黎曼-傑塔函數與伯努力數的關聯

Å C2`>×çbç`>çÍ

_î=d

Nû`¤

:

Ù−p

²=

-î•

_²=

-–

_{^*ƒbD+›‰bíÉ:}

û˝Þ

_:

_{˜/X ª}

2

M ¬ Å

_{þ  ý ~}

¿b

…û˝[N¬©/cb

_{4Ÿ¸Økƒb5ÞAƒb}

,

yŸ

Ô„-

–

_{^*ƒbD+›‰b}

,

1㯬 û˝-

-

^*ƒbD+›‰bíû˝A‹

,

Ôb_É5!…4”

,

½h8Jn

zp

_,

UŸ…ú€ç67k

_,

Ÿ…uŸÞ/T¿íï

,

?Ñ5Z‰

,

1ø-

–

_^*ƒbD+

›‰b8Jq“

,

Uۍ

6è<f¡ç3
/…û˝ÒN¬-

–

_^*ƒb

,

©/cb4Ÿ¸5

Øk

_ƒbDÞAƒb

,

_{Ô„+›‰b}

,

ª7‚à©/cb

_4Ÿ¸Økƒb

,



-

–

_^*ƒbD

+›‰b5óÉ4
Ĥ;W

ζ(2k) = (−1)k−122k−1B2kπ 2k (2k)! , k ∈ N,

£

S_2k0 (−1) = (−1) k−1_(2k)! 22k−1_(π)2k ζ(2k), S 0 2k+1(−1) = 0,

…û˝êÛ

_,

J-

_–

_^*ƒbÑ›a

_,

ªø

S_2k0 (−1) = B2k, B2k = 1 2k + 1 ( C_2k2k+1S₁0(−1) + k X i=1 C_2i+12k+1S_2k−2i0 (−1) ) , k ∈ N; Bk= S−10 (−1),

w2

_{, S}0 k(x)

u

©/£cbŸj¸5Økƒb

Sk(x)

íø¼ûb

, k ∈ N

Éœå

:

©/£cbŸj¸

,

-

–

_^*ƒb

,

_+›‰b

Abstract

This research hung over from the extended functions for the sum of powers of

con-secutive integers, we colleted the literatures of the related research about the functions of Riemann zeta and Bernoulli Number, both newly interpreted and predigested the

properties of the functions of Riemann zeta and Bernoulli Number. Thus we built the

connection between the functions of Riemann zeta and Bernoulli Number, according to

ζ(2k) = (−1)k−122k−1B2kπ 2k (2k)! , k ∈ N, and S_2k0 (−1) = (−1) k−1_(2k)! 22k−1_(π)2k ζ(2k), S 0 2k+1(−1) = 0,

Take the function of Riemann zeta as bridge, we find that

S_2k0 (−1) = B2k, B2k = 1 2k + 1 ( C_2k2k+1S₁0(−1) + k X i=1 C_2i+12k+1S_2k−2i0 (−1) ) ,

where S_k0(x) denotes the first derivative of Sk(x) for each positive integer k.

ñŸ

øı

é

. . . (1) ø û˝*
. . . (1) ù û_{˝ñíD&½æ . . . (2)} ú ¯Uì2 . . . (3) û û_{˝Ì„ . . . (4)}

ùı

d.«n

. . . (5) ø -–_{^*ƒb5½b4” . . . (5)} ù _{+›‰b5óÉû˝ . . . (16)} ú _{4Ÿ¸ØkƒbD-–^*ƒb5É: . . . (32)}

úı

3b!‹

. . . .(45) ø -–_{^*ƒb5½bô.4” . . . (45)} ù _{+›‰b5Ôy4” . . . (64)} ú _{:û-–^*ƒbD+›‰bíÇø¤ . . . (74)}

ûı

!D‡

. . . (81) ø ! . . . (81) ù _{‡ . . . (81)}

¡5d.

. . . (83)

øı é

ø

û˝*

Ék-

–

^*

(Riemann-zeta)

ƒbD+›‰b

(Bernoulli number)

5«n

,

â

k-

_{“;BDEu„jíØæ}

,

ĤӼ7-

–

_{^*ƒb5ªè4£‘D4}

,

]v

Vù–Öû˝6íE

,

/øòuû˝½-

,

û˝6ÕÕâ.°i8J~p1Ô

„

_,

¨ÿrÖO±íû˝

,

6û|7'Ö±ít

_,

à

ζ(n) = ∞ X n=1 1 nm = (2π)2mBm 2(2m)! , ζ(−n) = Bn+1 n + 1, x ex_{− 1} = ∞ X n=0 Bn xn n!, |B2n| ∼ 4 √ πn( n πe) 2n_{· · · ,}

Ĥ#AT¿.q7jí>g
óú7k©/cb4Ÿ¸uœñq/Ñ×VFQ§í

,

N©/cb

_{4Ÿ¸5ØkƒbDÞAƒb}

_,

ø-

_–

_{^*ƒbD+›‰b©!}

_,

u´

ªv|óɤ

,

@

_vuM)‚&í

úk¬ íóÉû˝

,

r…

(Euler)

ílk

1730

B

1750



ÈT|-

–

_^*ƒ

b

_,

U)-

–

_^*ƒbíû˝

,

Ê

_ƒb

₍

ԁucb

)

 b

_{ _ƒb žÆƒb}

rÖÜ

,

)ƒ7ªø¥íê

,

1k

₁₇₄₉

„p7

ζ(n) = ∞ X n=1 1 nm = (2π)2mBm 2(2m)! .

r…‚à7r…

-

_ïsŠ°¸t

, sin x = x ∞ Y n=1  1 − x 2 n2_π2 

¥_IA˚+íä

_,

‹Jùûô

_.

,

«˝74b¸-

–

^*ƒbíÉ[

,

1‹p

₊

›‰bíh1

,

_{!‹„|7úF£Xb-}

–

_^*ƒbíƒbM

,

éA.Šıqw˚

05n

,

Or…1Ý°|7-

–

_^*ƒb5FƒbM

,

úk

_n

ÑJb£

n < 0

ví

8”

,

†³j²

,

.

¬r…¢ù|-

–

_{^*ƒbíÇø_½bTà}

,

_¹Ñ-

–

_^*

ƒb„p”b}Óí4”

,

1

„p!k-

–

_{^*ƒbír…0}

(1.1),

⤪ør

…úbçíõ.

ζ(z) = ∞ X n=1 1 nz = ∞ Y k=1 (1 − P_k−z)−1_{, Re(z) > 1.} (1.1)

Ék+›‰bí–Ä

,

r…êÛ7£cbí

_Jbj¸

P∞ n=1 1 n2 ,

/yªø¥R

i

∞ X n=1 1 n2m = (−1)m−1(2π)2mB2m 2(2m!)

¥–Ä=+›‰k

1775

û˝

Bk

¸

Sk(n)

íÉ[

,

v|

Bk = lim n→0 Sk(n) n

Ó(

,

_{E–7+›‰bD©/cXb4Ÿ¸t5û˝
¡Ø#Ì †Åž -î•}

rÙ&

(2007)

©/cb

_{4Ÿ¸Økƒb5ÞAƒb}

,

N

_¬Z‚s}&

_,

½hÔ„-

–

^*ƒbD+›‰b
Ø#ÌA5û˝l|©/cb4Ÿ¸ØkƒbíÞAƒb

,

9õ,

_,

_{D+›‰bO”Ñò~íÉ[
ÇÕ}

_,

㯬 û˝-

-

_{^*ƒb³íû˝}

A‹

_,

.ØêÛ©/cb

_4Ÿ¸Økƒb

_,

Ê-

_-

_{^*ƒbD+›‰bíl,}

_,

rÆ

óç

_{½bíiH
…û˝!kô/Ø#ÌA5û˝}

_,

Ò‚à©/cb4Ÿ¸Øk

_ƒb

ћa

,

¿p«n-

-

_{^*ƒbD+›‰b5É:}

ù

û˝ñíD&½æ

!k,Hû˝*

,

…û

˝ÒN¬-

–

_^*ƒb

,

©/cb

_{4Ÿ¸5ØkƒbD}

ÞA

_ƒb

_,

yŸ

_{Ô„+›‰b}

_,

«n-

–

^*ƒbD+›‰b5óÉ4

,

ÖÍóÉí

H.lwb

,

Oú€ç67k

,

…bEu

_}ŸÞ/T¿

,

…û

_{˝v%âÉd.5}

½hûn

_,

1*wH2&â2Þh

,

ª7´Ç…bíÿ˜ÞÔ

,

7ø-

–

_^*ƒbD

+›‰b8Jq“

,

U€ç6è<f¡ç3
ã…û˝5û˝ñí

,

…û˝5&

½æà-

:

Ô

«n-

_–

_{^*ƒb5½b4”}

Ž «n+›‰b5½b4”

¡ ‚à©/cb

_4Ÿ¸Økƒb

,



-

–

_{^*ƒbD+›‰bÈ5É:}

ú

¯Uì2

…û˝5®2

,

cUàíóɯUì2à-

,

_{7®2F£5ÔyƒbCÖ}

Ô¯U

_,

†k®2y8JÌH£ì2

ø

_{R :}

[ýõbFA5Õ¯

ù

_{Z :}

[ýcbFA5Õ¯

ú

_{N :}

[ýAÍbFA5Õ¯

û

_{C :}

[ýµbFA5Õ¯

ü

_{Re(z) :}

[ý

_z

íõb¶M

ý

∞ :

[ýÌÌ×

þ

P∞ k=1ak :

[ý

P∞ k=1ak = a1+ a2+ · · · + a∞, k ∈ N

ÿ

ζ(z) :

[ý-

–

_^*ƒb



_{Sk(n) :}

[ý©/cb4Ÿ¸

,

w2

_{n ∈ N}



Sk(x) :

[ý

Sk(n)

5Øk

ƒb

,

w2

x ∈ R

ø

limx→af (x) :

[ýç

x

¡

a

v

f (x)

5

M

ù

d dxf (x) :

[ýƒb

f

5}

ú

R f (x)dx :

_[ýƒb

f

5 }

û

Γ(z) :

[ý

Gamma

_ƒb

ü

(a, b) :

_[ý.¨Ö

a

¸

_{b ,}

O

_k

_a

¸

_b

5ÈíFõb

ý

[a, b] :

_[ý¨Ö

a

¸

_b

£

_k

_a

¸

_b

5ÈíFõb

þ

Bm :

[ý+›‰b

, m = 0, 1, 2 · · ·

ÿ

P :

[ý£”bFA5Õ¯

û

û˝Ì„

…û˝5û˝Ì„à-

:

ø …û˝!kû˝qe5Ì„

,

úkªJN

_{¬bM}&‹Jð„5bM!‹}

_,

.d¿

p«n

ù …û˝!kû˝j¶5Ì„

,

úkìÜí„p

,

JsJ,5„pj¶v

,

†

¦

_{œq/R²ƒbj&턶}

_,

w…íj¶†.Ö‹

_ÅH

,

JTòdíø_

4Dªè4

ú …û˝!kû˝vÈíÌ„

,

úk-

–

_{^*ƒbíµ‰ƒbj&}

,

_ÉdÀ·H

„p

,

7.d¿p«n

,

/…û

_˝2F£5ƒb

_,

wì

_{2£M׶MÌÌ„}

Êõb¸ˇq8Jn

ùı d.«n

Ñ7«n-

–

^*ƒbD+›‰b5ÈíÉ:4
…ı¡5b¹Ék-

–

^*

ƒb +›‰b£©/cb4Ÿ¸5d.

