values of the riemann zeta function and integrals involving log
TRANSCRIPT
PacificJournal ofMathematics
VALUES OF THE RIEMANN ZETA FUNCTION ANDINTEGRALS INVOLVING log(2 sinh θ2 ) AND log(2 sin θ
2 )
ZHANG NAN-YUE AND KENNETH S. WILLIAMS
Volume 168 No. 2 April 1995
PACIFIC JOURNAL OF MATHEMATICS
Vol. 168, No. 2, 1995
VALUES OF THE RIEMANN ZETA FUNCTION ANDINTEGRALS INVOLVING log(2sinhf) AND log (2 sin f)
ZHANG NAN-YUE AND KENNETH S. WILLIAMS
Integrals involving the functions log (2 sinh(0/2)) andlog(2sin(0/2)) are studied, particularly their relationshipto the values of the Riemann zeta function at integral ar-guments. For example general formulae are proved whichcontain the known results
Γ log2 (2sin(0/2))d0 = 7π3/108,Jo
Γ 01og2 (2sin(0/2))d0 = 17τr4/6480,Jo
/ (log4(2sin(0/2)) - ^02log2(2sin(0/2)))d0 = 253π5/3240,Jo 2
/ (01og4(2sin(0/2)) - i-log(2sin(0/2)))d0 = 313π6/408240,Jo *
as special cases.
1. Introduction. Since the discovery of the formulae
(1.1)
21ogτ
=\ζ(3), where τ=ho 2
the relationship between the values of the Riemann zeta function
and integrals involving log ί 2 sin | J and Iogί2sinh| j has been
studied by many authors, see for example [2], [4], [5], [7], [9].
271
272 ZHANG NAN-YUE AND K.S. WILLIAMS
Recently Butzer, Markett and Schmidt [2] made use of centraland Stirling numbers to obtain a representation of ζ(2m + 1) by
integrals involving log ί 2 s i n h | J (see (2.13)). In §2 of this paper,
we reprove (2.13) and at the same time prove the analogous formulafor C(2m) (see (2.14)). Note that (1.2) is the special case of (2.13)when m = 1.
In [7], van der Poorten proves (1.1), as well as the formula
(1.3) / log2 ί 2 sin ~)dθ = —JO \ Δ/ l U o
and remarks that "It appears that (1.1) and (1.3) are not represen-tative of a much larger class of similar formulas". However in [9]Zucker establishes the two formulae
(1.4)
In §3, we prove the general formulae
(1.6)
4m(2ro -
22m+2 L ^ (2m + 1 ) 3 2 "
where the jB2m and E<ιm are the Bernoulli and Euler numbers respec-tively. We remark that (1.1), (1.3), (1.4) and (1.5) are all special
VALUES OF THE RIEMANN ZETA FUNCTION 273
cases of (1.6) and (1.7). Formula (1.6) is basically formula (3.10b)of [2] and formula (1.7) is essentially Theorem 4.1 of [1].
In §4, we establish the relations:
(1.8)
^Jnm+2(2nJ ml Jo V 2)
(1.9)
= . / ^!ogm (2sinh^ Jd»,
m! 7o 6 V 2/
which generalize the formulae (1.1) and (1.2). The formula (1.8)was given by Zucker [4, (2.5)]. Formula (1.9) can be established inan analogous fashion.
2. Representation of ζ(ή) by integrals involving
log ί 2 sinh f 1. For k > 1, x > 1, we have
since
Λ tit -t\ Λ £ . , θ
e — I = e* le* — e 2 J = 2 e 2 s i n h -,
-.
Similarly, we have
We set
(2.3)
274 ZHANG NAN-YUE AND K.S. WILLIAMS
Taking x = τ = | ( 1 + Λ/5) in (2.2) and (2.1) gives respectively
ί lθ£*ϊ l t) / 21ogτ(2 4) / w L ! 5 £ i a * ir2\ogτ Γ f)"\k
( 2 . 5 ) g ( k ) = l " & v" * ' d t = - [ \A{Θ) + - \ d θ .
