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, 5, 7,, 3, 7, 9, 3, 9, 3,...

.. a > p a p [ ] a bp, b p p cq, c, q, < q < p a bp bcq q a < q < p p p.. [ ] p, p,, p n A p p p n + A A p i p, p,..., p n A A A p. p A p A p i p p i p i+ p n + A p i p p i, i,..., n n p n p, p 3, p 3 5,....3. p n < n. [ ]. p n+ p p p n + p < 4 n n k n p k < k p n+ p p p n + < + + + n + n+ + < n+ + n+ n+.

π, 3, 5, 7 π 4, 3, 5, 7,, 3, 7, 9 π 8 π 5, π 68, π 9 π.4. π log log. [ ] 3 π > log log 3 > log log 3 n π p n < p n+, n.3 p n+ < n+ < n+ log < n+ log log log < n + log + log log n + log n + log + log log n + log + log + log log n + 3 log + log log < n π. s > ζs n s > + N n n + N s + n N n + s n n s. s d [ ] N s d + s s < + N s s. 3

n s n N n n N s n N n+ n [ s s d s ] N+ N+ s s d. N + s ζs s s > s ζs + s s > lim ζs, lim s + s ζs s + : ζs ζs.. f > f > n n k f + k f + fn d + n fk k n f d 4

[ ] g f g [ d g ] g + g g d g d g f + k g fk, g fk + f + k fk + fk + d f + k d. k n n k n k n k f + k f + fn d + f + k f + fn d + fk + fk + n fn + n fk k fk k n + f f d. + f + k d n k fk + f + fn + n k f + k d s > f s. B {} {} [] f s s n s s k f + k d n k n k + k s+ d s B + k d s + k s+ n n k B d, s+ B d + k s+ 5

n B s d + + n s+ s n k n k n k s k s n k n k d s [ ] n, s, s s [log ] n, s, k s s n s, s, log n, s, k φ n s s > n n ζs ks n s k ζs s φ ns s ϵ > n B φ n s s d + + n s+ s s n + + n s [ ] n + + n ϵ+ ϵ ϵ ϵ ϵ s n + + n. ϵ ϵ ϵ n s φ n s ϵ s δ φs φs s > s > φs ζs s φs s > n φ lim φ n lim n n k log n γ γ.57756649... k.. s > φs φ γ ζs s + φs γ 6

3 Γ Γ t e t dt 3. ϵ > t > t e t < t [ ] t e t dt < t dt ϵ ϵ t ϵ ϵ. > ϵ t e t dt <. ϵ t e t dt lim e t e t ϵ n ϵ t n n! t e t dt t > n e t > tn n! e t < n! t > tn n n > + t e t n! < t t n! t > n t n+ u u [ t e t n! n! dt < dt tn+ nt n ] u n! < n! n u n n. u t e t dt lim u u t e t dt 3. > 7

Γ e t dt lim u u e t dt lim u [ e t ] u lim u e u. Γ + Γ > 3. > u ϵ t e t dt [ e t t ] u u + t e t dt ϵ e u u + e ϵ ϵ + ϵ u ϵ t e t dt. n > + e u u < n! u u n e u u > ϵ e ϵ ϵ ϵ, u Γ + t e t dt 3. n > Γ + n + n Γ + n t e t dt Γ. + n + n Γ + n + n + n + Γ 3.3 Γn + n! Γ n!, Γn n! n n! 3.3 Γ Γ + n + + n + n > n,,..., n n Γ 8

3.. > f < <, t tf + tf f t + t y f, f, f y f t + t : f a + b f a + b + c a > 3.. > f g f+g f a >, b fa + b [ ], >, t. h f + g th + th t f + g + t f + g tf + tf + tg + tg f t + t + g t + t h t + t. h fa + b th + th tfa + b + tfa + b f ta + b + ta + b fa t + t + b h t + t. 9

3.3. > f log f f f g 3. log fg log f+log g fg Γ 3.4. a, b >, p, q >, p + q p ap + q bq ab a p b q [ ] > f q q + p ap a f q a a /q f f a /q q q /q p f a /q q a q q + p ap aa q q ap + p ap a p p + q a p. f a /q 3.5. p, q >, p + q [ ] f p d, f p p d g q q d f p f, g > g q d fg d. f p p d, g q g q q d a f f p, b g g q