,

}úªWn

,

…

_{ı2F×Û5bçRû}

¬˙ÌZŸCùAŸO
øÑ-

–

_{^*ƒb5½b4”}

,

_{ùÑ+›‰b5ó}

Éû˝

,

úÑ©/cb

_{4Ÿ¸5Økƒb-}

–

_{^*ƒbD+›‰b5óÉû˝}

ø

-

–

^*ƒb5½b4”

-

_–

_^*ƒbøOJ

_ζ(z)

V[ý

_,

?

_¹

ζ(z) = ∞ X n=1 1 nz, Re(z) > 1.

…Ô

ζ(z)

íb_½b4”

,

1‹Jùû„p

,

4ӈ-

: 4_{” 2.1.1. ζ(2m) =}P∞ n=1 1 n2m = (2π)2mBm 2(2m)! , m ∈ N.

„p

:

r…

(Euler)

k

₁₇₄₉

úF

_Xb

n ≥ 2

l

ζ(n) ,

FílT|

sin x = x ∞ Y n=1  1 − x 2 n2_π2  , sin x = x  1 − x 2 π2   1 − x 2 4π2  · · · , x 6= kπ, k ∈ Z,

log | sin x| = log |x| + ∞ X m=1 log     1 − x 2 m2_π2     .

Má}

cos x sin x = cot x = 1 x + ∞ X m=1 2x x2_{− m}2_π2, x cot x = 1 + ∞ X m=1 2x2 x2_{− m}2_π2, πx cot πx = 1 + ∞ X m=1 2π2_x2 π2_x2_{− m}2_π2 = 1 + ∞ X m=1 2x2 x2_{− m}2.

à‹

_{|x| < 1 ,}

úkL<

_{m ∈ N ,} x2 m2_{− x}2 = x2 m2 1 − _mx22 = ∞ X n=1 x m 2n ,

)ƒ

∞ X m=1 x2 m2_{− x}2 = ∞ X m=1 S2mx2m,

w2

S2m= ∞ X n=1 1 n2m, m ∈ N,

FJ

πx cot πx = 1 − 2 ∞ X m=1 S2mx2m.

;W,Hj¶

,

‚à

_{x cot x}

í¶}}Ç)ƒ

πx coth πx = 1 + 2 ∞ X m=1 (−1)m−1S2mx2m, (2.1)

y‚à

x coth x = x e2x_{− 1} e 2x_{+ 1
,}

w2

x e2x_{− 1} = 1 2  2x e2x_{− 1}  = 1 2 " 1 − x + ∞ X k=1 (−1)k+1Bk2x 2k (2k)! # ,

/

e2x+ 1 = 2 + ∞ X k=1 1 k!(2x) k ,

x coth x = 1 + ∞ X m=1 (−1)m−12 2m_B m (2m)! x 2m_, πx coth πx = 1 + ∞ X m=1 (−1)m−12 2m_B m (2m)! (πx) 2m , (2.2)

â

_(2.1)

£

_(2.2)

ø

S2m= (2π)2mBm 2(2m)! , m ∈ N,

Ĥ

∞ X n=1 1 n2m = (2π)2mBm 2(2m)! .  4_{” 2.1.2. ζ(z) =} P∞ n=1 1 nz = Q p∈P(1 − p −z₎−1 , Re(z) > 1 , P

[F”bFAÕ

¯

„p

: ζ

_ƒbír…0

ζ(z) = ∞ X n=1 1 nz = ∞ Y k=1 (1 − P_k−z)−1_{, Re(z) > 1.}

l

_5?¶}

Pm = m Y k=1 (1 − Pk−z) −1 = m Y k=1  1 + 1 Pkz + 1 Pk2z + · · ·  ,

¢

1 P1a1zP2a2z· · · Pmamz = 1 nz,

w2

_n

Ñ£cb

_,

/

_{n = P1}a1_{· · · P} mam, ai ≥ 0.

Ĥ

Pm = X 1 1 ns,

FJ

ζ(z) − Pm = ∞ X n=1 1 nz − X 1 1 nz = X 2 1 nz,

]

|ζ(z) − Pm| ≤ X n>Pm 1 nz,

ç

_{m → ∞}

v

_{, P n}−s

_Y¹k

_0,

_FJ

_P m → ζ(z), ζ(z) = ∞ Y k=1 (1 − PK−z) −1 .  4_{” 2.1.3. ζ(z)Γ(z) =}R∞ 0 xz−1 ex₋₁dx ,

w2

Γ(z) = R∞ 0 x z−1_e−x dx , Re(z) > 0

„p

:

íl„p

ζ(z) = R∞ 0 xz−1 ex₋₁dx Γ(z) , Γ(z) = Z ∞ 0 xz−1e−z_{dx, Re(z) > 0.}

úL<£cb

_n

£µb

z = λ + iσ ,

â

_|n−z_{| = |n}−λ_|

_{£ªœö¹¶ø−̤b}

P∞ n=1n −z

_Ê

Re(z) = λ > 1

v

,

u

"úY¹
ÇøjÞ

, ζ(1) = P∞n=1n −1

_uøêà

b

,

_¹

limz→1ζ(z) = ∞ ,

¥6[ý

ζ(z)

Ê

z = 1

ø”õ
ç

Re(z) > 1 ,

â

1 z − 1 = Z ∞ 0 t−zdt = ∞ X n=1 Z n+1 n t−zdt, z 6= 1,

ªø

ζ(z) = 1 z − 1+ ∞ X n=1  n−z− Z n+1 n t−zdt  = 1 z − 1+ ∞ X n=1 Z n+1 n (n−z− t−z)dt.

yq

φn(z) = Z n+1 n (n−z− t−z_)dt,

/

φn(z) = ∞ X n=1 φn(z).

Ĥ

_n−z_{− t}−z

_ú

_t

_í}u

z tz+1 ,

7

|φn(z)| ≤ max n≤t≤n+1    z tz+1   ≤ |z| tλ+1.

]

_φ(z)

Ê

_{Re(z) = λ ≥  > 0}

v

_,

6"úY¹
à¤Ê

_{Re(z) > 0}

v

_,

†

_{ζ(z) =}

1 z−1+ φ(z)

5?¼ ƒb

Γ(z) = R∞ 0 x t−1_e−x_dx (Re(z) > 0) ,

¥â }F¿|

í

_{¼ ƒbÊ£cbí¦M¹uøOí}

_{n! ,}

_¹

_{Γ(n + 1) = n! , n}

u£cb

ú

_{Re(z) > 1 ,}

†

Γ(z)ζ(z) = ∞ X n=1 n−z Z ∞ 0 xz−1e−xdx = ∞ X n=1 Z ∞ 0 ˜ xz−1e−n˜xd˜x (x = n˜x) = Z ∞ 0 xz−1 ex_{− 1}dx.

FJ

ζ(z) = R∞ 0 xz−1 ex₋₁dx Γ(z) , Γ(z) = Z ∞ 0 xt−1e−x_{dx, Re(z) > 1.}  4_{” 2.1.4. ζ(s) =}P∞ n=1 1 ns = 1 2− 1 s+1+ Pq k=1 [s]2k−1 (2k)! B2k− [s]2q (2q)!aq(1, s) (Re(s) > 1),

w2

_{aq(1, s) =}R∞ 1 B¯2q(t)t −2q−s_{dt, [s]}k = s(s + 1) · · · (s + k − 1), ¯Bq(t)

Ñ

+›‰ƒb

4_{” 2.1.5. δ(S) =} 1 S−1n 1−s ₋ 1 2n −s₊Pq k=1 [S]2k−1 (2k)! B2kn −s−2k+1 _{+ O} 1 ns+2q+1
,

w2

δ(S) =P∞ n=k+1 1 ks

Ñ

ζ(s)

íì¸

„p

_:

’.A

₍₁₉₉₅₎

@à

_µ‰ƒbÜ

_,

û|-

–

_^*ƒbíht

,

Fíl°

|4¸t

(

c’.A

(1994)

_{:û+›‰bíAÍb°Ÿ4¸ít}

):

ç

_{a 6= −1, q > a + 1}

v

_, n X k=1 ka= 1 a + 1n a+1₊ q X k=1 (−1)k k C a k−1Bqna−k+1+ γa+ O  1 nq−a−1  .

w2

γa = 1 − 1 a + 1− q X k=1 (−1)k k C a k−1Bk+ (−1)q+1Cqa Z ∞ 1 ¯ Bq(t)ta−qdt,

/

O  1 nq−a−1  = (−1)qC_aq Z ∞ a ¯ Bq(t)ta−qdt (n → ∞).

·<ƒ

B1 = −1₂ ,B2k+1 = 0(k ∈ N),

Z,tí

q

Ñ

2q ,

†‰$Ñ

:

ç

_{a = −1 , 2q > a + 1}

v

_, n X k=1 ka= 1 a + 1n a+1 + 1 2n a + q X v=1 1 2vC a 2v−1B2vna−2v+1+ γa+ C2qa Z ∞ n ¯ B2q(t)ta−2qdt, γa= 1 2− 1 a + 1− q X v=1 1 2vC a 2v−1B2v+ C2qa Z ∞ 1 ¯ B2q(t)ta−2qdt. (2.3)

ç

_{a = −P < −1 ,}

_¹

_{P > 1}

v

_,

úL<AÍb

q ,



_{2q > a + 1 ,}



_(2.3)

2

_,

I

n → ∞ ,

âk

lim n→∞ Z ∞ n ¯ B2q(t)ta−2qdt = lim n→aO  1 n2q+p−1  = 0.

])Y¹

P –

bíø_t

: ∞ X n=1 1 np = γ−p = 1 2 + 1 p − 1 − q X k=1 1 2kC −p 2k−1B2k+ C −p 2q Z ∞ 1 ¯ B2q(t)t−p−2qdt.

ùp,¼

_pU

_{[P ]}k _{= P (P + 1) · · · (P + K − 1) ,}

,¢ª‰$Ñ

_: ∞ X n=1 1 np = 1 2 + 1 p − 1− q X k=1 [P ]2k−1 (2k)! B2k− [P ]2q (2q)!aq(1, P ). (2.4)

w2

aq(n, P ) = Z ∞ n ¯ B2q(t)t−p−2qdt (n ∈ N).

â

_(2.4)

£

_(2.3),

q)ì¸

: δ(P ) = ∞ X k=n+1 1 kp = γ−p− n X k=1 1 kp,

Ĥ

δ(P ) = 1 P − 1n 1−p₋ 1 2n −p + q X v=1 [P ]2v−1 (2v)! B2vn −p−2v+1₋ [P ] 2q (2q)!aq(n, P ). (2.5)

…d@à

_µ‰ƒbÜ

_,

ԁu

_{j&ƒbíñø4ìÜ}

,

zt

(2.4) , (2.5)

2¡

b

_P

í<2ˆƒµbíø_ä–

,

Ä

7)ƒ-

–

_^*ƒb

ζ(s) = P∞ n=1 1 ns , (Re(s) > 1)

í[ýt£ì¸í:û

δn(s)

í

,l

F

_{‚ñø4ìÜucq}

_(2.3)

_ƒb

_f1(s)

¸

_ƒb

_f2(s)

Ê–

D

q

j&

, (2.4)

Ê

D

qø_Y¹k

a ∈ D

íõ

{Sn}(Sn6= a) ,

Êw,

f1(s)

¸

f2(s)

M

,

†

f1(s)

¸

_f2(s)

Ê

_D

qó

_,

ԁË

,

J

f1(s)

¸

f2(s)

ÊõWíø¨

M

,

†!A

(

Ì

cØ–

(1988)

µ‰ƒb

P 167

ìÜ

_{4.20 )}

q

D

ѵޖ

(

?[ý

_R2

_íó@–

_{),U = U (x, y, t), V = V (x, y, t)}

_Ñ

D × [a, ∞]

,íõ

_ƒb
q

_{S = x + yi ∈ D}

†

_{f (s, t) = U (x, y, t) + iV (x, y, t))}

6u

D × [a, ∞]

í

_©/ƒb

_,

dì,

[a, b]

í

_{µM }}

Z b a f (s, t)dt = Z b a u(x, y, t)dt + i Z b a v(x, y, t)dt.

