Titchmarsh's Method for the Approximate Functional Equations for ζ′(s)2, ζ(s) ζ"(s), and ζ′(s) ζ"(s)

被引:0
|
作者
Furuya, Jun [1 ]
Minamide, T. Makoto [2 ]
Tanigawa, Yoshio [3 ]
机构
[1] Hamamatsu Univ Sch Med, Dept Integrated Human Sci Math, Handayama 1-20-1, Hamamatsu, Shizuoka 4313192, Japan
[2] Yamaguchi Univ, Grad Sch Sci & Technol Innovat, Yoshida 1677-1, Yamaguchi 7538512, Japan
[3] Nagoya Univ, Grad Sch Math, Furo Cho, Nagoya, Aichi 4648602, Japan
关键词
derivative of the Riemann zeta function; approximate functional equation; exponential sum; ZEROS; DERIVATIVES; BEHAVIOR; MOMENT;
D O I
10.4153/CJM-2018-004-9
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Let zeta(s) be the Riemann zeta function. In 1929, Hardy and Littlewood proved the approximate functional equation for zeta(2)(s) with error term O(x(1/2) (sigma)((x + y)/vertical bar t vertical bar(1/4) log vertical bar t vertical bar), where -1/2 < alpha < 3/2, x, y >= 1, xy = (vertical bar t vertical bar/2 pi)(2). Later, in 1938, Titchmarsh improved the error term by removing the factor ((x + y)/vertical bar t vertical bar(1/4). In 1999, Hall showed the approximate functional equations for zeta'(s)(2), zeta((S) zeta ''(S), and zeta'(s)zeta ''(s) (in the range 0 < sigma < 1) whose error terms contain the factor ((x + y)/vertical bar t vertical bar)(1/4). In this paper we remove this factor from these three error terms by using the method of Titchmarsh.
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页码:1465 / 1493
页数:29
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