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Riemann-Siegel integral formula for the Lerch zeta function
Authors:
Eugenio P. Balanzario and Jorge Sánchez-Ortiz
Journal:
Math. Comp. 81 (2012), 2319-2333
MSC (2010):
Primary 11M35
Published electronically:
November 29, 2011
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Additional Information
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.
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0213309 (35 #4173)
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T. Goodwin, The evaluation of integrals of the form
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- 2.
- Deuring, M., Asymptotische Entwicklungen der Dirichletschen
 -Reihen., Math. Ann. 168, 1967, 1-30. MR 0213309 (35:4173)
- 3.
- Crouch, E.A.C.; Spiegelman, D., The evaluation of integrals of the form
 , application to logistic-normal models, J. Amer. Statist. Assoc. 85 (1990), no. 410, 464-469. MR 1141749 (92h:65032)
- 4.
- de Bruijn, N.G., Asymptotic methods in analysis, North-Holland, 1958. MR 0099564 (20:6003)
- 5.
- Edwards, H.M., Riemann's zeta function, Pure and Applied Mathematics, Vol. 58. Academic Press, New York-London, 1974. MR 0466039 (57:5922)
- 6.
- Galway, W.F., Computing the Riemann zeta function by numerical quadrature, Dynamical, spectral, and arithmetic zeta functions, San Antonio, TX, 1999, 81-91, Contemp. Math., 290, Amer. Math. Soc., 2001. MR 1868470 (2002i:11131)
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 ., Proc. Cambridge Philos. Soc. 45 (1949), 241-245. MR 0029281 (10:575f)
- 8.
- Lagarias, J.C.; Wen-Ching, Winnie Li, The Lerch zeta function I. Zeta integrals, Forum Mathematicum, published online.
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- Laurinčikas, A.; Garunkštis, R., The Lerch zeta-function, Kluwer, 2002. MR 1979048 (2004c:11161)
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- Lerch, M., Note sur la fonction
 , Acta Math. 11 (1887), 19-24. MR 1554747
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- McNamee, J., Error-bounds for the evaluation of integrals by the Euler-Maclaurin formula and by Gauss-type formulae, Math. Comp. 18 (1964), 368-381. MR 0185804 (32:3264)
- 12.
- Turing, A.M., A method for the calculation of the zeta-function, Proc. London Math. Soc. (2) 48 (1943), 180-197. MR 0009612 (5:173a)
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Additional Information
Eugenio P. Balanzario
Affiliation:
Instituto de Matemáticas, Unidad Morelia, Universidad Nacional Autónoma de México
Email:
ebg@matmor.unam.mx
Jorge Sánchez-Ortiz
Affiliation:
Apartado Postal 61-3 (Xangari), 58089, Morelia Michoacán, México
Address at time of publication:
Facultad de Matemáticas, Av. Lázaro Cárdenas S/N, Ciudad Universitaria, Chilpancingo Gro., México
Email:
jsanchez@matmor.unam.mx
DOI:
http://dx.doi.org/10.1090/S0025-5718-2011-02566-4
PII:
S 0025-5718(2011)02566-4
Keywords:
Lerch zeta function,
Riemann-Siegel,
saddle point
Received by editor(s):
September 27, 2010
Received by editor(s) in revised form:
April 15, 2011
Published electronically:
November 29, 2011
Article copyright:
© Copyright 2011
American Mathematical Society
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