3.4 f p p f p + g q p q g q q fg f p g q. p f p p f p d + q g q q g q d f p g q fg d. p f p f p p + p q a q g q q q p + q f p g q f p g q fg d, fg d. 3.6. Γ [ ], >, p >, q >, p + q 3.5 Γ p Γ q t p e t p p p dt t p e p t t q e q t dt t p + q e t dt Γ p + q. t q e q t q dt q p log Γ + q log Γ log Γ p Γ q log Γ p + q. Γ + Γ Γ 3.7. > f 3 f Γ f + f.

f 3 f. [ ] Γ 3 3 n fn n! < f Γ > f Γ < f Γ < n + n n + n + n log fn + log fn + log f + n, log fn + log fn log f + n log fn, log fn + log fn n + + n + + log f + n log fn. log fn + log f + n log fn, + + log fn + log f + n + log fn, log fn log fn log f + n log fn log f + n log fn fn n!, fn + n! logn log f + n logn! log fn log fn. log fn + log fn. log n, n n! f + n n n!. 3.4 f + n + n f + n + n + n f + n + n + f. 3.4 + n + n + n n! + + n f n n! + + n n n! + + n + n + n n.

n n n + n n! + + n f n n! + n + + n n. n + n f n n! + + n f. n n + n lim n n n! + + n f Γ f f Γ Γ lim n n n! + + n 3.8. [ ] Γ lim n n n! + + n. 3.7 < Γ lim n n n! + + n Γ n n n! + + n Γ n + n + n! + + + n + n n! n + + n + n + Γ n n + n +. lim n Γ n Γ lim n Γ n + Γ Γ + > lim n Γ n Γ <,,,,... 3

3.8 n e log n n n! Γ n + + n e log n n e + e log n n e + e + ne e + n + n e n +. n lim + + + n log n γ n lim n e log n n e γ Γ e γ lim n n ν e ν + ν e γ e ν +. 3.5 ν Γ 3. > Γ > log Γ 3.5 n e ν log Γ γ log + log lim n + ν ν n e ν γ log + lim log n + ν ν n e ν γ log + lim log n + ν ν n γ log + lim n ν log +. ν log Γ γ log + ν ν ν ν ν log +. 3.6 ν γ + ν γ ν + + νν + < r r > νν + < ν r ν 4 ν

r ν ν r ν r + + 3 + < r < r d d log Γ γ + ν ν + r > > d d log Γ γ + ν ν + log Γ Γ e log Γ ν ν Γ Γ d log Γ, d Γ Γ d d log Γ γ + ν. 3.7 ν + ν + ν + ν ν + > ν d log Γ d d d Γ Γ + ν ν ν +. log Γ d k Γ k k! 3.8 d k Γ ν + k 5

4 sin c Γ > + f c Γ Γ logc log logc c 3.6 3. Γ Γ + f f c Γ Γ + f + c Γ Γ + c + Γ Γ + c Γ Γ f 3.7 f Γ 4.. c Γ + Γ Γ c Γ. sin φ ΓΓ sin π 4. φ + φ. 4. φ + Γ + Γ sinπ + π ΓΓ sin π Γ Γ sin π ΓΓ sin π φ. Γ Γ c Γ. 4.3 6

4.3 + φ φ Γ Γ sin π + Γ + Γ Γ Γ Γ c c Γ Γ sin π c φ. Γ π + sin sin π π cos + φ φ c φ 4.4 Γ sin φ 3. sin π Γ + φ Γ sin π Γ + Γ π π3 3! + π5 4 5! φ π φ φ m φm π φ 4.4 4. < < φ > 4. φ g d log φ d φ g 4.4 + log φ + log φ log c + log φ. g + g 4 + g 4.5 g M > g M g g M 4.5 g g + + g M 4. 7

n,,... g M n g g g log φ log φ log φ log φ φ φ π π φ π 4.. 4.3. [ ] 3.5 4.4. Γ [ ] 4. π. sin π ΓΓ sin π π ν π ΓΓ Γ e γ π sin π.. ν Γ e γ sin π π ν ν π ΓΓ. e ν +, ν ν e ν ν. ν 4. / Γ π sin π π. > Γ > 4.4 4. + π Γ Γ Γ 4.6 8

5 ζ 73 ζ n n + + 3 +.644934 735.64493466848643647 748 ζ π 6. ζ4 π4 π6 π8, ζ6, ζ8 9 945 945,, n, 4, 6, 8,..., 6 ζn 75 n ζn π n 4.3 sin π 4.3 3 π ν πζ ν sin π n sin π n +! πn+ π 3! π3 3 + 5! π5 5 + n 3 πζ 3! π3, ζ π 6 4.3 5 π µ ν π π ζ ζ4 ν ν 4 µ>ν ν ν sin π 5 π ζ ζ4 5! π5, ζ4 π4 36 π4 6 π4 9. 9