Now we evaluate f(k) and g(k). We have
(2.6)
and
(o U - u
-du
o 1-r io 1-ίi
2k Jo 1 — u Jo 1 — u
1 r f1 log* u 1
2k L «/τ2 1 — u J
t h a t is
1/I= (2* -
In the two integrals, using the substitution t = £ and noting τγz
Y3J + j , we have
•^Tlog^r
VALUES OF THE RIEMANN ZETA FUNCTION 275
and
= {-l)kI(k) + (-1)*+1 J7 jz^dt
Hence we have
/ - \ ^ 1 i b Λ 4*4-1
(—1) [7* log t log ^ r+ 2 k Jo 1-t t ~ k + 1 '
Thus
τ~ιt \ogzmr
-1 2m
Prom (2.6), we obtain
(2.7) /(2m) + 9{2m) = φ, j; >-f-fdt - ^ - ^ ,
(2.8)
/(2m - 1) - ^(2m - 1) =2^1 - ^ π V ( 2 m - 1)
1 AT* log 2" 1" 1 ί l o g 2 m r+ 2 2 m ~ 1 Jo 1 - t + 2 m
Next we evaluate the integral / o ' τ 1-ψ-£dt. Since
1 — ί
276 ZHANG NAN-YUE AND K.S. WILLIAMS
it suffices to evaluate
- u )i: -duu
= / log*(l -u)d\oguJo
= log*(l — u) log u + k }~**"' duo o 1 — u
logfc+1 r + A;(-l) fc+1 /~"° θk~1A(θ)dθ.=K ~ι J.
Hence we have
(2.9)o g i 2 2 " ι log 2 m + 1 τ, ± dt =/(2m) + — o
6 ,\-t κ ' 2m +1/ 21ogτ
+ 2m / θ2m-1A{θ)dθ,Jo
(2.10)
1 2mι 2 log Tι 2 log T
- (2m - 1) / θ2m-2A(θ)dθ.Jo
Substituting (2.9), (2.10) into (2.7), (2.8) respectively, we obtain
(2.11) /(2m) + g(2m) =
(2.12)
/(2m - 1) - g(2m - 1) = 2 ( 1 - ^ ) / ( 2 m - 1)
Combining (2.11), (2.12) with (2.3), (2.4), (2.5) gives
VALUES OF THE RIEMANN ZETA FUNCTION 277
THEOREM 1. For m = 1,2,...
(2.13)
2 2 m / 21ogτ (Γ ( θ\ 01 2 m
' (2m)! Λ
- [ l o g ( 2 s i n h ^
and
(2.14)
/.21ogr
As previously remarked (2.13) is due to Butzer, Markett andSchmidt [2], while (2.14) appears to be new. We note that (2.13)can be rewritten as
8 T
(2m - 1)!
and (2.14) as
(2.16)
i 7Γ ( 2
4m
278 ZHANG NAN-YUE AND K.S. WILLIAMS
since
Taking m — 1, 2 in (2.15), we have
0(2.18) ^ 0 1 o g ( 2 s i n h £ ) d 0
(2.19)
= -fc(5).flMog ( 2 sinh£)]d0 f
Taking m = 1,2 in (2.16) gives
/ 21ogτ / 0\ π2
(2.20) yo l o g ( 2 ώ ι h - J d » = - - ,
(2.21)
3. Representation of ζ(n) by integrals involving
logΓ2sin|j. We set
B = B(Θ)= log (2sin^j,
so that
We consider the integral
(3.1) J(k) = Γ ^ ^ - " f f A . "H" if*(B+ >2Ji u Jo \ I
where the first integral is along the arc of the unit circle \u\ = 1 fromu = ltou = ω = em/3 in a counter-clockwise direction. Makingthe substitution t — i(l — u), u = i(t — i), du = idt, we obtain
( 3 . 2 ) it / Λ + fo t — i Jo t — i Ji t — i
VALUES OF THE RIEMANN ZETA FUNCTION 279
We now evaluate the two integrals on the right side of (3.2). Wehave
/•! log* t , f1 t + i , u/ — — dt = / log* t
Jo t-i Jo t2 + l 6 dt
1 /-i l o g * ί , , /-i l o g * t
2io 1 + ί2 Λ 1 + ί2+ + ί2
Since
O O / , x n
where S(s) = Σ (2n+na (^ e ( 5 ) > 0) is a particular Dirichlet
function (see (4.4) in [8]) and
we have
(3 3) I ' Ί~Jdt = ¥{1~ ¥For the second integral in (3.2), we have
Jo 2J 2iθ \2J 2
•k+i ,5 I cos 5 + ιyί + sm^J^θkdθi:
-L ϊ _ I _ (i\k+1 • / f 1 + s i n f~[k~+\\?>) +%Jo cos f
1 / π \ f c + 1 ifc /"I ^* cos -
TiyUJ * + ~¥+iJo l-sini
2 f e+2 7o cos f
280 ZHANG NAN-YUE AND K.S. WILLIAMS
Since
— / \dθ = / 0*dlog 1 - sin - )2 Jo 1-sin § 7o 6V 2/
-ήn-)dθ
= -( | ) f c log2 - k j ί f θk~ι log (l - sin?),»,
we have
(3.4)1
t — i
From (3.1)-(3.4), we obtain
(3.5)
Taking A; = 2m — 1 and k = 2m in (3.5) gives respectively
(3-6)
2m
Iog2
VALUES OF THE RIEMANN ZETA FUNCTION 281
(3.7)
22m+l y 2 /
Taking the real part of (3.6) yields
( - l ) m / π \ 2 m 22 m /"I
() + X2(2m)!