ì

_{2Öµ¡¾2 }}

Z ∞ a f (s, t)dt = lim A→∞ Z A a f (s, t)dt (V A > a).

ⵉbbÜ

, R_a∞f (s, t)dt

Y¹kø_µbíkb‘KuúLø]Ó/k

∞

íõb

{AK} , A1 = a ,

b

P ∞ k=1 RAk+1 Ak f (s, t)dt

Y¹

,

ÊY¹v

,



Z ∞ a f (s, t)dt = ∞ X k=1 Z Ak+1 Ak f (s, t)dt.

ùÜ 2.1.1.

q

_D

uøµÞ–

, f (s, t) , f_s0(s, t)

u

_{D × [a, b]}

,í

_©/ƒb

_,

†

_Ö

µ¡¾ }

F (s) =R_abf (s, t)dt ,

u

_D

qí

_j&ƒb

„p

:

q

f (s, t) = U (x, y, t) + iV (x, y, t) ,

ku

Z b a f (s, t)dt = Z b a u(x, y, t)dt + i Z b a v(x, y, t)dt.

I

_{F (s) = U (x, y) + iV (x, y) ,}

¥³

u, v, U, V

·uõƒb

,

ªœõ¶D™¶

,



U (x, y) = Z b a u(x, y, t)dt V (x, y) = Z b a v(x, y, t)dt,

Ä

_f0 s(s, t) (S ∈ D)

æÊ

,

FJúL<

t ∈ [a, b] , f (s, t)

Ê

D

,

j&

,

â

µ‰ƒbû

bt£

_{C − R}

‘K

_,



f_s0(s, t) = Ux(x, y, t) + iVx(x, y, t) = Uy(x, y, t) − iVy(x, y, t),

Ä

_f0 s(s, t)

Ê

D × [a, b]

©/

,

Ä7

Ux, Uy, Vx, Vy

·©/

,

â

Ö¡¾ }í©/4£ª

4ìÜ

,

Ê

_{D ⊂ N}2

_q

_,

_ú

_{x, y}

_í

_Rûb©/

_,

_/

Ux = Z b a ux(x, y, t)dt = Z b a vy(x, y, t)dt = Vy, Ux = Z b a uy(x, y, t)dt = Z b a [−vx(x, y, t)]dt = −Vx.

â

_{j&ƒbíø_k}‘K}

₍

cØ–

(1988)

µ‰ƒb

)

ø

F (s)

Ê

_D

qj&

,

)

„



q

_C

ѵÞ

_{, G = {Re(s) > 1, S ∈ C} ,}

†

_G

Ñø_µÞ–

,

úL<

t ∈ (0, ∞) ,

_dìNbƒb

ts

Ê

_{S ∈ G}

,¦3

_M

_,

_¹

_ts _{= e}s ln t _{( ln t}

_ÑAÍúb

_),

_J

,

_dìíƒb

_ts

_Ê

_G

_,À

_Mj&

_,

_/

_(ts₎0 s= tsln t

6Ê

G

,À

Mj&

(

-dà

G

í

Ö2£

ts

_¦3

_Mídì

₎

q

f (s, t) = ¯B2q(t)t−s−2q,

†

fs0(s, t) = ¯B2q(t)t−s−2qln t ,

âk

B¯k(t) = Bk(t−[t]) ,

ç

_{k ≥ 2}

vuU‚Ñ

₁

í

_©/ƒb

₍

Ìc‚µ

,

Ù

EÄ5bç}&íj¶£Wæ²

è

),

]

_{f (s, t) , f}0 s(s, t)

Ê

G × (0, ∞)

,©/

,

úLø]Ó/k

∞

íõb

{Ak} , A1 = n ( n ∈ N ),

ì2

Fk(s) = Z Ak+1 Ak ¯ B2q(t)t−s−2qdt (k ∈ N).

†âùÜ

2.1.1 ,

ª)

ùÜ 2.1.2. Fk(s) (k ∈ N)

Ê

G

q

j&

ùÜ 2.1.3.

_µMƒbb

P∞ k=1Fk(s)

Ê

G

q"úY¹

„p

:

L

_{S ∈ G , x = Re(S) > 1 ,}

q

max| ¯Bk(t)| = Mk (k ≥ 2) ,

†

|Fk(s)| = Z Ak+1 Ak ¯ B2q(t)t−s−2qdt ≤ Z Ak+1 Ak | ¯B2q(t)||t−s−2q|dt ≤ M2q Z Ak+1 Ak t−s−2qdt.

7

∞ X k=1 M2q Z Ak+1 Ak t−s−2qdt = M2q Z ∞ n t−s−2qdt = M2q 1 − x − 2q t 1−x−2q  ∞ n = M2q 2q + x − 1n 1−x−2q_.

Y¹

,

â

_{S ∈ G}

íL<4

,P∞ k=1Fk(s)

Ê

G

,"úY¹

,

)„

 ùÜ 2.1.4.

q

P∞ k=1

Ê

G

qY¹k

ƒb

aq(n, s) ,

†

aq(n, s) = R∞ n B¯2q(t)t −2q−s_dt

úL<AÍb

n, q ,

Ê

_{aq(n, s)}

q

j&

„p

_:

úL<ä£Õ

_{D ⊂ G ,}

q

_{x0 = minx∈D{Re(S)}}

†

_{x0 > 1 ,}

Ê

_D

,

_, −Re(S) < −x0

|Fk(s)| =     Z Ak+1 Ak ¯ B2q(t)t−s−2qdt     ≤ Z Ak+1 Ak | ¯B2q(t)||t−s−2q|dt ≤ M2q Z Ak+1 Ak t−s−2qdt ≤ M2q Z Ak+1 Ak t−x0−2q_dt.

7

∞ X k=1 M2q Z Ak+1 Ak t−x0−2q_{dt = M} 2q Z ∞ n t−x0−2q_dt = M2q 2q + x0− 1 n1−x0−2q_.

Y¹
â

M –

_‡¶

, P∞ k=1F (s)

Ê

D

qø_Y¹

,

¹

P∞ k=1F (s)

Ê

G

qq£ø_

Y¹

,

¢âùÜ

2.1.2 ,

Ê

_G

q

j&

,

]â

_{j&ƒbMá°ûí&gÔ…gìÜ}

₍

c

Ø–

(1988)

_µ‰ƒb

P.146

ìÜ

_{4.9 ), aq(n, s)}

Ê

_G

q

j&

,

)„

 ìÜ 2.1.1.

q

_C

ѵÞ

_{, G = {S|Re(S) > 1, S ∈ C}}

Ñø–

,

†-

–

_^*ƒb

ζ(S) =P∞ n=1 1 ns

Ê

G

q

j&

,

/

ζ(S) = 1 2+ 1 s − 1− q X k=1 [S]2k−1 (2k)! B2k− [S]2q (2q)!aq(1, s). aq(n, s) = Z ∞ 1 ¯ B2q(t)t−2q−sdt. (2.6)

dìNbƒb

ns

_£

_4Nƒb

_t−2q−s _{(t > 0)}

_Ê

_G

_q¦3

_M

„p

_:

_"ÎùÜ

_2.1.4

í„pj¶

_,

ª)

P∞ n=1 1 ns

Ê

G

qq£ø_Y¹

,

7

1 ns

Ê

_G

qÀ

_Mj&

_,

]âMá°ûí&gÔ…gìÜ

,

øÊ

G

q

j&

,

âùÜ

2.1.4 ,

aq(1, s)

Ê

G

qj&

,

7,¼ ƒb

[s] k = s(s + 1) · · · (s + k − 1)

u

_s

í

_k

ŸÖá



_,

éÍ

_,

…D

1 s−1

·Ê

G

q

j&

,

â

j&ƒbíû†«4”

, ζ(S) = 1 2 + 1 s − 1 − q X k=1 [S]2k−1 (2k)! B2k− [S]2q (2q)!aq(1, s).

Ê

_G

q

j&

, s = x ∈ (1, ∞)

v

_,

ât

(2.4) ,



_{ζ(x) = ξ(x)}

]â

_j&ƒbíñø

4ìÜ

,

)Ê

G

q

_{, ζ(x) = ξ(x) ,}

_¹t

_(2.6)

A

,

)„

 ìÜ 2.1.2. ζ(s)

íì¸

_{δ(s) =}P∞ k=n+1 1 ks

úL<AÍb

n

Ê

G

qj&

,

/

δ(S) = 1 S − 1n 1−s₋1 2n −s + q X k=1 [S]2k−1 (2k)! B2kn −s−2k+1₋ [S] 2q (2q)!aq(n, S), aq(n, S) = Z ∞ n ¯ B2q(t)t−2q−Sdt. (2.7)

„p

:

éNìÜí„p

,

ât

(2.5),

ª)„

 ìÜ 2.1.3.

Ê

_G

q

_ζ(s)

íì¸

δ(s) =P∞ k=n+1 1 ks

à-

:ûí,l

δ(S) = 1 S − 1n 1−s₋1 2n −s + q X k=1 [S]2k−1 (2k)! B2kn −s−2k+1 + O  1 ns+2q+1  .

„p

_:

Û

_,lç

_{n → ∞}

v

_{, −}[S]2q (2q)!aq(n, S)

í

,

q

{fn(s)} , {gn(s)}

Ñ

µƒb

,

J

_{limn→∞fn(s) = limn→∞gn(s) = 0 ,}

/

  fn(s) gn(s)    ≤ L ,

†p

fn(s) = O(gn(s)) , (n → ∞),

øt

_(2.7)

yǃ

q + 1

á

,

@à

: −[S] 2q (2q)!aq(n, S) = [S]2q+1 (2q + 2)! ¯ B2q+2n−s−2q−1− [S]2q+2 (2q + 2)!aq+1(n, S).

âk

|aq+1(n, S)| =     Z ∞ n ¯ B2(q+1)(t)t−2(q+1)−sdt     ≤ Z ∞ n | ¯B2(q+1)(t)||t−2(q+1)−s|dt ≤ M2(q+1) Z ∞ n t−2(q+1)−Re(S)dt = M2(q+1) ReS + 2q + 1n −Re(S)−2q−1 = M2(q+1) |s + 2q + 1|n −s−2q−1_.

Ä7

      −[S]_(2q)!2qaq(n, S) ns+2q+1       ≤      [S]2q+1 (2q + 2)!B¯2q+2      + M2(q+1) |s + 2q + 1| = L.

]

−[S] 2q (2q)!aq(n, S) = O  1 ns+2q+1  (n → ∞),

FJ

δ(S) = 1 S − 1n 1−s₋ 1 2n −s + q X k=1 [S]2k−1 (2k)! B2kn −s−2k+1 + O  1 ns+2q+1  .