6 6. sinnπ, cosnπ n,,... f f a + f + f. a n cosnπ + n b n sinnπ 6. a n, b n 6. f f f f 6. f a + a n cosnπ n n b n sinnπ n f f + f a + a n cosnπ n a n a cosnπ d f d a. a + a d + [ nπ sinnπ ] a n cosnπ n n, a n cosnπ d cosα + β cos α cos β sin α sin β, cosα β cos α cos β + sin α sin β, cosα + β + cosα β cos α cos β

n, m cosnπ cosmπ d { cos n + mπ + cos n mπ d, n m,, n m. f cosmπ d a + a n cosnπ cosmπ d n a cosmπ d + n a m. a n cosnπ cosmπ d a a m f d, 6. f cosmπ d, m,,... 6. f ˆfy ˆfy fcos πy + i sin πy d. 6.3 i f f f sin πy ˆfy f cosπy d t f e πt f ˆfy e πt cosπy d

ˆfy d dy ˆfy d e πt cosπy d dy πe πt sinπy d πte πt sinπy d t [ ] e πt sinπy e πt cosπyπy d t t πy e πt cosπy d πy ˆfy. t t d y e πt dy πy e πt y t ˆfy e πt y ˆfy Ce πt y, C y C ˆf e πt d u/ πt 4.4 C πt u e u du πt Γ e πt d.. t f e πt t > ˆfy t e πt y 6.4 θt θt e πtn + e πtn 6.5 n n f e πt g f + n n g fn e πtn θt 6.6 n n

g n m f g n f + n m f m m g g m a m a n g a + g d a m cosπm m n f + n d f d ˆf. g cosmπ d n n+ n n f + n d n n f + n cosmπ + n d f cosmπ d f cosπm d ˆfm. g ˆf + m g ˆf + n+ n ˆfm cosπm. m f + m g. f d f + n cosmπ d ˆfm. 6.7 6.4 ˆf t, ˆfm t e πt m 3

g t + 6.6 6.8 6.. e πt m θ t m θt t θ. t. 6.8 t 6.3 θt Ψt n e πtn θt + Ψt Γs s e d s > n πn t d πn dt Γs s s/ π s n s πn t s e πnt πn dt π s n s t s e πnt dt. s Γ π s n s t s e πnt dt. π s s Γ n t s s e πnt dt. n π s Γ s ζs n t s e πnt dt t s n e πnt dt t s Ψt dt 4

, ] [,, ] t /u t s Ψt dt u s + Ψ u du u u s Ψ du. u 6. + Ψu θu u θ Ψ s > π s Γ s ζs + + Ψ u u u Ψu + u u. t s Ψt dt + t s Ψt dt u s Ψ du + t s Ψt dt u + s > π s s Γ ζs u s u Ψu + u u s Ψu du + t s Ψt dt [ t s Ψt dt + s + t s Ψt dt du +. u u s du u s + t s s + t Ψt dt + s s s s. t s s + t Ψt dt t s s + t ] t s Ψt dt Ψt dt t s s u s du [ s u s ] 6.9 s s s 6.9 5

6.. ζs + ζs π s Γ s ζs s s s π s Γ s ζs ζ s ζs π s sγ s s Γ ζ s. π s sγ s s Γ ζs π s sγ s Γ 4. s s Γ s s Γ Γ s s Γ π sin πs ζ s. 4.6 sγ s s Γ π sγ s s π Γ s s π s π sin πs ζs s π s ζs π s sin πs π Γ sζ s, s Γ s ζ s. 7 s > N > N p P N s p N p s 6

< p s < p s k p s k + p s + p s + p 3s + P N s N n n s < P Ns N n N lim P Ns ζs N n < s n ζs. s n 7. ζs. ζs p p s s > p 7. s > log ζs log lim P Ns lim log P Ns N N lim log p s N p N p log p s. log log log ζs p p k k k k < k p ks p + s p k k p. ks 7

s < p k k p ks p p n p ks p k k p k p p pp p nn n n. n < log ζs p p s < s > P N P N n N < log P N p N p < N n > N+ n d logn + > log N n 7.. p N p p > log log N N >. p 8 896 8.. log e.788... π. log 8

t e u Li Li dt log t log log t dt e u log u du. e u + log Li log u + u k du k! k [log u] log log + [ ] u k log kk! k log log k log log + + c. kk! k c log log k u k k! log log log k k u du + kk! k log u k [ ] t Li log t dt t dt log t log t t log log + log t dt. log t dt log t dt + log t dt log dt + log + log log + 4 log log + 4 log 9 log dt log log C log. k! du

