3 A B

34 MeBio : ψ 4 ψ L [ψe t e t ] 5 L [ψe t ] ψ B ψe t e t t ψe t e t e t t Re > A ψe t e t t ζ ζ 3 ζ Re Re ζ Re lim + L[ψe t e t ] ψe t e t e t t ζ ζ + ψe t e t e t t + ψe t e t t ψ ζ ζ + ψe t e t t Laplace Tauber lim L[It] Newman ψe t e t e t t Re > It t A Wiener 3 Chebyhev von Mangolt 5 3 > n Chebyhev ϑ ϑ log p p< ϑ ϑ log ϑ Chebyhev ψψ log{l.c.m,,, [], []} ψ log ψ 94.4 ψ ϑ ψ ψ ϑ k ϑ + ϑ + ϑ 3 + k ϑ ψ

35 MeBio : von Mangolt ΛnΛn { log p n p m p : Λn Chebyhev Λn ψn ψn Λn Dirichlet ζ ζ ζ ζ n Λn n p, m log p p m 4 λnλn { log p n p p : λn Chebyhev λn ϑn ϑn λn Dirichlet Φ Φ n λn n p log p p ζ ζ Re > n 5 Möbiu µnµn n p p p r n 6 Euler φnφn #{k k >, k, n } 5 3 Möbiu fn, gn fn n g gn n n µ f fn, gn { n µ n \ n { n n \ fn n, gn φn n n n g φ φn n n µ 3 fn log n, gn Λn n Λ log n, n n µ log Λn ζ n Möbiu µn Euler µn n µn Dirichlet ζ ζ p p p µ + µpp + µp p + n µn n

36 MeBio : 5 36 fn, gn, hn Dirichlet F H n hn n H F G hn n n n fg fn n, G n gn n, fn n gn n g F ζg, n µ f G ζ F ζ ζ n Λn n log n n Λ Dirichlet ζ ζ ζ log n n log ζ p Λ Λn n log p n µ m, p n Λn n n log mp m log n n n ζ n Λn n ζ Λn n n log n n ζ ζ p p log p p p + p + log p log p p p p Λp p + Λp p + Λp3 p 3 + p n + log p p Λn n + log p p ψ ϑ ψ ϑ 5 4 ϑ O Zagier n < p < n n C n + n n nc k > nc n > k n<p< n p e ϑn ϑn

37 MeBio : 5 37 ϑn ϑn < n log 4 ϑ ϑ < ϑ [] + log ϑ [] < [] log 4 + log < log 4 + ϑ ϑ < log 4 + ϑ ϑ < 4 log 4 + ϑ < log 4 + ϑ O 5 4 ψ π log [, ] [, w] w, ] w ψ log m 3 m p π m π < π log... m i, p i i < < log p πw log + log w log < < w<p< πw log + log log w w<p< πw log + log log w ψ log p ψ π log πw log < < + log log w ψ πw log w log log w w log ϑ π log ϑ p< log p < log π log p< ϵ ϑ > ϵ <p< log p > ϵ <p< ϵ log ϵ π π ϵ log ϑ ϵ > π π ϵ log π log < ϵ ϑ + π ϵ log π ϵ < ϵ π log < ϵ ϑ + log ϵ

38 MeBio : 5 38 ϑ < π log < ϵ ϑ + log ϵ ϑ, log ϵ < lim π log < ϵ ϑ ψ ϑ ϑ < M M n > log /n < ϑ /n log < N < + log N ψ ϑ + ϑ / + ϑ /3 + + ϑ /N < ϑ + M / + M /3 + + M /N < ϑ + M / log N N N < log ϑ < ψ < ϑ + M log / ψ ψ < M M ϑ < ψ ϑ < M ψ M log / < ϑ < ψ 5 B 5 5 B S S ϑ ψ S lim up S k S k k > λ lim up S > S k > λ k λ λ u u u S S > λ > < < > λ S [ k, λ k ] λk k S > λk k λ log λ lim up k S λ k < λk k S λ λ k k u k u k u k 4 ψ > ϑ 3 ψ