)„



ù

_{+›‰b5óÉû˝}

…

_{ùHb¹+›‰b5óÉû˝}

_,

_{Ž¤7j+›‰bí½b4”£@à}

Ô

¸Ä

(2006)

©/£cb5Ÿj¸tí«n

¸ÄÊd2Tƒ

: Sn(k) = 1n+ 2n+ · · · + (k − 1)n, k ∈ N, k > 1 = Z k 0 Bn(x)dx,

¢

Z k+1 k Bn(x)dx = Z k+1 0 Bn(x)dx − Z k 0 Bn(x)dx = Sn(k + 1) − Sn(k) = kn,

ø

_k

à

_x

¦H

Z x+1 x Bn(t)dt = xn,

øsi}

Bn(x + 1) − Bn(x) = nxn−1,

é

x = 0

ƒ

_{x = k + 1}

Ú‹

Bn(k) − Bn(0) = nSn−1(k) = n Z k 0 Bn−1(x)dx, Sn(k) = 1 n + 1[Bn+1(k) − Bn+1(0)], Bn(k) = n Z k 0 Bn−1(x)dx + Bn(0),

/

Bn = Bn(0) = Sn0(0),

é

Bx(0) = 1 Bn(x) = n Z x 0 Bn−1(t)dt + Bn, n ∈ N,

I

_{n = 1, 2, 3, · · · ,} B1(x) = x + B1, B2(x) = x2+ 2B1x + B2, B3(x) = x3+ 3B1x2 + 3B2x + B3, .. .

ª)

Bn(x) = n X i=0 C_inBixn−i, Sn(k) = 1 n + 1 n X i=0 C_in+1Bikn+1−i,

ÇøjÞ

,

é

x = 0 Bn(1) − Bn(0) = 0.

FJ

Bn= Bn(0) = Bn(1) = n X i=0 C_inBi.

]

Bn+1 = n+1 X i=0 C_in+1Bi = n X i=0 C_in+1Bi+ Bn+1,

Ĥ

n X i=0 C_in+1Bi = 0,

FJ

Bn= − 1 n + 1 n−1 X i=0 C_in+1Bi.

Ž Q2

(2003)



_{É+›‰bís_]Rt}

F

_{‚+›‰buâ+›‰Öá7V}

_,

ì2Ñ

: z ez_{− 1} = ∞ X n=0 Bn n! , |z| < 2π.

Bn

Ñ

+›‰b

,

…uø_ݽbíb

,

/˛ø

B0 = 1 , B1 = −1₂ , B3 = 1₆ , B4 = −₃₀1 , · · · , B2k+1= 0 (k ≥ 3) ,

Ék+›‰bí]Rt˛

Bn = n X k=0 C_nk (n ≥ 2), B2n = In− X (p−1)|2n 1 p,

w2

_In

uø_cb

,

7¸uúFU

p − 1

cÎ

2n

íÖb

p

T¸

+›‰b…´rÖí4”

,

øòurÖç6û˝í½õ

,

)ƒ7'Ö

ïí!‹

,

…d

_¦¬û˝ƒb

_{y =} 1 sinh x

4b

,

)ƒ7

+›‰bø í0



,

_¹-Þí!

ø ìÜ£R

ìÜ 2.2.1. a, b, k

Ñ£cb

_,

†

X 2a+b−1=2k 22a_B 2a (2a)!(b)! = 1 (2k)! − 1 (2k + 1)!. ìÜ 2.2.2. a ≥ 1, k

Ñ£cb

_{, b}

ÑÝŠcb

,

†

X a+b=k 22a_B 2a (2a)!(2b)! = 2k − 1 + (2 − 22k_)B 2K (2K)! . R_{ 2.2.1. a, b, k}

Ñ£cb

_,

/

_{b ≥ 3 ,}

†

B2k =  1 − 1 2k + 1  1 22k − (2k)! 22k X a+b−1=2k 22aB2a (2a)!(b)!. R_{ 2.2.2. a, b}

Ñ£cb

_,

†

B2k = (2k)! 2 − 22k+1 X a+b−1=2k 22aB2a (2a)!(2b)! + 2k − 1 22k+1_{− 2}.

ù !í„p

ìÜ

_2.2.1

í„p

_:

5?

_{Y =} 1 sinh x ,



Y = 2 ex_{− e}−x = 2e x e2x_{− 1} = 2x e2x_{− 1} = ∞ X n=0 Bn2n n! x n ∞ X n=0 1 n!x n−1 = 1 − x + ∞ X k=0 B2k22k 2k! x 2k ! _∞ X n=0 1 n!x n−1 =  e x x − e x₊B2k22k 2k! x 2k  ∞ X n=0 1 n!x n−1 = 1 x+ ∞ X n=1  1 (n + 1)! − 1 n!  xn+ ∞ X n=1 X 2a+b−1=n 22aB2a (2a)!(b)!x n = 1 x+ ∞ X n=1 1 (n + 1)! − 1 n! + X 2a+b−1=n 22a_B 2a (2a)!(b)! ! xn = 1 x+ ∞ X k=1 1 (2k + 1)! − 1 (2k)! + X 2a+b−1=2k 22a_B 2a (2a)!(b)! ! x2k + ∞ X k=1 1 (2k)! − 1 (2k − 1)! + X 2a+b−1=2k 22a_B 2a (2a)!(b)! ! x2k−1, (2.8)

¢

_{Y =} 1 sinh x

í4bÑ

1 x − 1 6x 1 + 7 360x 3₋ 31 15120x 5 + · · · −2(2 2k_{− 1)B} 2k (2k)! x 2k−1 + · · · , (2.9)

â

_(2.8)



_(2.9)

ªœsi[b

_,

Ä

_(2.9)

í

_XŸá[bÑÉ

_,

FJ)㓆

_2.2.1 X 2a+b−1=2k 22a_B 2a (2a)!(b)! = 1 (2k)! − 1 (2k + 1)!. 

â

_(2.8)



_(2.9)

íJŸá[bó

_,

†

1 (2k)! − 1 (2k + 1)! + X 2a+b−1=2k 22a_B 2a (2a)!(b)! = − 2(22k_{− 1)B} 2k (2k)! ,

Ĥ

X 2a+b−1=2k 22aB2a (2a)!(b)! = 2k − 1 + (2 − 22k)B2K (2K)! ,

âk

_{2a + b = 2k ,}

_¹

_b

Ñ

_Xb

_,

]I

b

Ñ

_2b

Hp,

_,

FJ)㓆

2.2.2 X a+b=k 22a_B 2a (2a)!(2b)! = 2k − 1 + (2 − 22k_)B 2K (2K)! . 

R

2.2.1

í„p

:

âìÜ

2.2.1

ø

_,

ç

_{b ≥ 3}

v

X 2a+b−1=2k 22a_B 2a (2a)!(b)!+ 22k_B 2k (2k)! = 1 (2k)! − 1 (2k + 1)!, B2k = (2k)! 22k 1 (2k)! − 1 (2k + 1)! − X 2a+b−1=2k 22a_B 2a (2a)!(b)! ! ,

FJ

B2k =  1 − 1 2k + 1  1 22k − (2k)! 22k X a+b−1=2k 22a_B 2a (2a)!(b)!. 

R

2.2.2

í„p

:

âìÜ

2.2.2

ø

_,

ç

_{b ≥ 1}

v

22k_B 2k (2k)! + X a+b=k 22a_B 2a (2a)!(2b)! = 2k − 1 + (2 − 22k_)B 2K (2K)! , (2 − 22k+1_)B 2K (2K)! = X a+b=k 22a_B 2a (2a)!(2b)! − 2k − 1 (2k)! ,

FJ

B2k = (2k)! 2 − 22k+1 X a+b−1=2k 22a_B 2a (2a)!(2b)! + 2k − 1 22k+1_{− 2}.



ú lÔW àR

2.2.1

_{#|]Rtl+›‰b}

B2n , (n ≥ 2) B4 =  1 − 1 5  × 1 24 − 4! 24  22_{× B} 2 2! × 3!  = 4 5× 1 16− 24 16  4 ×1 6 2 × 6  = − 1 30, B6 =  1 − 1 7  × 1 26 − 6! 26  24_{× B} 4 4! × 3! + 22B2 2! × 5!  = 6 7× 1 64− 720 64 16 × −₃₀1
24 × 6 + 4 ×1₆ 2 × 120 ! = 1 42, B8 =  1 − 1 9  × 1 28 − 8! 28  26_{× B} 6 6! × 3! + 24_B 4 4! × 5! + 22_{× B} 2 2! × 7!  = 8 9× 1 256 − 40320 256 64 × −₄₂1
720 × 6 + 16 × −₃₀1
24 × 120 + 4 ×1₆ 2 × 5040 ! = − 1 30, B10= ₆₆5, · · ·

°Üª)ƒFí

+›‰b

B2n

àR

2.2.2

_{#|]Rtl+›‰b}

B2n , (n ≥ 2) B4 = 4! 2 − 25  22_B 2 2! × 2!  + 3 25_{− 2} =  −24 30  × 1 6+ 3 30 = − 1 30, B6 = 6! 2 − 27  24_B 4 4! × 2! + 22_B 2 2! × 4!  + 5 27_{− 2} =  −720 126  × 1 360 + 5 126 = 1 42, B8 = 8! 2 − 29  26B6 6! × 2! + 24B4 4! × 4! + 22B2 2! × 6!  + 7 29_{− 2} =  −40320 510  × 24 40320 + 7 510 = − 1 30,

B10= ₆₆5, · · ·

°Üª)ƒFí

+›‰b

B2n

¡ rÁœ

₍₁₉₈₈₎

-

_–

_{^*ƒbD+›‰b}

úkõ¶×k

1

í

µb

s ,

ì

2-

–

_^*ƒbà-

: ζ(s) = ∞ X n=1 1 ns.

ì

_{2+›‰bÑ-Þƒbœ Ç2í[b}

_Bk: s es_{− 1} = 1 − 1 2s + ∞ X k=1 (−1)k−1 Bk (2k)!s 2k_{, (2.10)}

â

_(2.10)



_,

ªR|©ø_

Bk

·uÜb

Í(ú

_ζ(s)

dj&ô

(analytic continuation) ,

U)

_ζ(s)

AÑì2Êc

_

_{µbÞ,íšÓƒb}

_{(meromorphic function) ,}

1/

_ÉÊ

_{s = 1}

¥øõ

ø_šÑ

1

í

Àõ
7/%¬j&ô5(

, ζ(s)

Å

_{—-Þíƒbj˙}

_: π−12sΓ(1 2s)ζ(s) = π −1 2(1−s)Γ(1 2(1 − s))ζ(1 − s).

J,Ék

ζ(s)

í4”

,

ªJʯ
g

(Ahlfors)

z

³vƒ

,

¥.u¥¹dıF

bní3æ

-

–

_{^*ƒbD+›‰bà-½bíÉ[}

: 4_{” 2.2.1.}

úL<£cb

_{k ≥ 1 ,}



ζ(2k) = ∞ X n=1 1 n2k = 2 2k−1 Bk (2k)!π 2k .