C elog + 4 log log Li log log log log + C log, Li / log log + C log. Li / log 8.. 8.. π Li. 9 n n! zn + z +! z + 3! z3 + z + iy C z e z e z n n! zn. e z +z e z e z z, z C z + yi, y R e yi n n n! yin n n n! yn + i cos y + i sin y. e z e e yi. n! yin + n n n n +! yn+ n +! yin+ 3

e z e cos y + i sin y, e z e z + iy s s σ + iτ σ, τ R t t s e s log t t s e σ log t+iτ log t e σ log t t σ s σ + iτ σ > σ σ n n s n n σ n ζs n n s n σ ζσ Rs σ σ N n N s n n s log n e s ζs Rs > σ σ > ζs Rs > s σ + iτ < σ < σ σ σ σ t s e t dt t s e t dt t σ e t dt t σ e t dt t σ e t dt t σ e t dt t s e t dt σ Rs σ R ϵ t s e t dt 3 t σ e t dt Γσ, t σ e t dt Γσ

s Γs t s e t dt σ < Rs < σ s σ < σ Γs Rs > n Γs Γs + n ss + s + n s + n Γs Rs > n s,,..., n n Γs C s,,,... 4. s ΓsΓ s π sin πs s Γs /Γs C ζs 6.9 t s s + t Ψt dt t C ξs 6.9 ξs ss ζs ss C ss π s s Γ ζs 9. t s s + t ξs ξ s ζs π s s Γ ξs + s Ψt dt t + 9. ζs C s lims ζs s 3

s ζs ζs p p s Rs > Rs > ζs Γs ξs Rs > ξs ξ s ξs Rs < ξs Rs ζs Newman / log Hadamard de la Valée Poussin 896 ζs Rs Mertens 949 Selberg Erdös D. J. Newman Tauber Cauchy lim f/g f g C f C g f Og π.. π 3 log ζs n n s, Φs p log p, ϑ log p s C, R p s p p ζs Φs Rs > δ > Rs + δ n s e s log n e Rs log n n Rs n +δ. 33

n n s n + +δ n n n + n + lim R n R n +δ d + +δ d + lim +δ R +δ d [ δ δ log < ] R + δ. log δ/ < δ/, log < δ δ/ Rs + δ p log p p s p δ log p p Rs p p p +δ/ δ log p p +δ n δ p n +δ/ p δ/ p +δ δ. ζs Φs Rs >.. Rs > ζs p p s [ ] ζs ζs r 3 r 3 p rs Rs >. s p s p r p r,r 3,.3. ζs Rs > s [ ] Rs > R d lim s R ζs s n n+ n n s n [ d lim s R s s ] R d s n s n s s 34 n n s n+ n s d d..

Rs > [ s du ] us+ u s n n n n + s s n n+ n n n+ d s s n n n+ n ma n u n+ s du d us+ s du d u s+ s u s+ n s n Rs+. n s n Rs+.4. ϑ O. [ ] <. Rs > n n + n n r n r n n n < p n p n n n n n<p n p n n n<p n p log p log p ϑn ϑn n<p n n log log n<p n n log p ϑn ϑn n n<p n [/] n n < n +, [] n n + { ϑn ϑn, ϑ ϑ/ ϑ[] ϑ[/] ϑn + ϑn, ϑ ϑ/ ϑn ϑn + log n log + log log + log. 35

C > log C log C log ϑ ϑ/ C r / r > / r+ ϑ/ k ϑ/ k C k,,..., r + k ϑ ϑ/ r+ C r+ k C k k C, k ϑ C + ϑ/ r+ C + ϑ C + O..5. Rs ζs Φs s [ ] ζ s ζs p Rs >. ζs log p p s p log p p s + p log p p s log p Φs + p s p log p p s p s. Rs >.3 ζ s/ζs Rs > s ζs Φs Rs > s ζs ζs φs s Rs > φs ζ s ζs s + φ s s + φs s φ s s + s φs lim s s ζ s ζs s + iα α R, α ζs µ s + iα ν.3 µ, ν.6 s iα µ s iα ν ζ s Φs s ζs s ± iα µ s ± iα ν lim ϵφ + ϵ, lim ϵ + ϵφ + ϵ ± iα µ, lim ϵ + ϵφ + ϵ ± iα ν. ϵ + 36