39 MeBio : 5 39 λ k y S y λ y k λ k lim inf k S k k S < λ lim inf S < λ < < < < λ S [λ k, k ] S < S k < λ k k λ k S k λ < k λ k k u λ k λ k u k u k λ + + log λ lim inf S < k λ k λ S λ u u u k lim S 5 5 A ζ ζ ζ ζ n Λn n n n Λn n ψ ψn ψn n ψn n n + ψn n n n+ n n + ψn n n ψn n ψ + n ψn n ψe t e t t n ψn n + ψe t ζ ζ ζ ζ Re > ψe t It ψe t e t Laplace It LI Ite t t ψe t e +t t e t t + ζ ζ + + ζ ζ LI Re > ψ O It It t LI It t ψe t e t t ψe t e t e t t ψ

40 MeBio : 5 4 Stieltje ζ ζ n Λn n [ ] ψ ψ ψ + ψ + ψ Heaviie H ψ ΛnH n n ζ ζ n [ n Λn n n [ ψ Λn ΛnH n ] δ n + ψ + ] + n ψ + ΛnH n + ζ ζ ψe t e t t ψe t Laplace ψet t ψ Λn H p m ψe t He t p m ψet t n< m, p m, p [ ψe t L t ] { } δe t p m e t e t t m, p ζ ζ Laplace ζ ζ p, m δ p m p, m m, p δe t p m e t p m ζ ζ 6 A 5 6 Wienner A LI It It Laplace Ite t t Re > Re > lim lim T T lim T T Ite t t lim It t It t LI lim T T Ite t t It t, lim LI Newman g Ite t t Re > g Ite t t

41 MeBio : 5 4 Re < g T g T T It t lim T g T g T Ite t t g T R R δr g { C < R, Re > δr} C Cauchy g g T πi C g g T e T + R R T F T lim F T T 4 F T, R, F T, R, F 3 T, R, δ, F 4 T, R, δ F T, R < B R 3 F T, R < B R B T, R 4 F 3 T, R, δ < CRδ CR R T 5 4 R, δ lim T F 4T, R, δ ε> B R < ε 4 R T F T, R < ε 4 F T, R < ε 4 3 R F 3 T, R, δ < ϵ 4 δ T 4, 3 R, δ T > T F 4 T, R, δ < ϵ 4 T T > T F T < F T, R + F T, R + F 3 T, R, δ + F 4 T, R, δ < ϵ F T Im Ri C C C δr O Re Ri δ

42 MeBio : 5 4 C C { C Re > } F T, R g g T e T + πi C R g g g T T Ite t t ma It B Ite t t < B e t t B T T e Ret t B [ e Ret Re ] T Be ReT Re Re C Re iθ π < θ < π + z R + e iθ e iθ e iθ + e iθ co θ Re R F T, R < πi π g g T e T + C R Be ReT Re e ReT Re R πr R B R C C { C Re < } g g T F T, R g T e T + πi C R g T C C { C R, Re < } C Re < g T T Ite t t < B T e t t B T e Ret t B [ e Ret Re ] T < Be ReT Re F T, R πi g T e T + C R < B R ge T + πi C R < δ < δr δ C 3 C { C δ < Re < }, C 4 C { C Re < δ } C 3 R F 3 T, R, δ ge T + πi C 3 R, F 4 T, R, δ ge T + πi C 4 R F 3 T, R, δ C g + R MR et <, < R arcin δ C 3 R R, arcin δ R F 3 T, R, δ < π MR R arcin δ R < CRδ

43 MeBio : 5 43 CR R T T R δ F 3 T, R, δ F 4 T, R, δ g + R < MR, et < e δt, < πr C 4 F 4 T, R, δ < π MR e δt πr < DRe δt DR R T δ e δ T T R, δ lim T F 4T, R, δ ψ ϑ Zagier 5 6 ζ ζ n ϑ Λn n p, m Ψ ζ ζ log p p m Ψ Ψ p, m> n λn n p log p p ζ ζ log p p m Re > Ψ Re > Re > Ψ Abel λn ϑn ϑn ϑn Ψ n n n n n n ϑn n n + ϑn n n n+ n + ϑn n n ϑ + n ϑn n n ϑn n + ϑe t e t t ϑe t Laplace Ψ Jt ϑe t e t Laplace LJ Jte t t ϑe t e +t t e t t + Ψ + + Ψ + LJ > Jt t LJ Jt t ϑe t e t t ϑe t e t e t t ϑ