à‹Ê4”

2.2.1

2¦

_{k = 1 ,}

†âk

_{B1 =} 1 6 ,

Ĥ)ƒ

∞ X n=1 1 n2 = 1 6π 2_{, (2.11)} (2.11)

í!‹u×bçðr…F„pí

,

Å˝JVøò·¿¿ËùAp
4

”

2.2.1

í!‹çͪ

(2.11)

´b¿prÖ
4”

2.2.1

_.°í„p

,

ø

O·.Øñq

,

&àªJ¡5¯
gCÉJïD

(Titchmarsh)

íz

¥¹dı

íñíu

b#ø_ªœ»éòQí4”

2.2.1

í„p

,

çTç3-)

,

ıú©è

¥¹dıíè6}Œï

ÛÊÇá„p4”

2.2.1

I

_{f (s) =} 1 es_{− 1}· 1 s2k(s ∈ C),

5?-Þí }

I(x) = 1 2πi Z |s|=(2N +1)π f (s)ds, (2.12)

w2

_N

Ñø£cb
.Ø„pæÊø_D

N

ÌÉí£b

δ > 0 ,

U)ÊÆ

|s| = (2N + 1)π

,

_,



|es_{− 1| ≥ δ,}

FJç

_{|s| = (2N + 1)π} |f (s)| = O (N π)−2k
,

Ĥ }

I(N ) = O (N π)−2k+1
, (2.13)

ÇøjÞ

,

úk©øÝÉícb

m , f (s)

Ê

_{s = 2mπi}

¥øõø_š

Ñ

_(2mπ)−2k

_íÀõ
7/â

_(2.10)



_,

_ø−

_{f (s)}

_Ê

_{s = 0}

_¥õíšu

(−1)k−1 Bk (2k)!

FJ }

(2.12)

°vÑ

I(N ) = N X m=1 2 (2mπi)2k + (−1) k−1 Bk (2k)!, (2.14)

ÛÊI

n → +∞ ,

â

_(2.13)

¸

(2.14)



_,

…)ƒ

∞ X m=1 2 (2mπi)2k + (−1) k−1 Bk (2k)! = 0,

FJ

∞ X n=1 1 n2k = 2 2k−1 Bk (2k)!π 2k

Ĥ4”

_2.2.1

)„



 Ø#Ì

(2006)

+›‰bD-

–

^*ƒb

Ø#Ìk©/cb4Ÿ¸5ÞAƒbû˝ød2Tƒ+›‰bD-

–

_^*

ƒb

,

w2

_{Ü+›‰bÑÅ—-}

_{Bk, k ∈ N,} t et_{− 1} = ∞ X k=0 Bktk k! , |t| < 2π.

7Ék+›‰bí–Ä

,

uâr…êÛ7£cbí

_Jbj¸

P∞ k=1 1 k2 = π 2 6 ,

F

àíj¶u£ý

_{ƒb5;D[bÉ[
q}

_m



_n

u£cb

_,

_ƒb

sin x

x

íÉPÊ

x = nπ(n 6= 0) ,

_7ª[ýÑ̤

sin x x = ∞ Y k=1  1 − x 2 k2_π2  .

7ÇøjÞ

,

_{¥ƒbí4bÇÑ}

sin x x = 1 − x2 3! + x4 5! + · · · + −1n_x2n (2n + 1)! + · · · .

ªœ[b()|

− ∞ X k=1 1 k2_π2 = − 1 3!,

]

∞ X k=1 1 k2 = π2 6 .

7r…M/lƒ
ùŸjíJb¸

,

/ªø¥Ri

∞ X k=1 1 k2m = (−1)m−1(2π)2mB2m 2(2m)! .

¥

_{B2m , m ∈ N}

uø_Üb

_,

9õ,

_B2m

6ÿu,H5

_{+›‰b
¥–Ä=}

U7

_+›‰û˝

_Bk

D

Sk(n)

íÉ[

,

;W

t et_{− 1} = ∞ X k=0 B2ktk k! ,

J£

et− 1 = ∞ X m=1 tm m!,

ª)

t = ∞ X k=0 B2ktk k! ∞ X m=1 tm m!.

FJ

Bk(k ∈ N)

Å—í]cì2Ñ

B0 = 1, C₁kBk−1+ C2kBk−2+ · · · + CkkB0, k ≥ 2.

Ĥ

B1 = 1 2, B2 = 1 6, B3 = 0, B4 = − 1 30.

7,Hí]cì2ªŸA

_{(B + 1)}k_{= B}k _,

_{àùáìÜÇ}

_{(B + 1)}k _,

¾ 

Bn

)

C₁kBk−1+ C₂kBk−2+ · · · + C_kkB0 = 0,

z,Þä2í,™²A-™

_{¹u,Þíì2
¥²,™Ñ-™íG¶}

6àk̤b,

,

ԁuàk

eBt ₌ P∞ k=0 Bk_tk k!

,™²A-™(

,

¹u+›‰

bíì2
Uà¤pU

,

†

t et_{− 1} = e Bt_,

à¤

t = eB+1− eBt_,

Ä7

(B + 1)k= Bk, k ≥ 2.

yƒ

Sk(n)

í°¸½æ

, em et_{− 1} − 1 et_{− 1} = 1 t  ten et_{− 1}− t et_{− 1}  = 1 t e nt_eBt_{− e}Bt = 1 t e (B+n)t_{− e}Bt _.

]

_¥ƒbÊ

_{t = 0}

Ç2

_tm

_í[bÑ

1 (m + 1)!(B + k) m+1_{− B}m+1 _.

¹

Sk(n) = 1 k + 1 k X j=0 C_jk+1Bj(n + 1)k+1−j,

·<ƒ

Bm

|ÛÊ

Sk(n)

2í|(øá

,

ªJpéõ|

Bk = lim nto0 Sk(n) n . 

'QO6Tƒ-

–

^*ƒb

,

úkõ¶×k

1

5µb

s ,

ì2-

–

^*ƒ

bÑ

ζ(s) = ∞ X k=1 k−s.

r…F)ƒíäwõ

_ßÿu

ζ(s)

Ê£

_{Xbí¦M
¥_ƒbª%âj&ô}

ˆì2Ê

_{µbÞ,
¥_ƒbÊŠcbí¦MN¬œ }

e−kt = 1 − kt + k 2_t2 2! + · · · + (−1)n_kn_tn n! + · · · ,

I

_{k = 1, 2, · · · L − 1 ,}

1ó‹)ƒ

L−1 X k=1 e−kt = (L − 1) − S1(L)t + S2(L) 2! t 2 + · · · + (−1) n_S n(L) n! t n + · · · .

Oç

_{t > 0}

v

_, P∞ k=1e −kt

_{uøY¹íªb}

_,

_w¸u

e−t 1 − e−t = 1 et_{− 1}.

Ä7

ζ(−m) = (−1)mm! ×  1 et_{− 1}

Ç2

t m

_í[b

 = (−1)mm! Bm+1 (m + 1)! = (−1)mBm+1 m + 1.

·<ƒ

F (t) = t et_{− 1}+ t 2,

u

_øXƒb

_,

w4bÇ2.}|ÛJbá

,

¥[ý

B2k+1, k ∈ N ,

Ĥ

ζ(−2k) = B2k+1 2k + 1 = 0.

¥<ŠXbíÉõ˚Ñ

ζ(s)

íéÍÉõ
ÄÑ

ζ(s)

Å

_—˜ƒj˙

_: π−s2Γ s 2  ζ(s) = π−(1−s)2 Γ  1 − s 2  ζ(1 − s),

FJ

ζ(1 − 2m) = −B2m 2m ,

]

ζ(2m) = (−1) m−1_(2π)2m_B 2m 2(2m)! . 

,Ô

_{ËŸ|+›‰bD-}

–

_^*ƒb5É:

,

¤

_{¹Å˝JVøò¿¿}

ËùAp5!‹

' ˜0ˆ Ó{M

(1993)

-

“;

˜0ˆ Ó

_{{Mk-“;øz2Tƒúk}

_{n ∈ N ;}



ìÜ 2.2.3. Bn= _(2π)2(2n)!2nS2n ,

w2

Bn

Ñ

+›‰b

, S2n =P ∞ m=1 1 m2n

„p

:

_{íl✠Çø}

ex = 1 + x +x 2 2! + · · · + xn n! + · · · ,

FJ

x ex_{− 1} = x x + x_2!2 + · · · + x_n!n + · · · = 1 1 + _2!xx_3!2 + · · · + xn−1_n! + · · ·,

ycq¥_¼Býúk—Düí

x

_Mª[Ab

x ex_{− 1} = 1 + ∞ X n=1 βn n!x n_,

1øw[b¦A

βn n!

í$

,

¥cuÑ7üì[bvœÑjZ
Ĥ;WÉ[

 1 + x 2!+ · · · + xn−1 n! + · · ·  ×  1 + β1 1! + · · · + βn n!x n + · · ·  = 1,

I˝

_{V®_j4}

_xn (n ∈ N)

í[bkÉ
ª)|j˙

1 n!βn+ 1 (n − 1)!2!βn−1+ · · · + 1 (n − k + 1)!k!βn−k+1+ · · · + 1 (n + 1)! = 0,

si

_J

_{(n + 1)!}

)

C₁n+1βn+ C2n+1βn−1+ · · · + Ckn+1βn+1−k+ · · · + Cnn+1β1+ 1 = 0,

‚àDâùáóNíÉ[

,

¥<j˙¯Uí$,ªJŸA

: (β + 1)n+1− βn+1 = 0 (n ∈ N),

Í(zùáÇ

,

¾ |òá

βn+1

₍

_,

_4j

_βk

_Tà

_β k

H

,

)ƒüì

βn , (n ∈ N)

í̤j˙

2β1+ 1 = 0, 3β2+ 3β1+ 1 = 0, 4β3+ 6β2+ 4β1+ 1 = 0, 5β4+ 10β3 + 10β2+ 5β1+ 1 = 0, (2.15)

â

_(2.15)

ª)

β1 = − 1 2, β2 = 1 6, β3 = 0, β4 = − 1 30, β5 = 0, β6 = 1 42, β7 = 0, β8 = − 1 30, β9 = 0, β10= 5 66, β11 = 0, β12= − 691 2730, β13 = 0, β14= 7 6, (2.16)

Í(;W

úk

_{|x| < 1,} x coth x = 1 + ∞ X n=1 (−1)n−12 2n_B n (2n)!x 2n_,

¥ê

Bn

Ñ

+›‚b

,B1 = 1 6, B2 = 1 30, · · ·

(2.17)

ªø

x ex_{− 1}+ x 2 = x 2 ex+ 1 ex_{− 1} = x 2 ex2 + e− x 2 ex2 − e− x 2 = x 2coth x 2 = 1 + ∞ X m=1 (−1)m−1 Bm (2m)!x 2m_.

âk

_(2.16)

2

_{βn(n > 1)}

íJbáÌÑÉ

_,

â

x ex₋₁ + x 2

íÇø

x ex_{− 1} + x 2 = 1 + ∞ X n=2 βm m!x m ,

úk

_Xb—™í

_{β ,}



β2n= (−1)n−1Bn,

ku

B1 = 1 6, B2 = 1 30, B3 = 1 42, B4 = 1 30, B5 = 5 66, B6 = 691 2730, B7 = 7 6, · · · .

â-H5t

:

úk

_{|x| < 1,} πx coth πx = 1 + 2 ∞ X m=1 (−1)m−1S2mx2m,

¥ê

S2m= ∞ X n=1 1 n2m

£

_{(2.17) ,}

ªø

πx coth πx = 1 + 2 ∞ X n=1 (−1)n−1S2nx2n, πx coth πx = 1 + ∞ X n=1 (−1)n−1(2π) 2n_B n (2n)! x 2n_,

y*,Þsªù|

S2n = (2π)2n 2(2n)!Bn. 

