ACTA ARITHMETICA LXIX.2 (1995)

ACTA ARITHMETICA LXIX.2 (1995)

A χ -analogue of a formula of Ramanujan for ζ(1/2)

by

Shigeki Egami (Toyama)

To the memory of Professor Norikata Nakagoshi

In his famous Notebooks ([4]) Ramanujan stated the following formula for ζ(1/2): For τ > 0,

X ∞

n=1

1

e ^{τ n}

²

−1 = 1 6τ + 1

2 r π

τ ζ

 1 2

 + 1

4

+ 1 2

r π τ

X ∞

n=1

√ 1 n

 sinh(2π p

πn/τ ) − sin(2π p πn/τ ) cosh(2π p

πn/τ ) − cos(2π p

πn/τ ) − 1

 .

Berndt and Evans ([2], see also [1]) gave a proof of this formula by using the Poisson summation formula. The purpose of this paper is to show a similar formula for the value L(1/2, χ) of Dirichlet L-functions. Our proof based on the Mellin transform is substantially different from [2].

The motivation for this work came from a discussion with Masanori Katsurada. The author would like to thank him.

Let q be a positive integer, χ a primitive Dirichlet character modu- lo q, and L(s, χ) the Dirichlet L-function for χ. Furthermore, we will use the following standard notation:

E(χ) =

 1 if χ is principal, 0 otherwise, W (χ) =

 √ qg(χ) ⁻¹ for χ(−1) = 1, i √

qg(χ) ⁻¹ for χ(−1) = −1, where

g(χ) = X

a mod q

χ(a)e ^2πia/q . Then our result can be stated as follows:

[189]

190 S. Egami

Theorem. For τ > 0, X

k mod q

χ(k) X ∞

n=1

e ^−kn

²

^τ 1 − e ^−qn

²

^τ

= E(χ)π ² 6τ − 1

2 L(0, χ) + 1 2

r π τ L

 1 2 , χ

 + 1

2 r π

τ W (χ)

× X ∞

n=1

χ(n) √ n

 sinh(2π p

πn/(qτ ) − χ(−1) sin(2π p

πn/(qτ )) cosh(2π p

πn/(qτ )) − cos(2π p

πn/(qτ )) − 1

 .

P r o o f. First we express the left hand side of the above equation by the inverse Mellin integral:

I(τ, χ) = X

k mod q

χ(k) X ∞

n=1

e ^−kn

²

^τ

1 − e ^−qn

²

^τ = X

k mod q

χ(k) X ∞

m=1

X ∞

n=1

e ^−(kn

²

^{τ +qmn}

²

^{τ )}

= X ∞

n=1

X ∞

m=1

χ(m)e ^{−τ n}

²

^m = 1 2πi

R

(c)

Γ (s)ζ(2s)L(s, χ)τ ^−s ds, where c > 1 and R

(c) denotes the integral along the line <s = c. Shifting the line of integration to <s = 1/2 − c and changing the variable s ↔ 1/2 − s we have

I(τ, χ) = 1 2πi

R

(c)

Γ

 1 2 − s



ζ(1 − 2s)L

 1 2 − s, χ



τ ^−1/2+s ds + R(τ, χ),

where R(τ, χ) denotes the sum of the residues at s = 1, 1/2, and 0,

R(τ, χ) = E(χ)π ² 6τ + 1

2 r π

τ L

 1 2 , χ



− L(0, χ).

Using the functional equations for ζ(s) and L(s, χ) (see e.g. [3], p. 59 and p. 71) we have

I(τ, χ) − R(τ, χ) = W (χ) πi

R

(c)

Γ (2s)ζ(2s)L

 1 2 + s, χ



×

 cos

 πs 2



− χ(−1) sin

 πs 2

 qτ (2π) ³

 _s ds

= W (χ) 2πi

X ∞

m,n=1

R

(2c)

Γ (s)

r qτ

(2π) ³ m ² n

 _s

×

 cos

 πs 4



− χ(−1) sin

 πs 4



ds.

A formula of Ramanujan 191

In order to calculate each integral in the above double series we note that 1

2πi

R

(c)

Γ (s)

 cos(πs/4) sin(πs/4)



x ^−s ds = e ^−x/

√ 2

 cos(x/ √ 2) sin(x/ √

2)

 ,

which can easily be obtained from the well known formula:

e ^−(x+iy) = 1 2πi

R

(c)

Γ (s)(x + iy) ^−s ds (x, c > 0).

Then we observe I(τ, χ) − R(τ, χ)

= W (χ) X ∞

n=1

χ(n) √ n

X ∞

m=1

e ^−2π √

_πn

qτ

×

 cos

 2πm

r πn qτ



− χ(−1) sin

 2πm

r πn qτ



= W (χ) X ∞

n=1

χ(n) √ n

 sinh(2π p

πn/(qτ )) − χ(−1) sin(2π p

πn/(qτ )) cosh(2π p

πn/(qτ )) − cos(2π p

πn/(qτ )) −1

 , which completes the proof of the Theorem.

References

[1] B. C. B e r n d t, Ramanujan’s Notebooks, Part II , Springer, 1989.

[2] B. C. B e r n d t and R. J. E v a n s, Chapter 15 of the Ramanujan Second Notebook:

Part II , Modular forms, Acta Arith. 47 (1986), 123–142.

[3] H. D a v e n p o r t, Multiplicative Number Theory, 2nd ed., Graduate Texts in Math.

74, Springer, 1980.

[4] S. R a m a n u j a n, Notebooks, 2 Vols., Tata Institute, Bombay, 1957.

FACULTY OF ENGINEERING TOYAMA UNIVERSITY 3190 GOFUKU, TOYAMA CITY 930 JAPAN

Received on 16.2.1994 (2564)