Riemann-Siegel integral formula for the Lerch zeta function

Balanzario, Eugenio; Sánchez-Ortiz, Jorge

doi:10.1090/S0025-5718-2011-02566-4

Riemann-Siegel integral formula for the Lerch zeta function
HTML articles powered by AMS MathViewer

by Eugenio P. Balanzario and Jorge Sánchez-Ortiz PDF

Math. Comp. 81 (2012), 2319-2333 Request permission

Abstract:

Here we present a Riemann-Siegel integral formula for the Lerch zeta function. Proceeding as in Turing’s method for computing the Riemann zeta function, our integral formula allows for the numerical computation of the Lerch zeta function by numerical quadratures.

References

Eugenio P. Balanzario, A Riemann-Siegel formula for the Hurwitz zeta function, Bol. Soc. Mat. Mexicana (3) 10 (2004), no. 1, 1–13. MR 2071998
Max Deuring, Asymptotische Entwicklungen der Dirichletschen $L$-Reihen, Math. Ann. 168 (1967), 1–30 (German). MR 213309, DOI 10.1007/BF01361542
Edmund A. C. Crouch and Donna Spiegelman, The evaluation of integrals of the form $\int ^{+\infty }_{-\infty }f(t)\exp (-t^2)\,dt$: application to logistic-normal models, J. Amer. Statist. Assoc. 85 (1990), no. 410, 464–469. MR 1141749
N. G. de Bruijn, Asymptotic methods in analysis, Bibliotheca Mathematica, Vol. IV, North-Holland Publishing Co., Amsterdam; P. Noordhoff Ltd., Groningen; Interscience Publishers Inc., New York, 1958. MR 0099564
H. M. Edwards, Riemann’s zeta function, Pure and Applied Mathematics, Vol. 58, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0466039
William F. Galway, Computing the Riemann zeta function by numerical quadrature, Dynamical, spectral, and arithmetic zeta functions (San Antonio, TX, 1999) Contemp. Math., vol. 290, Amer. Math. Soc., Providence, RI, 2001, pp. 81–91. MR 1868470, DOI 10.1090/conm/290/04575
E. T. Goodwin, The evaluation of integrals of the form $\int ^\infty _{-\infty } f(x) e^{-x^{2}} dx$, Proc. Cambridge Philos. Soc. 45 (1949), 241–245. MR 29281, DOI 10.1017/s0305004100024786
Lagarias, J.C.; Wen-Ching, Winnie Li, The Lerch zeta function I. Zeta integrals, Forum Mathematicum, published online.
Antanas Laurinčikas and Ramūnas Garunk tis, The Lerch zeta-function, Kluwer Academic Publishers, Dordrecht, 2002. MR 1979048
M. Lerch, Note sur la fonction ${\mathfrak {K}} \left ( {w,x,s} \right ) = \sum \limits _{k = 0}^\infty {\frac {{e^{2k\pi ix} }}{{\left ( {w + k} \right )^s }}}$, Acta Math. 11 (1887), no. 1-4, 19–24 (French). MR 1554747, DOI 10.1007/BF02418041
John McNamee, Error-bounds for the evaluation of integrals by the Euler-Maclaurin formula and by Gauss-type formulae, Math. Comp. 18 (1964), 368–381. MR 185804, DOI 10.1090/S0025-5718-1964-0185804-1
A. M. Turing, A method for the calculation of the zeta-function, Proc. London Math. Soc. (2) 48 (1943), 180–197. MR 9612, DOI 10.1112/plms/s2-48.1.180