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Computing the truncated theta function via Mordell integral

Author: A. Kuznetsov
Journal: Math. Comp. 84 (2015), 2911-2926
MSC (2010): Primary 11Y16, 11M06
Published electronically: April 9, 2015
MathSciNet review: 3378853
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Abstract: Hiary has presented an algorithm which allows us to evaluate the truncated theta function $ \sum _{k=0}^n \exp (2\pi \mathrm {i} (zk+\tau k^2))$ to within $ \pm \epsilon $ in $ O(\ln (\tfrac {n}{\epsilon })^{\kappa })$ arithmetic operations for any real $ z$ and $ \tau $. This remarkable result has many applications in Number Theory, in particular, it is the crucial element in Hiary's algorithm for computing $ \zeta (\tfrac {1}{2}+\mathrm {i} t)$ to within $ \pm t^{-\lambda }$ in $ O_{\lambda }(t^{\frac {1}{3}}\ln (t)^{\kappa })$ arithmetic operations. We present a significant simplification of Hiary's algorithm for evaluating the truncated theta function. Our method avoids the use of the Poisson summation formula, and substitutes it with an explicit identity involving the Mordell integral. This results in an algorithm which is efficient, conceptually simple and easy to implement.

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Additional Information

A. Kuznetsov
Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, M3J 1P3, Canada

Keywords: Truncated theta function, Mordell integral, Riemann zeta function
Received by editor(s): June 25, 2013
Received by editor(s) in revised form: February 11, 2014, and March 13, 2014
Published electronically: April 9, 2015
Additional Notes: This research was supported by the Natural Sciences and Engineering Research Council of Canada
Article copyright: © Copyright 2015 American Mathematical Society

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