ú

_{4Ÿ¸ØkƒbD-}

–

^*ƒb5É:

Ô úL<£cb

_{k ,}

©/cb4Ÿ¸

_Sk(x)

Ñ

_©/ƒb

_,

/ª}

˛ø©/cb4Ÿ¸

Sk(n) = 1k+ 2k+ · · · + nk ,

ªà

Çø[ý

Sk(x) = 1 k + 1(x + 1) k+1_{− C}k+1 2 Sn(x) − · · · − Ckk+1S1(x) − x − 1 .

ç

_{k = 1}

v

S1(x) = 1 1 + 1(x + 1) 1+1_{− x − 1} = 1 2{x 2_{+ 2x + 1 − x − 1}} = 1 2(x 2_{+ x),} S1(x)

Ñ

x

íÖ

áƒb

,

]x©/4
†ç

k = n + 1

v

,

ªJ)ƒ

: S(n+1)(x) = 1 n + 2(x + 1) n+2_{− C}n+2 2 Sn(x) − · · · − Cn+1n+2S1(x) − x − 1 ,

ÄÑ

_{S1(x), S2(x), · · · , Sn(x)}

îÑ

_x

íÖ

_{áƒb/îx©/4}

_,

/

_Cn+2 2 , C₃n+2 , C_n+1n+2

îÑb[b

,

Ĥ

_S(n+1)(x)

6Ñ

_x

íÖ

_{áƒb1x©/4}

FJ

,

âbç¦Ñ¶„p

,

ªJ)ƒ

Sk(x) = 1 k + 1{(x + 1) k+1_{− C}k+1 2 Sk−1(x) − · · · − Ckk+1S1(x) − x − 1}.

x©/54”

J

_D

Ñ£/ä5Õ¯

_,

/

_ƒb

_{f : D → R}

kÕ¯

_D

2©/

_,

†

_ƒb

_f

xÌG©/54”
Ĥç

_{f : D → R}

kÕ¯

D

.ÌG©/

,

†æÊ

_{ε0 > 0} ,

U)³

_{δ > 0}

ªJÅ—-

, |f (u) − f (v)| < ε0

úkFí

u

¸

v

˘k

D

Õ¯2

, |u − v| < δ

I

_m

ÑAÍb

,

UæÊ<õ˘k

D

Õ¯2íõ

u



_v,

yU

_{|u − v| <} 1 m ,

Ou

_{|f (u) − f (v)| ≥ ε0 ,}

ø¥<õ

u



_v

°v²¦

,

1ø…b™pÑ

un

D

vn

¥øì2k

D

Õ¯2íå

{un}



{vn} ,

y

%⚌Û

-

&gÔ…gìÜ

(Bolozano-Weierstrass)

ªJ)ƒ

{un}



{vn}

íäå

{unk}



{vnk}

ø}Y¹B

D

Õ¯/s_õ

,

OuúkFíAÍb

m

Vz

, |unk− vnk| < 1 mk ≤ 1 k ,

Ĥä

å

{unk}



{vnk}

}Y¹B

D

Õ¯2í/øõ

u

£õ

v

yâ

f

k

D

Õ¯íõ

u

©/ªJ)ƒ

{f (unk)}

¸

{f (vnk)}

·}Y¹B

f (u) ,

FJ

{f (unk)−f (vnk)}

ø}Y¹B

0,

¥upeí

,

ÄÑúkFíAÍb

m , |f (unk) − f (vnk)| ≥ ε0 ,

Ĥ

_ƒb

_f

kÕ¯

D

2ÌG©/
]ç

a < b

/

_{i = [a, b]}

Ñ

_R

25'KÕ

,

†

úL<£cb

_{k , Sk : I → R}

ÑÌ

_{G©/ƒb
FJ;W,Hzp)ø}

_,

úL<

£cb

_{k , Sk(x)}

k

_I

–È2ÑÌ

_G©/ƒb

˛ø

_Sk(x)

k

_I

–È2ÑÌ

_G©/ƒb

,

†úL<£cb

_{k ,}

©/cb4Ÿ

¸

_Sk(x)

Ñ

_x

5ª}

_ƒb
ÄÑ

Sk(x) = 1 k + 1(x + 1) k+1_{− C}k+1 2 Sk−1− · · · − C_kk+1S1(x) − x − 1 ,

FJúkL<íAÍb

_{k , k ∈ N , Sk}(x)

Ñ

_x

íÖ

_{áƒb
¢ÄÑÖáƒ}

bxª}54”

_,

Ĥ

_Sk(x)

Ñ

_x

5ª}

_ƒb

Ž

Sk(n)

5ÞA

ƒb

J5?

_Sk(n)

5ÞA

_ƒb

P∞ k=0Sk(n)Tk ,

ªN¬ÀílêÛ

: ∞ X k=0 Sk(n)Tk = ∞ X k=0 n X l=1 lk ! Tk = ∞ X k=0 n X l=1 (lT )k ≤ ∞ X k=0 n X l=1 (nT )k = n 1 − (nT ) ∞ 1 − nT  .

ç

_{|nT | < 1 ,}

_¹

_{|T | <} 1 n

v

,

;Wªb

,

ªR)

P∞ k=0Sk(n)Tk

5,ä

, ∞ X k=0 Sk(n)Tk ≤ n 1 − nT,

]ªø

P∞ k=0Sk(n)T k

_5Y¹šÑ

_{|T | <} 1 n

Í7

,

ʤY¹š¸ˇq

,

ªJl)

∞ X k=0 Sk(n)Tk = ∞ X k=0 n X l=1 lkTk = ∞ X k=0 n X l=1 (lT )k = n X l=1 ∞ X k=0 (lT )k = n X l=1 1 1 − lT,

Ĥ

_,

çø−áb

n

v

_,

I

_{|t| <} 1 n ,

¹ª|ú@5ÞAƒbM

¡

_Sk(x)

5NbÞA

_ƒb

J5?

_Sk(n)

5NbÞA

_ƒb

P∞ k=0Sk(n)T k k! ,

N¬ÀílªJêÛ

: ∞ X k=0 Sk(n) tk k! = ∞ X k=0 ( n X l=1 lk)T k k! = n X l=0 ∞ X k=0 (lT )k k! = n X l=1 elT = e T (n+1)_{− e}T eT _{− 1} .

ø

_Sk(n)

ØkB

_R

øƒ

_R

5

_ƒb

_,

wì2Ñ

S1(x) = x(x + 1) 2 . Sk(x) = 1 k + 1{(x + 1) k+1 − x − 1 − k X i=2 C_ik+1Sk−i+1(x)}, k ≥ 2.



_Sˆ_k(x)

5œ

 _Ç

Ñ7l

Sk(x)

5NbÞA

ƒb

,

íl

,

û˝6ø

e T (x+1)_−eT eT₋₁

õAu

T

í

ƒ

b

_,

Í(5?

eT (x+1)_−eT eT₋₁

Ê

T = 0

5œ

 Ç
I

eT (x+1) _{− e}T eT _{− 1} = ∞ X k=0 ˆ Sk(x) Tk k!,

w2

ˆ Sk(x) = dk dTk  eT (x+1)_{− e}T eT _{− 1}  |T =0 .

'péË

,

à‹û˝6ªJ„p

:

úL< ìí

k

D

x , ˆSk(x) = Sk(x) ,

†×Š

¹ªµA

Ê´³£„p

_Sˆ_{k(x) = Sk(x)}

5‡

,

û˝6Êõø_

T

í

_ƒb5œ 

Ç

_,

6ÿu

T eT₋₁

Ê

T = 0

5Ç

,

Í(I

T eT₋₁ = P∞ k=0Bk Tk k! ,

¥w2í

Bk = d k dTk( T eT₋₁) |T =0

õÒlêÛ

B2k+1= 0, ∀k ∈ N

' ©/cb4Ÿ¸5Økƒb

úk

_{Sk(n) = 1}k_{+ 2}k_{+ · · · + n}k

_ít

_,

_wõ

_{Éb‚àùáìÜ}

_: (j + 1)k+1 = jk+1+ C₁k+1jk+ · · · + C_kk+1+ 1.

z,A

j = 1

‹ƒ

_{j = n ,}

%¬“

,

ª)ƒ

_: (n + 1)k+1− 1 = (k + 1)Sk(n) + C2k+1Sk−1(n) + · · · + Ckk+1S1(n) + n. (2.18)

Ä¤à‹˛%ø−

S1(n), S2(n), · · · , Sk−1(n)

ít

,

ÿªJ‚à,)ƒ

Sk(n)

íøOj

: Sk(n) = 1 k + 1n k+1₊1 2n k₊k 2B2n k−1₊k(k − 1)(k − 2) 2 × 3 × 4 B4n k−3_{+ · · · ,}

w2

_{B2 =} 1 6, B4 = − 1 30, B6 = 1 42, B8 = − 1 30, · · · .

,H¥<

Bi

˚Ñ+›‰b

(Bernoulli number)

;W

Sk(n)

íøOt

,

ç

l|/_

Sˆ_k(n)

5

M(

,

?´Ê.l+›‰

b5‡T-

,

A§#|

S_k+1ˆ (n)

5

Má

?

Ê ì

k

í‘K-

,

ø

_Sk(·)

õAø_â

_N

øƒ

_N

í

_ƒb
ÓÇ;W

_(2.18),

Øk

_ƒb

Sk(·)

5ì2Ñ

R ,

ì2¶à-

: Sk(x) = 1 k + 1{(x + 1) k+1_{− C}k+1 2 Sk−1(x) − · · · − Ckk+1S1(x − x − 1)},(2.19) S1(x) = x(x + 1) 2 . (2.20)

;W¥šíØk;¶

,

û˝6'ñqªJ)ƒ

S2(x) = 1 2 + 1{(x + 1) 2+1_{− C}3 2S1(x) − x − 1} S2(x) = x(x + 1)(2x + 1) 6 , (2.21)

°Ü)ƒ

S3(x) = x2_{(x + 1)}2 4 . (2.22)

Ð

_Sˆ_k(x)

5óÉ4”

4_{” 2.3.1.} d2 dx2Sˆk(x) = _dxdk ˆSk−1(x), ∀k ∈ N.

„p

:

ÄÑ

∞ X k=0 ˆ Sk(x) Tk k! = eT (x+1)_{− e}T eT _{− 1} ,

FJ

∞ X k=0 d dx ˆ Sk(x) Tk k! = d dx  eT (x+1)_{− e}T eT _{− 1}  = T e T (x+1) eT _{− 1} , ∞ X k=0 d2 dx2Sˆk(x) Tk k! = d2 dx2  eT (x+1)_{− e}T eT _{− 1}  = T 2_eT (x+1) eT _{− 1} .

¢ÄÑ

_Sˆ_{0(x) = x ,}

FJ

∞ X k=0 d2 dx2  ˆSk(x) Tk k! = ∞ X k=1 d2 dx2 ˆSk−1(x) Tk k! = T ∞ X k=0 d2 dx2 ˆ_Sk+1(x) k + 1 ! Tk k!.

Ĥ

∞ X k=0 d2 dx2 ˆ_S k+1(x) k + 1 ! Tk k! = ∞ X k=0 d dx ˆSk(x) Tk k!.

¹

d2 dx2 ˆSk+1(x)  = (k + 1) d dx ˆ Sk(x), ∀k ∈ N,

])„



4_{” 2.3.2.} d dxSˆ2k+1(x) = (2k + 1) ˆS2k(x), ∀k ∈ N.