44 MeBio : Re n Λn n ζ ζ Re ψ ζ Re 5 7 ζ + it n σ+it n σ e it log n n σ {cot log n i int log n} Re n σ+it n σ cot log n σ > ζ ζ n Λn n Re ζ ζ σ + it n Λn cot log n n σ > 3Re ζ ζ n n n σ 4Re ζ ζ σ + it Re ζ ζ Λn cot log n + cot log n n σ Λn + 4 cot log n + co t log n n σ Λn + cot log n n σ σ + it + it a a > + it b b > lim σ + σ ζ ζ lim σ { 3 ζ σ + ζ ζ lim z it z +it ζ z lim ζ σ σ + ζ ζ lim σ σ + ζ σ σ 4 ζ ζ σ + it ζ ζ σ + it a σ + it b } σ + it 3 4a b < σ + ζ ζσ σ ζ ± it ζσ ± it Oσ ζ ± it ζσ ± it O

45 MeBio : 5 45 ζσ + itζσ + it 4 ζσ 6 ζσ it 4 ζσ it O σ σ + Re > ζ p log ζ p p log { ζσ + itζσ + it 4 ζσ 6 ζσ it 4 ζσ it } m> m> > p p m p mσ m log p m> p mσ+it + 4p mσ+it + 6p mσ + 4p mσ it p mσ it 4 p imt + p imt p p m m 8 Wiener n a n n< X a n Dirichlet n a n n Tauber Wiener 5 8 n > a n > Dirichlet L n a n n L N> L Π { C Re > N} Π Π L a, b a L Π L N > L Π N N A L A N + b m L m Π L m N A m A > L m L m Π N A N + a α, b α m a n n< X A NΓα XN log X α

46 MeBio : Wiener f > f O g f Re > g Re > ρ f ρ It fe t e t ρ t > It Laplace LI fe t e t ρ e t t f ρ Re > g + + ρ lim LI f ρ + f ρ f f ρ f ρ + g + + ρ G Re > G G T lim T G T G T Re > Re > f ρ + f ρ + Re < G T R R δr G { C < R, Re > δr} C Cauchy G G T πi C G G T T + R R T F T lim F T T

47 MeBio : 5 47 Im Ri C C C δr O Re Ri δ C C { C Re > } F T, R πi G G G T B f ρ + T T f ρ + G G T T + C R ma f ρ B Re t B t < B T [ Re Re ] T BT Re Re Re C Re iθ π < θ < π + z R + e iθ e iθ e iθ + e iθ co θ Re R F T, R πi G G T T + C R < π BT Re Re T Re Re R πr R B R C C { C Re < } G G T F T, R G T T + πi C R G T C C { C R, Re < } C Re < T f ρ T G T + < B T ] T t B Re t B [ Re Re BT < Re Re F T, R πi G T T + C R < B R

48 MeBio : 5 48 GT + πi C R < δ < δr δ C 3 C { C δ < Re < }, C 4 C { C Re < δ } C 3 R F 3 T, R, δ GT + πi C 3 R, F 4 T, R, δ GT + πi C 4 R F 3 T, R, δ C G + R MR T <, < R arcin δ C 3 R R, arcin δ R F 3 T, R, δ < π MR R arcin δ R < CRδ CR R T T R δ F 3 T, R, δ F 4 T, R, δ G + R < MR, T < T δ, < πr C 4 F 4 T, R, δ < π MR T δ πr < DRT δ DR R T δ T δ T R, δ lim T F 4T, R, δ

49 MeBio : 49 6 ζ π ζ ζ Weiertra orer genu Haamar Jenen ζ ˆζ ˆζ, ξ ˆζ ξ ξ ξ < Re < ζ 6 3 ξ ξ ρ ρ ρ ζ Imρ ρ Jenen 6 fz z < R log f π π log fre iθ θ Cauchy log f πi z R logz z z πi π logre iθ Re iθ ire iθ θ π π logre iθ θ Laplacian + y + i y log fz i y Cauchy Riemann log fz + log fz + i y fz, Riemann

50 MeBio : 6 ζ 5 i y fz log fz log fz fz 6 Jenen ρ, ρ,, ρ n fz z < R log f n log ρ j R j π π log fre iθ θ R ρz z R ρ ρ < R Rz ρ ρ R ρz z R Rz ρ zz ρ Rz ρ gz fz n k R ρ k z Rz ρ k R ρ > R gz z < R z R gz fz π π log fre iθ θ π π log gre iθ θ log g log f n log ρ j R j Haamar