Since
- 1\ /Λ\2/c+l
2k ) β2k+l r>2m-2k-2l r>2
2 2 m ~ 1
we have
m-2)!7o2(2m)!32™-1 (2m-2) !7o ^ 0 (2ife
Recalling the formula (2.17), (3.9) gives the following result.
282 ZHANG NAN-YUE AND K.S. WILLIAMS
THEOREM 2. For m>2,
(3.10)
π2™ r/l\*"-i
-l)KβJ "" 4m(2m-l)
Taking m = 2,3 in (3.10) we obtain
(3.Π,
(3.12)
Taking the imaginary part of (3.7) yields
(3.13)/•f / jβ\2m (—λ\m /7Γ\ 2 m + 1
Since
f)and
2 m + 1
(3.14) (2m)!5(2m + 1) = ^
where the m are the Euler numbers, we have from (3.13)
THEOREM 3. For m > 1,
(3-15) / 3 Σ M i Γ 2 ^ Iog2m"2fc (2 sin Ik=0
( - l ) m π 2 m + 1 Γ 122m+2
VALUES OF THE RIEMANN ZETA FUNCTION 283
Taking m = 1,2,3 in (3.15), we have
(3.16) j jV (»-»?)*=
(3.17)
(3.18)
; [ V (2-.?) - %>>V* (j-π?) + ί log (ί-oί)]d»_ 77821ττ7
~ 26 3 6 7"
Taking the imaginary part of (3.6), we obtain
(3.19)2m-χ
= -(2m - l)!5(2m)
Since
/ , 0\2m-l
( %)
we have
(3.20)
^ d ( ) los2
(2m - 2)!22 m Jo
TO-l ( —
+2 2 f c
1 (2sin-)^.
284 ZHANG NAN-YUE AND K.S. WILLIAMS
Taking m = 1 in (3.20) we obtain
5(2) = -^rlog2 -- Γ log fl - siJ-λdθ - Γ log (2ύJ\IΔ 4 JO \ U JO \ IJ
where 5(2) = Σ ή^ψ = 0.915965... is Catalan's constant.
Taking the real part of (3.7), we have
ΊL
3 I m
+ -2
In view of
we obtain
(3.21)
2 m
Taking m = 1,2 in (3.21), we obtain
(3.22)
(3.23) j ί § [βlog3 (2sin 5) - J*3log (2sin
VALUES OF THE RIEMANN ZETA FUNCTION 285
4. Relations between integrals involving log (2 sin f) and
logί2sinh|J and certain series. The power series expansion of
3ftf ^ given by
arcsin z ^ 22nx2n~
Integrating and differentiating this equality, we have
(4.1) (arcsinα;)2 = f; K 'ά5 2»'(ϊ)
(2x)2n x2 x arcsin x
Next, we apply the method of constructing polylogarithms to thefunction
„ . . 2x arcsin x ^ (2x)2n
We set
τ = (arcsin x) =
In general, we have
(4.3)
Km[x)=
Taking x = \ in (4.3) gives
-dx.2) h X
286 ZHANG NAN-YUE AND K.S. WILLIAMS
Then, using integration by parts (m — 1) times, we obtain
-dx
= ~ jί* Km_2{x)d\og\2x) = A Jj
(-1)" 1- 1 y l . m_1/n ,2arcsina; , / . θ\= . . . = 7—-—ΓT / logm 1(2x) r—=dx [x = sin -
(m-l)!yo V ; x / Γ ^ ^ V 2/
that is
(4.4)-j / -I \mn
=
m!
Taking a; = | in (4.2), (4.3) and m = 0,1,2 in (4.4), we obtain
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
VALUES OF THE RIEMANN ZETA FUNCTION 287
Changing x into ix in (4.1) (4.2) and (4.3) with m = 0, we have
n _ x .
~ ! + 2( 2
n
n )
(4.13) y ; ( 1 } W = (sinh-1 α;)2.22(2")n = i
Now, taking a; = \ in (4.11), (4.12), (4.13), we deduce (as sinh"1 \ =logr)
( 4 1 4 )
(4.16) Σ<-±>
Analogous to the construction of Km(x), we set
„ , x 2xsinh~1 x „ , ,F ( ) F ( ) = ( s i n h
and, from (4.11) and (4.12), we have
(4.17)
o x
After integration by parts, we obtain
/Il o g
288 ZHANG NAN-YUE AND K.S. WILLIAMS
that is
(4-18)
Σ ( ΓiC\ = ( " 1 ) ? 2 " 1 /2 I θ 8 T^ogm (2ώΛ?)«»,m = 0,1,2,... .
