Linearized product of two Riemann zeta functions - Project Euclid

No. 8]

Proc. Japan Acad., 90, Ser. A (2014)

123

Linearized product of two Riemann zeta functions By Debika B ANERJEE and Jay M EHTA Harish Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, Uttar Pradesh, India (Communicated by Masaki KASHIWARA,

M.J.A.,

Sept. 12, 2014)

Abstract: In this paper, we elucidate the well-known Wilton’s formula for the product of two Riemann zeta functions. A proof of Wilton’s expression for product of two zeta functions was given by M. Nakajima in [5] using the Atkinson dissection. On the similar line we derive Wilton’s formula using the Riesz sum of the order  ¼ 1. Key words: zeta function.

Wilton’s formula; Riesz sums; Atkinson dissection; Dirichlet series; Riemann

1. Introduction. In [5], M. Nakajima derived an expression for the product of two Dirichlet series. From this expression he gives a proof for the well-known Wilton’s formula [6] and Bellman’s formula [2] for the product of two Riemann zeta functions in particular. In this method, he uses the Atkinson dissection [1]. The Atkinson1 dissection involves splitting of P the double sum as m;n¼1 1 X

¼

1 X X

þ

m¼1 n  ;

where fn g and fn g are increasing sequences of real numbers and n and an are complex numbers. Assume that they are continued to meromorphic functions over the whole plane and that they satisfy the growth condition ’ð þ itÞ  ðjtj þ 1Þs’ ðÞ ; ð þ itÞ  ðjtj þ 1Þs ðÞ in the strip b <  < c ðb > 0; c > 0Þ. In case of the Riemann zeta function, s ðbÞ ¼ 12 þ b. In this paper, we consider an integral of the following form for Dirichlet series  and ’, (for c > 0 and   0), F ðcÞ ð’ðuÞ; ðvÞÞ ¼ F ðcÞ ð’ðuÞ; ðvÞ; xÞ Z 1 ðwÞ ¼ ’ðu þ wÞðv  wÞxwþ dw; 2i ðcÞ ðw þ  þ 1Þ and its counterpart F ðcÞ ððvÞ; ’ðuÞÞ under the condition ’ þ c;

x

0

doi: 10.3792/pjaa.90.123 #2014 The Japan Academy

where A ðxÞ ¼ A0 ðxÞ ¼

 þ c:

In the next section, we consider  and ’ as the Riemann zeta functions in particular. The Atkinson dissection is the special case of the Riesz sum A ðxÞ with  ¼ 0 in the sense that ð1Þ F ðcÞ ð’ðuÞ; ðvÞÞ þ F ðcÞ ððvÞ; ’ðuÞÞ

124

D. BANERJEE and J. MEHTA

¼

1 X0 X 1  v am  m n u n ðm x  n Þ ð þ 1Þ m¼1 n m x

þ

1 X X0 1 v n nu a m m ðn x  m Þ ð þ 1Þ n¼1 m n x

implies 1 X v ð2Þ m u m an n m;n¼1 1 X X

¼

v m u m a n n

m¼1 n