51 MeBio : 6 ζ 5 3 ζ ξ ξ ρ ρ ζ ξ ˆζ π Γ ζ π Γ ζ π Γ + ζ ξ Γζ Γ, ζ ξ ζ π ξ Γ { π e γ + n π e γ { n n + n e n } e n } ρ ρ ρ ρ Re log ζ log π + γ log log + n log + n n + ρ log ρ ρ Imρ ζ ζ log π + γ + n + n n + ρ ρ + [ log ζ Imρ> [ log ] ] log ρ [ log ] + n + log ρ log + n n

52 MeBio : 5 7 Riemann π Π Li Imρ> { Li ρ + Li ρ} + Π π Li u uu log u log log π ρ ζ Riemann Riemann Π Möbiu 7 Π n n π n Π π + π + 3 π n π n + π + π + 3 π π π π. + 7 π Π π 7 k n µn Möbiu µk { g f n< n n \ n f µng n n< n n > n n p r pr pr m µk µ + {µp + + µp m } + {µp p + + µp m p m } + + µp p p m k n m + m C + m m

53 MeBio : 7 53 µng µn f n n< n< m< n n mn µnf f k k< n k mn k Σ k f µmg µmg g n n< n< mn k m< n k< m k 7 3 π µn n Π n n µn n Π n n µn n m m π nm µn k π k π k n k Π ζ p n p n log ζ log p n n p p p p n Π log n n p p p p n n np n n n π n log ζ Mellin Laplace e t log ζ Π Πe t e t t L[Πe t ] Laplace Πe t πi c+i log ζ e t c e t Π πi c+i log ζ

54 MeBio : 7 54 ζ Π πi πi c+i c+i log ζ { log π + γ log log + n log + n n + ρ log ρ } πi c+i log π + γ Γ Σ Π πi πi πi πi + πi πi πi c+i [ log ζ log log log log log log ζ log c+i c+i c+i c+i c+i ] c+i πi [ log ζ [ ] log [ log n Imρ> c+i ] ] log + n log ρ [ log ζ n ] + log log ρ Li 7 Li log ε lim ε + log + +ε log C + O ε + ε Re O π log ε + ε C Re

55 MeBio : 7 55 log ε C + > Im > C + log Li πi C > Im < C log Li + πi log log + log, n ρ, log α 7 fα πi Reα < α \, c c+i log α log log α i Imα > {α + it t > } log ii Imα < {α it t > } log α α Im Im α O c Re O c Re α Reα > fα C+ u α C u α log u u Imα > log u u Imα < C +, C 7 4

56 MeBio : 7 56 Reα < fα u α log u u fα α α log α α logα log α α α αα f α α πi πi πi log c+i log c+i [ [ αα log ] c+i αα log α πi ] c+i αα c ir, c + ir α f α α α log α [ ] α + α 3 3α 3 + α 3α α 4 lim α fα Re c > Reα i Reα > [ f α α u α α α ] C u α u fα u lim Imα C+ α log u u lim Imα C+ α log u ii Reα < [ f α α u α α α ] u α log u u + C lim fα 7 3 α u, lim u πi, lim u α u fα u Imα C α log u u πi, u Imα C α log u u u α log u u + C C lim fα 7 4 C α log log α nπi + α 3α lim fα nπi 4α4 α

57 MeBio : Reα > lim Imα C+ u α log u u lim Imα C+ α log u u lim Imα C+ α log u u, lim u πi, lim u Imα C α log u u πi, u Imα C α log u u α a + bi a >, b > b + z α log u I u C+ α log u C e z z C C3 z e z z z Im Im Im u z α log a log + b log i z α log C + C C 3 O Re O Re O a log Re α log C C a + bi a + bi +a + bi ez log a + bi z C C 3 z C a + bi log a + bi C 3 α log 3 [ e z z ]C C3 I eα log α log C C3 e z b C e z z z < e z C z [ ] e z z z z z < a log e α log b C3 e z z e + C C 3 C C3 z z e α log α log z R R e a log α log e a log a + b log z < e z C 3 z z e < a log a + b log π a + b log b lim b I 3 a a + b log a log e a + b log πe a log a + b log

