Relations and positivity results for the derivatives of the Riemann ξ function

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:166

Article

Coffey, MW

关键词: theta function;    functional equation;    Riemann zeta function;    Riemann xi function;    derivatives;    Li criterion;   

DOI  :  10.1016/j.cam.2003.09.003

来源: Elsevier

【 摘 要 】

We present and evaluate the integer-order derivatives of the Riemann xi function. These derivatives contain logarithmic integrals of powers multiplying a specific Jacobi theta function and as such can be alternatively viewed as certain Mellin transforms at integer argument. We describe how the derivatives at s=0, s=1/2 and s=1 can be evaluated exactly. We further show, based upon a novel representation, that the even order derivatives at s=1/2 are all positive, as are all derivatives at s=1. An expression is presented for the derivatives on the critical line, which may be useful in studying the zeros of the function Xi(t)=xi(1/2+it). (C) 2003 Elsevier B.V. All rights reserved.

【 授权许可】

Free