r 4 Φ + ϵ + irα + r p log p p +ϵ p iα/ + p iα/ 4 6 8µ ν 6 8µ + ν 8µ µ ζ + iα Φs ζ s ζs p log p p s p s ζ s ζs s Rs Φs Rs s.6 Schwarz. D fz D D f z fz [ ] F z f z fz F z z D fz F z f z fz fz F z fz f z fz.7. ft t gz fte zt dt Rz > Rz ft dt g.8. ϑ d [ ] n n p λn log p λn λn ϑn ϑn Rs > 37

log p p s λn n s Φs p n ϑn ϑn ϑn n s n s n n ϑn ϑn n s n + ϑn s n n n+ s ϑn d s s+ s s n n ϑ s+ d et e st ϑe t dt. n n+ n n ϑn n s n s n + s n ϑ d s+ ft ϑe t e t t, gz Φz + z + z Rz >.5 Φz + z hz + z gz z + hz Rz z hz z + z + Rz Rz > fte zt dt.7 ft dt.9. ϑ. ϑe t e t e zt dt ϑe t e z+t dt [ ] z e zt Φz + z + ϑe t e t dt e zt dt Φz + z + z gz. ϑ d 38

ϑ [ ] lim sup λ > ϑ λ ϑ t ϑt ϑ λ λ ϑt t t dt λ λ.8 < ϵ < λ λ u u λ t dt t u t λ u du >. u ϑt t t dt ϵ du R > y > y > R ϵ < y y ϑt t t dt < ϵ > R y, y λ ϵ < λ λ u u du λ lim sup ϑ lim inf ϑt t t ϑ dt < ϵ λ < ϑ λ ϑ t ϑt ϑ λ λ ϑt t t dt λ λ λ t t dt t u λ u u du <..8 ϵ < ϵ < R > y > y > R ϵ < y y ϑt t t dt < ϵ > R y λ, y ϵ < λ λ u u du 39 λ ϑt t t λ dt < ϵ λ u u du

ϑ lim inf lim sup ϑ. ϑ lim inf ϵ > ϑ ϑ ϑ lim lim sup ϵ <p ϑ p log p π log lim ϵ <p π log lim inf π log log p p ϑ log π log, ϵ log ϵ π π ϵ log ϑ ϵ + π ϵ log lim ϵ > ϑ ϵ + log. ϵ ϑ ϵ + log ϵ ϵ lim ϑ + ϵ lim inf π log lim sup π log, lim π log.7 T > g T z T fte zt dt g T z z lim g T T ft dt g R δ > R {z C z R, Rz δ} C gz C C 4

C + C δ R C g g T πi C gz gt z e zt + z R C + C {Rz > } gz gt z e zt + z R z B R, dz z B ma t ft z C + gz g T z fte zt dt B e zt dt T T [ B e Rzt dt B Rz e Rzt T Be RzT Rz Rz >. z Re iθ ezt + z R z erzt + z R z erzt + e iθ e iθ R e RzT e iθ + e iθ z + z RzT Rz erzt e R R R B R π gz gt z e zt + z dz C + R z B π R πr B R. 4 ] T

C C {Rz < } gz g T z g T z C C {z C z R, Rz < } z C g T z B T T T fte zt dt B Be RzT Rz e Rzt dt B ezt + z R z Rz <. e zt dt [ Rz e Rzt erzt Rz R B R πi g T ze zt + z C R dz z πi C ] T g T ze zt + z R dz z B R. C gz gze zt + z dz C R z T gz + z R z T C {Rz } C gz + z R z M gze zt + z C R dz z C M e zt dz M e RzT dz C MRe δt + M MRe δt + M arcsinδ/r arcsinδ/r e RT sin θ R dθ e RT θ/π R dθ [ MRe δt + MR π ] arcsinδ/r RT e RT θ/π MRe δt + πm e RT arcsinδ/r/π T MRe δt + πm T T. 4

g g T gz gt z e zt + z dz πi C R z gz gt z e zt + z dz πi C + R z + gz gt z e zt + z dz πi C R z B R + g T ze zt + z dz πi C R z + gze zt + z dz πi C R z B R + B R + MR π e δt + M T. lim sup g g T B T R. R R lim g g T T 43