58 MeBio : 7 58 u lim Imα C α log u u u lim Imα C+ α log u u πi, lim u Imα C α log u u πi 7 4 Reα < lim Reα u α log u u α a + bi a < a z α log u Im Im u C 4 z O α log a log + b log i Re O Re I a u α log u C4 e z [ e z z z z e α log α log a ] e + C 4 C4 z z z eα log α log + e a log a + b log C 4 C4 e z e z z z z z < e a log a + b log 7 5 πi log c+i [ log ] log πi log c+i [ log ] πi log [ log ] c+i πi c+i log log 7 6 πi log c+i 7 f πi log [ log c+i ] Li [ log ] C + log u u

59 MeBio : 7 59 log log log + πi f πi πi πi πi πi log log log log log c+i c+i c+i c+i c+i [ log ] [ log + πi [ log [ log [ log ] ] + ] + ] πi log log c+i ] c+i [ [ ] c+i Li C + log u u Li πi 7 7 πi log c+i 7 πi u ρ z C+ log ρ log c+i u ρ ρ log u u + log ρ log ρ z Liz ρ a + bi < a <, b > Li ρ + Li ρ C+ u ρ log u u log z z Liρ πi u ρ ep {ρ log u} ep {a log u + b log ui} u a {cob log u + i inb log u} 4, ρ + 4i u z z ρ ρ O O u u : ε z z : ρε z : ρε

60 MeBio : 7 6 πi u : + ε z z : + ρε ρ [ b log π ] b πi Li ρ u ρ u + πi log u C+ lim b Liρ πi Liz z Li [ ] b log n nπi π Li ρ b lim b Liρ nπi ρ log z ρ log z z z nπi nπi log log ρ ρ log ρ ρ ρ ρ ρ πi m log c+i 7 πi πi m m log log log + m c+i c+i m log + m log + m m uu log u u m u m log u u u 3 u log u u uu log u u 4 Riemann ρ ρ

61 MeBio : Π 7 9 Π Li Imρ> { Li ρ + Li ρ} + u uu log u log Li { Li ρ + Li ρ} Imρ> Riemann Riemann

62 MeBio : 6 8 Γ Gau, Legenre 8 n k ζ ep n k in k n π πi n in k n n π k ζ nn πi ep n n n n ζ k ζ k i nn n k ζ k i n ζ k ζ nn i n k i n ζ k n k n ζ k n k n + n n ζ k n k n in n n n f n X 4 n X n + + nx in n f n in i n f, n 3 f 4X 3 + 3X ii n k, k k 3 ink + + ink in k co ink + f k in in f k in n n π, n 3 n π,, n π,, n f n X X in n n 3 π, in n n 4 4 k in k+ + + {k k } in 4 k+ in k+ + + k + in n π, 4 n π, n n π,, in n π,, in n n n X n in n n π in n 3 n π in n π in n n π in 4 n π π n in n in n n π, in 4 n π, in n n π π n n 4 n

63 MeBio : 8 63 n in k n π k n n n n m n n k in k m n π m k k in in k m π m k m m π in k m k k m m π in k m π m k in k m m π co k m m m m n n k m π a n Dirichlet n σ c n σ c lim up n lim up n log n log n log a n + a n+ + log n n n n a k k n a k k Zagier 8 in in π [ lim ] ϵ + in + lim ϵ + ϵ ϵ in t t t π R π fz + R ϵ lim ϵ C π 8 3 e iz z ϵ R e iz z in co in C ϵ π D R in z z i ϵ z πi [ θ R e π θ π R e R θ] π π D eiz z z e iz z z < D e iz z π R e R lim R R ϵ > n T log[]! nn #{ >, n} 3 k σ k nσ k n n k eiz C ϵ R D in t t t z z π R in θ π z e R in θ e Rθ Rθ < R R D

64 MeBio : n φ Dirichlet n n n n n ζ ζ n ζ ζ ζ + n φn n + n n φn n φn n n n n n ζ n φn n Re > Re 6 π n< φn n 6 π

Z: Q: R: C: sin 6 5 ζ a, b

Z: Q: R: C: sin 6 5 ζ a, b Z: Q: R: C: 3 3 7 4 sin 6 5 ζ 9 6 6............................... 6............................... 6.3......................... 4 7 6 8 8 9 3 33 a, b a bc c b a a b 5 3 5 3 5 5 3 a a a a p > p p p, 3,