„p

_:

â4”

_2.3.1

ø

d2 dx2Sˆ2k+1= (2k + 1) d dxSˆ2k(x), ∀k ∈ N.

ÄÑ

∞ X k=0 ˆ S_k0(−1)T k k! = T eT _{− 1}, (2.23)

FJ

ˆ S_k0(−1) = Bk,

/

ˆ S_2k+10 (−1) = B2k+1 = 0, ∀k ∈ N.

¢ÄÑ

∞ X k=0 ˆ Sk(−1) Tk k! = 1 − eT eT _{− 1} = −1,

FJ

ˆ Sk(−1) = 0, ∀k ∈ N. (2.24) =⇒ Z x −1 ˆ S_2k+100 (y)dy = Z x −1 (2k + 1) ˆS_2k0 (y)dy, =⇒ Sˆ_2k+10 (x) − ˆS_2k+10 (−1) = (2k + 1)h ˆS2k(x) − ˆS2k(−1) i , ∀k ∈ N, =⇒ Sˆ_2k+10 (x) = (2k + 1) ˆS2k(x), ∀k ∈ N.

])„



yJbç¦Ñ¶

_,

„p

_Sˆ k(x) = Sk(x), ∀x ∈ R, k ∈ N

5„pà-

:

ç

_{k = 1}

v

_,

;W4”

2.3.2 ,

ªJ)ƒ

ˆ S3(x) − ˆS3(−1) = 3 Z x −1 ˆ S2(y)dy = 3 Z x −1 y(y + 1)(2y + 1) 6 dy = x 2_{(x + 1)}2 4 ,

Ä

_Sˆ_{3(−1) = 0 (}

;Wt

(2.24)) ,

/

_{S3(x) =} x2_(x+1)2 4 ,

]

Sˆ3(x) = S3(x) , ∀x ∈ R

cq

k = p > 1

v

_{, ˆSp(x) = Sp(x), ∀x ∈ R}

A
ÛÊ5?

k = p + 1

v

_,

;Wt

(2.23)

DrÁœ

(1988)

-

–

_{^*ƒbD+›‰b}

,

J£

_(2.21),

û˝

6ªJ)ƒ

ˆ S_l0(−1) = S_l0(−1) = B1, ∀l ∈ N.

¢;W4”

2.3.1 ,

û˝6

ˆ S_p+10 (x) − ˆS_p+10 (−1) = p Z x −1 ˆ S_p0(y)dy = p{ ˆSp(x) − ˆSp(−1)} = p{Sp(x) − Sp(−1)} = S_p+10 (x) − S_p+10 (−1),

FJ

Sˆ_p+10 (x) = S_p+10 _{(x), ∀x ∈ R.}

yŸ })

ˆ Sp+1(x) − ˆSp+1(−1) = Sp+1(x) − Sp+1(−1), ∀x ∈ R.

ÇÕ

,

;W

(2.24)

DrÁœ

(1988)

-

–

_{^*ƒbD+›‰b}

,

J£

_{(2.19) ,}

ª

)

ˆ Sp+1(−1) = Sp+1(x), ∀x ∈ R,

]

ˆ Sp+1(x) = Sp+1(x), ∀x ∈ R.

Ĥ

_,

;Wbç¦Ñ¶

,

)„

Sˆk(x) = Sk(x), ∀x ∈ R, k ∈ N. 

yYW

eT (x+1)_−eT eT₋₁

Ê

T = 0

5œ

 Ç

,

.Øl|

ˆ S0(x) = x, ˆ S1(x) = x(x + 1) 2 , ˆ S2(x) = x(x + 1)(2x + 1) 6 .

Í7ú

_{k ≥ 3 , ˆSk(x)}

5l

_,

N

_{˛ÿ‰)'õ¦
ÖÍMøl}

_Sˆ_{k(x) ,}

y

øl!‹D

_Sk(x)

5ì2úÎ

_,

ªJÀUð„

Sˆk(x) = Sk(x)

59õ

,

Oà°

‡H

_,

ú

_k

'×ív`

,

Møªú

Sˆk(x)

D

Sk(x)

uÎn‘ví T

F

_,

û˝6)ƒ

_Sˆ_k(x)

x4”

2.3.1

D4”

2.3.2

Í

7;W

Sk(x)

5

ì2

_,Sk(x)

?x4”

_2.3.1

D4”

_2.3.2

W¤

_,

ªJ)„

Sˆk(x) = Sk(x), ∀x ∈ R, k ∈ N

6ÿu

∞ X k=0 Sk(x) Tk k! = eT (x+1)_{− e}T eT _{− 1} . 

1 ‚à

Sk(x)

¨Æ5U‚ƒb

Ck(x)

úL< ì£cb

k ,

I

_+›‰b

_{Bk(x) =} d dxSk(x) = S 0 k(x − 1) ,

†û˝

6ø}-Þ4”

: 4_{” 2.3.3.}

J

k

Ñ×kk

2

5£cbv

_,

†

_{Bk(0) = Bk(1).}

„p

_:

ÄÑ

∞ X k=0 Sk(x) Tk k! = eT (x+1)_{− e}T eT _{− 1} ,

FJ

∞ X k=0 S_k0(x)T k k! = T eT (x+1) eT _{− 1} ,

]

∞ X k=0 Bk(x) Tk k! = T eT x eT _{− 1},

Ĥ

∞ X k=0 Bk(1) Tk k! = T eT et_{− 1}, ∞ X k=0 Bk(0) Tk k! = T eT _{− 1}.

› U‚

_ƒb

Ck(x)

5

Z s‰²

â†/ð

(1991)

5

_{Z s}&»ød}

_,

ªøU‚

_ƒb[ýÑ

Ck(x) = X m∈Z an,keinx, n ∈ Z, k ≥ 2,

w2

einx = cos (nx) + i sin (nx),

an,k = 1 2π Z 2π 0 e−inxCk(x)dx.

;W

Ck(x)

5ì2D }d‰b‰²

,

ª)

an,k = Z 1 0 e−2πinxBk(x)dx, n ∈ Z, k ≥ 2.

QO;Wcb

n

5¦

_Mª}Ñ

_{n 6= 0}

D

_{n = 0}

s8$Vl

_an,k

ç

_{n = 0}

v

_, Z 1 0 e−2nπxiBk(x)dx = Z 1 0 Bk(x)dx = Sk(0) − Sk(−1) = 0, k ≥ 2,

]

_{a0,k = 0, ∀k ≥ 2.}

ç

_{n 6= 0}

v

_,

ÄÑ

Z 1 0 e−2πinxB0(x)dx = Z 1 0 e−2πinxdx = 0, Z 1 0 e−2nπxiB1(x)dx = Z 1 0 e−2nπxi  x − 1 2  dx = Z 1 0 xe−2nπxidx −1 2 Z 1 0 e−2nπxidx(= 0) = −xe −2nπxi 2nπi | 1 0 + 1 2nπi Z 1 0 e−2nπxidx(= 0) = − 1 2nπi,

l

R1 0 e −2πinx_B k(x)dx

5NbÞA

ƒb

,

ª)

∞ X k=0 Z 1 0 e−2nπxiBk(x)dx  Tk k! = Z 1 0 e−2nπxi ( _∞ X k=0 Bk(x) Tk k! ) dx = Z 1 0 e−2nπxi T e T x eT _{− 1}  dx = T eT _{− 1} Z 1 0 e(T −2nπi)xdx = T eT _{− 1}· 1 T − 2nπie (T −2nπi)x _|1 0  = T T − 2nπi,

FJ

Z 1 0 e−2nπxi T e T x eT _{− 1}  dx = − T 2nπi + ∞ X k=2 an,k Tk k!,

]

an,k = − k! (2nπi)k, n ∈ Z, n 6= 0, k ≥ 2.

;W

_{Z s‰²}

Ck(x) = P n∈Zan,ke−2nπxi ,

FJ

Bk  x 2π  = X n∈Z,n6=0 − k! (2nπi)ke −inx , k ≥ 2.

¹

,

úL<

_{k ≥ 2, x ∈ [0, 1],} Bk(x) = X n∈Z,n6=0 − k! (2nπi)ke 2nπxi = − k! (2πi)k X n∈Z,n6=0 1 nke 2nπxi = − k! (2πi)k X n∈Z,n6=0 1 nk[cos (2nπx) + i sin (2nπx)] = − k! 2k_(πi)k ( _∞ X n=1 1 nk [cos (2nπx) + i sin (2nπx)] + ∞ X n=1 1 (−n)k[cos (2nπx) − i sin (2nπx)] ) . (2.25)

â

_(2.25),

ªJ)ƒ

_k

Ñ£

_Xbv

_, S_k0(x − 1) = − k! 2k−1_(−πi)k ∞ X n=1 1 nkcos (2nπx), ∀x ∈ [0, 1],

7ç

_k

Ñ×k

₁

5Jbv

, S_k0(x − 1) = − k! 2k−1_(−πi)k ∞ X n=1 1 nksin (2nπx), ∀x ∈ [0, 1].

1

S_k0(x)

D-

–

_^*ƒb5É:

ç

_k

Ñ£

_Xbv

_, S_k0(0) = − k! 2k−1_(πi)k ∞ X n=1 1 nk = − k! 2k−1_(πi)kζ(k), (2.26)

7ç

k

Ñ×kø5Jbv

, S_k0(0) = 0. (2.27)

ç

_k

Ñ£

_Xbv

_, ζ(k) = −2 k−1_(πi)k (k + 1)! {k − C k+1 2 S 0 k−1(0) − C k+1 3 S 0 k−2(0) · · · − C k+1 k S 0 1(0)}.(2.28)

‚à

_(2.21)



_{(2.26) ,}

ª)

ζ(2) = ∞ X n=1 1 n2 = S 0 2(0)π2 = π2 6 .

QOy‚à

(2.20)



_(2.21)



_(2.22)



_(2.27)



_(2.28)

ª)

ζ(4) = ∞ X n=1 1 n4 = −π 4 154 − C 5 2S 0 3(0) − C35S 0 2(0) − C45S 0 4(0) = π 4 90.

./‚à

_(2.26),

)ƒ

S₄0(0) = − 1 30.

°Ü

_,

;W

(2.28)

ª)

ζ(6) = ∞ X n=1 1 n6 = 2π 6 3156 − C 7 3S 0 4(0) − C 7 5S 0 2(0) − C 7 6S 0 1(0) = π 6 945.

Y¤éR

,

ç

_k

×k

₆

5

_Xbv

_,

ªJ

_"Τj¶

_,

lMøl|

S₁0(0) , S₂0(0) , · · · , S_k−20 (0),

Í(yªø¥‚à

S₁0(0) , S₂0(0) , · · · , S_k−20 (0) ,

_¹ª)

ƒ

_ζ(k)

5

_M

úı 3b!‹

;Wùıd.«n

,

Ñ7U×Vyqk7j-

–

_{^*ƒb +›‰bDØkƒ}

b

_,

…û˝øT|yÖóÉí!…4”5ÌzpD„p

,

1ø…ı}Ñú

,

ø

-

–

^*ƒb5½bô.4”

,

ù+›‰b5Ôy@à4”

,

ú:û-

–

^*ƒbD+›‰bíÇø¤

,

Joé|,H

_{ƒbÊbçä,í˜4£½b}

4

ø

-

–

^*ƒb5½bô.4”

…

^k¦Ñb_-

–

_{^*ƒbí½bô.4”}

,

1‹Jzpð„

,

_v€ç6

?y7j-

–

_^*ƒb

,

*7¿pû˝
4ӈ-

: 4_{” 3.1.1.}

J

_{ζ(s) =}P∞ n=1 1 ns, s ∈ (1, ∞) ,

†

ζ 0_{(s) = −}P∞ n=1 ln n ns

./}(ªø

ζ(k)_{(s) = (−1)}kP∞ n=1 (ln n)k ns , k ≥ 2

„p

:

˛ø

ζ(s) = ∞ X n=1 1 ns,

ø,˝¬®¦úb

_{ln n ,}

Ĥ

ln ζ(s) = ∞ X n=1 −s ln n,

y‚àúb}¶

,

ú

_s

}

,

°)

d ds{ln ζ(s)} = d ds ( _∞ X n=1 −s ln n ) , 1 ζ(s)ζ 0 (s) = ∞ X n=1 − ln n, ζ0(s) = ζ(s) ∞ X n=1 − ln n, ζ0(s) = ∞ X n=1 − ln nζ(s), ζ0(s) = ∞ X n=1 − ln n ns ,

FJ)„

ζ0(s) = − ∞ X n=1 ln n ns . 

