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An alternative to Riemann-Siegel type formulas


Author: Ghaith A. Hiary
Journal: Math. Comp. 85 (2016), 1017-1032
MSC (2010): Primary 11M06, 11Y16; Secondary 68Q25
Published electronically: October 15, 2015
MathSciNet review: 3434892
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Abstract: Simple unsmoothed formulas to compute the Riemann zeta function, and Dirichlet $ L$-functions to a powerfull modulus, are derived by elementary means (Taylor expansions and the geometric series). The formulas enable the square-root of the analytic conductor complexity, up to logarithmic loss, and have an explicit remainder term that is easy to control. The formula for zeta yields a convexity bound of the same strength as that from the Riemann-Siegel formula, up to a constant factor. Practical parameter choices are discussed.


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Additional Information

Ghaith A. Hiary
Affiliation: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210

DOI: https://doi.org/10.1090/mcom/3019
Keywords: Riemann zeta function, Dirichlet $L$-functions, algorithms
Received by editor(s): March 20, 2014
Received by editor(s) in revised form: July 31, 2014
Published electronically: October 15, 2015
Additional Notes: Preparation of this material was partially supported by the National Science Foundation under agreement No. DMS-0932078 (while at MSRI) and DMS-1406190, and by the Leverhulme Trust (while at the University of Bristol).
Article copyright: © Copyright 2015 American Mathematical Society