Computing the truncated theta function via Mordell integral
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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.References
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Additional Information
- A. Kuznetsov
- Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, M3J 1P3, Canada
- MR Author ID: 757149
- Email: kuznetsov@mathstat.yorku.ca
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Math. Comp. 84 (2015), 2911-2926
- MSC (2010): Primary 11Y16, 11M06
- DOI: https://doi.org/10.1090/mcom/2953
- MathSciNet review: 3378853