Geodesic
Fast methods to compute the Riemann zeta function
Ghaith Ayesh Hiary1
1 Pure Mathematics<br/> University of Waterloo<br/>Waterloo, Ontario<br/> Canada N2L 3G1
Annals of mathematics, Tome 174 (2011) no. 2, pp. 891-946.

Voir la notice de l'article provenant de la source Annals of Mathematics website

The Riemann zeta function on the critical line can be computed using a straightforward application of the Riemann-Siegel formula, Schönhage’s method, or Heath-Brown’s method. The complexities of these methods have exponents 1/2, 3/8, and 1/3 respectively. In this article, three new fast and potentially practical methods to compute zeta are presented. One method is very simple. Its complexity has exponent 2/5. A second method relies on this author’s algorithm to compute quadratic exponential sums. Its complexity has exponent 1/3. The third method, which is our main result, employs an algorithm developed here to compute cubic exponential sums with a small cubic coefficient. Its complexity has exponent 4/13 (approximately, 0.307).
DOI : 10.4007/annals.2011.174.2.4

Ghaith Ayesh Hiary 1

1 Pure Mathematics<br/> University of Waterloo<br/>Waterloo, Ontario<br/> Canada N2L 3G1 
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Ghaith Ayesh Hiary. Fast methods to compute the Riemann zeta function. Annals of mathematics, Tome 174 (2011) no. 2, pp. 891-946. doi : 10.4007/annals.2011.174.2.4. http://geodesic.mathdoc.fr/articles/10.4007/annals.2011.174.2.4/

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