A generalization of the Riemann–Siegel formula

被引：0
作者：
Cormac O’Sullivan
机构：
[1] The CUNY Graduate Center,Department of Mathematics
来源：
Mathematische Zeitschrift | 2023年 / 303卷
关键词：
11M06;
D O I：
暂无
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摘要：
The celebrated Riemann–Siegel formula compares the Riemann zeta function on the critical line with its partial sums, expressing the difference between them as an expansion in terms of decreasing powers of the imaginary variable t. Siegel anticipated that this formula could be generalized to include the Hardy–Littlewood approximate functional equation, valid in any vertical strip. We give this generalization for the first time. The asymptotics contain Mordell integrals and an interesting new family of polynomials.
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共 25 条
[1]  
Arias de Reyna J(2011)High precision computation of Riemann’s zeta function by the Riemann–Siegel formula. I Math. Comput. 80 995-1009
[2]  
Berry MV(1995)The Riemann–Siegel expansion for the zeta function: high orders and remainders Proc. R. Soc. Lond. Ser. A 450 439-462
[3]  
Bober JW(2018)New computations of the Riemann zeta function on the critical line Exp. Math. 27 125-137
[4]  
Hiary GA(1979)On the zeros of the Riemann zeta function in the critical strip Math. Comput. 33 1361-1372
[5]  
Brent RP(1967)Asymptotische Entwicklungen der Dirichletschen Math. Ann. 168 1-30
[6]  
Deuring M(2017)-Reihen Acta Arith. 177 1-37
[7]  
Dixit A(2005)Error functions, Mordell integrals and an integral analogue of a partial theta function J. Math. Soc. Jpn. 57 513-521
[8]  
Roy A(2013)On a mean value formula for the approximate functional equation of Int. Math. Res. Not. IMRN 20 4712-4733
[9]  
Zaharescu A(1923) in the critical strip Proc. Lond. Math. Soc. 2 39-74
[10]  
Feng S-J(1933)Zeros of a family of approximations of the Riemann zeta-function Acta Math. 61 323-360
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