â,H„pø

ζ0(s) = − ∞ X n=1 ln n ns ,

./yú

s

}øŸ

d ds{ζ 0 (s)} = d ds ( − ∞ X n=1 ln n ns ) ,

‚àNb}¶)ƒ

ζ(2)(s) = ∞ X n=1 (ln n)2 ns ,

°Üª°)

ζ(3)(s) = − ∞ X n=1 (ln n)3 ns , ζ(4)(s) = ∞ X n=1 (ln n)4 ns , .. . ζ(k)(s) = (−1)k ∞ X n=1 (ln n)k ns .  4_{” 3.1.2. ζ(s) =} n1−s₋₁ 1−s + C(s) + O 1 ns
, s > 1,

w2

C(s) = lim n→∞ n X k=1 1 ks − n1−s− 1 1 − s !

„p

:

ílé

f (x) = _x1s, x ∈ [1, ∞), s 6= 1 ,

/˛ø

limx→∞f (x) = 0 ,

°vy

I

_pn=Pn k=1f (k), qn= Rn 1 f (x)dx, rn = pn− qn ,

Ĥª‚à

qn+1 = Z n+1 1 f (x)dx = n X k=1 Z k+1 k f (x)dx ≤ n X k=1 Z k+1 k f (k)dx = n X k=1 f (k) = pn,

¢˛ø

_{f (n + 1) = pn+1− pn≤ pn+1− qn+1 = rn+1 ,}

Ĥ)ƒ

0 < f (n + 1) ≤ rn+1,

y‚à

rn− rn+1 = qn+1− qn− (pn+1− pn) = Z n+1 n f (x)dx − f (n + 1) ≥ Z n+1 n f (n + 1)dx − f (n + 1) = 0,

FJø−

0 < f (n + 1) ≤ rn+1 ≤ rn ≤ r1 ≤ f (1) ,

¢ÄÑ

0 ≤ rn− rn+1 ≤ Z n+1 n f (n + 1)dx − f (n + 1) = f (n) − f (n + 1),

]ªø

0 ≤ ∞ X n=k (rn− rn+1) ≤ ∞ X n=k (f (n) − f (n + 1)) , k ≥ 1,

Ĥ

0 ≤ n X k=1 f (k) − Z n 1 f (x)dx − C ≤ f (n),

w2

_{C = limn→∞rn ,}

/

_{0 ≤ C ≤ f (1) ,}

6ÿuz

n X k=1 f (k) = Z n 1 f (x)dx + C + O (f (n)) n X k=1 1 ks = Z n 1 1 xsdx + C + O (f (n)) = n 1−s_{− 1} 1 − s + C + O (f (n)) ,

w2

_{C = limn→∞rn = limn→∞(p(n) − q(n)) = limn→∞}Pn k=1 1 ns − n1−s₋₁ 1−s  ,

])

„

ζ(s) = n 1−s_{− 1} 1 − s + C(s) + O  1 ns  , s > 1,

w2

C(s) = lim n→∞ n X k=1 1 ks − n1−s_{− 1} 1 − s ! . 

4_{” 3.1.3. ζ(s) =} 1 s−1 + P∞ k=0γk(s − 1)k, s > 1,

w2

γk = lim s→1 (−1)k k! " _∞ X n=1 (ln n)k ns − k! (s − 1)k+1 # ,

?ª[Ñ

γk = (−1)k k! m→∞lim " _m X n=1 (ln n)k n − (ln m)k+1 k + 1 #

„p

:

cq

ζ(s) = 1 s−1 + P∞ k=0γk(s − 1) k_{, s > 1,}

_Ĥ

ζ(s) = 1 s − 1 + ∞ X k=0 γk(s − 1)k = 1 s − 1 + γ0+ γ1(s − 1) + γ2(s − 1) 2 + · · · ,

])ƒ

ζ(s) − 1 s − 1 = γ0+ γ1(s − 1) + γ2(s − 1) 2_{+ · · · ,}

ĤI

_{s → 1} γ0 = lim s→1  ζ(s) − 1 s − 1  = lim s→1 " _∞ X n=1 1 ns − 1 s − 1 # = lim s→1(−1) 0 " _∞ X n=1 (ln n)0 ns − 1 s − 1 # ,

Í(˝¬ú

s

}

ζ0(s) + 1 (s − 1)2 = γ1+ 2 × γ2(s − 1) + 3 × γ3(s − 1) 2_{+ · · · , (3.1)}

/â4”

3.1.1

ø

ζ0(s) = − ∞ X n=1 ln n ns ,

Hp,

_,

Ĥ)ƒ

− ∞ X n=1 ln n ns + 1 (s − 1)2 = γ1+ γ2(s − 1) + γ3(s − 1) 2_{+ · · · ,}

yI

_{s → 1} γ1 = lim s→1 " − ∞ X n=1 ln n ns + 1 (s − 1)2 # = lim s→1(−1) 1 " _∞ X n=1 (ln n)1 ns − 1 (s − 1)2 # ,

°šyø

_(3.1)

ú

s

}

ζ00(s) − 2! (s − 1)3 = 2! × γ2+ 3 × 2 × γ3(s − 1) + 4 × 3 × γ4(s − 1) 2 _{+ · · · ,} ∞ X n=1 (ln n)2 ns − 2! (s − 1)3 = 2! × γ2+ 3 × 2 × γ3(s − 1) + 4 × 3 × γ4(s − 1) 2_{+ · · · ,}

øšI

_{s → 1} γ2 = 1 2!lims→1 " _∞ X n=1 (ln n)2 n − 2! (s − 1)3 # = 1 2!lims→1(−1) 2 " _∞ X n=1 (ln n)2 n − 2! (s − 1)3 # ,

./ú

s

}

.. . γk = 1 k!lims→1(−1) k " _∞ X n=1 (ln n)k n − k! (s − 1)k+1 # , γk = (−1)k k! lims→1 " _∞ X n=1 (ln n)k n − k! (s − 1)k+1 # . 

¢*4”

3.1.2

ø

ζ(s) = n 1−s_{− 1} 1 − s + C(s) + O  1 ns  , s > 1,

˝¬°v

_J

s − 1 (s − 1)ζ(s) = − n1−s− 1
+ (s − 1)C(s) + (s − 1)O 1 ns  ,

¢˛ø

ζ(s) = 1 s − 1 + ∞ X k=0 γk(s − 1)k,

°š˝¬°v

_J

_{s − 1} (s − 1)ζ(s) = 1 + ∞ X k=0 γk(s − 1)k+1,

â‡s)ƒ

1 + ∞ X k=0 γk(s − 1)k+1= − n1−s− 1
+ (s − 1)C(s) + (s − 1)O  1 ns  ,

Ĥ

∞ X k=0 γk(s − 1)k+1 = −n1−s+ (s − 1)C(s) + (s − 1)O  1 ns  ,

¢ÄÑ

O 1 ns  ≤ M 1 ns

/ç

_{n → ∞ ,}

ªø

_O 1 ns
≈ 0 ,

]

∞ X k=0 γk(s − 1)k+1 ≈ −n1−s+ (s − 1)C(s),

Ĥø−

m X k=0 γk(s − 1)k+1 = −m1−s+ (s − 1)Cm(s),

6ÿuz

m X k=0 γk(s − 1)k+1 = −m1−s+ (s − 1) " _m X n=1 1 ns − m1−s_{− 1} 1 − s # ≈ (s − 1) m X n=1 1 ns + m 1−s_{− 1,}

γ0(s − 1) + γ1(s − 1)2+ · · · + γm(s − 1)m+1 = (s − 1) m X n=1 1 ns + m 1−s_{− 1,}

Í(˝¬ú

s

}

γ0+ 2γ1(s − 1) + · · · + (m + 1)γm(s − 1)m = m X n=1 1 ns + (s − 1)(−1) 1 m X n=1 ln n ns + (−1)1(ln m) m1−s,

yI

_{s → 1} γ0 = m X n=1 1 n + (−1) 1 (ln m) ,

w2

_γ0

\˚5Ñr…

-

ïn˚−b

(Euler − Mascheroni constant),

?ª[Ñ

γ0 = lim m→∞ m X n=1 1 n − (ln m) ,

./ú

s

}

2!γ1+ 3 × 2γ2(s − 1) + · · · + (m + 1) × mγm(s − 1)m−1 = 2(−1)1 m X n=1 ln n ns + (s − 1)(−1) 2 m X n=1 (ln n)2 ns + (−1) 2_{(ln m)}2 ,

yI

_{s → 1} 2!γ1 = 2(−1)1 m X n=1 ln n n + (−1) 2_{(ln m)}2 ,

ku

γ1 = 2 2!(−1) 1 m X n=1 ln n n + (−1)2 2! (ln m) 2 ,

½µ,HT

,

./ú

s

}

3!γ2+ 4 × 3 × 2γ3(s − 1) + · · · + (m + 1) × m × (m − 1)γm(s − 1)m−2 = 3(−1)2 m X n=1 (ln n)2 ns + (s − 1)(−1) 3 m X n=1 (ln n)3 ns + (−1) 3_{(ln m)}3 ,

yI

_{s → 1} 3!γ2 = 3(−1)2 m X n=1 (ln n)2 n + (−1) 3_{(ln m)}3 ,

ku

γ2 = 3 3!(−1) 2 m X n=1 (ln n)2 n + (−1)3 3! (ln m) 3 , .. .

M/ú

s

}

,

ç}

k + 1

Ÿ(

_,

ô/,H5d

,

ª)ƒ

(k + 1)!γk = (k + 1)(−1)k m X n=1 (ln n)k ns + (s − 1)(−1) k+1 m X n=1 (ln n)k+1 ns + (−1)k+1(ln m)k+1,

yI

_{s → 1} (k + 1)!γk = (k + 1)(−1)k m X n=1 (ln n)k n + (−1) k+1_{(ln m)}k+1 ,

Ĥ

γk = (k + 1) (k + 1)!(−1) k m X n=1 (ln n)k n + (−1)k+1 (k + 1)!(ln m) k+1 = (−1) k k! " _m X n=1 (ln n)k n − (ln m)k+1 k + 1 # ,

]

_m

ªJô

_.B

∞ ,

ku

γk = (−1)k k! m→∞lim " _m X n=1 (ln n)k n − (ln m)k+1 k + 1 # .

)„

 4_{” 3.1.4.}

q

C(n)

Ñ

_n

í£Äb_b

,

†

_ζ2_{(s) =}P∞ n=1 C(n) ns , s > 1