Taking m = 1 in (4.18) and appealing to (2.18), we have the well-known formula
( 4 1 9 ) 2
Taking m = 3 in (4.18) yields
and substituting into (2.18) gives
oo (_]_)n-l 5 r21ogτ
Since
log
from (4.21) we obtain
(4.22)
20 (-!ΓC(2n)83 ^ n{n + 2)π2
REFERENCES
[1] P.L. Butzer and M. Hauss, Integral and rapidly converging series repre-sentations of the Dirichlet L-functions L\{s) and L_4(s), Atti Sem. Mat.Fis. Univ. Modena, 40 (1992), 329-359.
VALUES OF THE RIEMANN ZETA FUNCTION 289
[2] P.L. Butzer, C. Markett and M. Schmidt, Stirling numbers, central fac-torial numbers, and representation of the Riemann zeta function, Resultsin Mathematics, 19 (1991), 257-274.
[3] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products,Academic Press, New York (1980).
[4] D.H. Lehmer, Interesting series involving the central binomial coefficient,Amer. Math. Monthly, 92 (1985), 449-457.
[5] D. Leshchiner, Some new identities for ζ(k), J. Number Theory, 13(1981), 355-362.
[6] L. Lewin, Poly logarithms and associated functions, North Holland, NewYork, 1981.
[7] A.J. Van der Poorten, Some wonderful formulas . . . an introduction topolylogarithms, Queen's papers in Pure and Applied Mathematics, 54(1979), 269-286.
[8] Kenneth S. Williams and Zhang Nan-Yue, Special values of the Lerchzeta function and the evaluation of certain integrals, Proc. Amer. Math.Soc, 119 (1993), 35-49.
[9] I.J. Zucker, On the series ]Γ (2^) k~n and related sums, J. Number
Theory, 20 (1985), 92-102.
Received November 5, 1992 and in revised form November 16, 1993. The first
author's research was supported by the National Natural Science Foundation
in China. The second author's research was supported by Natural Sciences and
Engineering Research Council of Canada Grant A-7233.
CARLETON UNIVERSITY
OTTAWA, ONTARIO
CANADA K1S 5B6
E-mail address: [email protected]
CONTENTS
HL Aelaksen* E,«C. Tm and C. Zhu, Invariant theory of special
orthogonal groupe ........................ 207
M. Brίt tenham* Bs&ential kmrnatione and Haken normal form 217
J Biirbea and S *Y. Π , Weighted Hadamard products of holomor-
phie ftietioi^ ϊn the ball ,...,*<>, — ...„„....„„.... 235
Z* Nan-Yne and K«S. Williams, Values of the Riemaim ^eta fiiπc-
tion md integrate ixiYolviag log (2 sinh #02) and log (2 sin #02) . . . . . . . 271
M* O^uchi, A diίferβBtiable etnictiire for a bundle of C*~aJgebras a ^ >
dated with a dynamical system 291
E. Paolettί^ Generalized Wahl maps aEd adjoint line bundles on a ge-
neral curve. , , , , . . « — ...*** — . . . . . . . . . . . . . . . . 313
V. Pati , M. Shahshahanί and A Sitaram t The spherical meaE
value operator for compact symmetric epaces. 335
J, Shurman, Fourier coefficieats of an orthogonal Eisenstein series 345
S*N* Ziesler? I/~botmdedπes8 of the Hubert transform and maximal
function along flat curves in W1 „ , . . . . 383
^^^^^pill ^^^^^^^^^^^Hll
PACIFIC JOURNAL OF MATHEMATICS
Volume 168 No. 2 April 1995
207Invariant theory of special orthogonal groupsHELMER ASLAKSEN, ENG-CHYE TAN and CHEN-BO ZHU
217Essential laminations and Haken normal formMARK BRITTENHAM
235Weighted Hadamard products of holomorphic functions in the ballJACOB BURBEA and SONG-YING LI
271Values of the Riemann zeta function and integrals involvinglog(2 sinh θ2 ) and log(2 sin θ
2 )
ZHANG NAN-YUE and KENNETH S. WILLIAMS
291A differentiable structure for a bundle of C∗-algebras associated with adynamical system
MOTO O’UCHI
313Generalized Wahl maps and adjoint line bundles on a general curveROBERTO PAOLETTI
335The spherical mean value operator for compact symmetric spacesVISHWAMBHAR PATI, MEHRDAD MIRSHAMS SHAHSHAHANIand ALLADI SITARAM
345Fourier coefficients of an orthogonal Eisenstein seriesJERRY MICHAEL SHURMAN
383L p-boundedness of the Hilbert transform and maximal function alongflat curves in Rn
SARAH N. ZIESLER
PacificJournalofM
athematics
1995Vol.168,N
o.2