Geodesic
A nearly-optimal method to compute the truncated theta function, its derivatives, and integrals
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. 859-889.

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

A poly-log time method to compute the truncated theta function, its derivatives, and integrals is presented. The method is elementary, rigorous, explicit, and suited for computer implementation. We repeatedly apply the Poisson summation formula to the truncated theta function while suitably normalizing the linear and quadratic arguments after each repetition. The method relies on the periodicity of the complex exponential, which enables the suitable normalization of the arguments, and on the self-similarity of the Gaussian, which ensures that we still obtain a truncated theta function after each application of the Poisson summation. In other words, our method relies on modular properties of the theta function. Applications to the numerical computation of the Riemann zeta function and to finding the number of solutions of Waring type Diophantine equations are discussed.
DOI : 10.4007/annals.2011.174.2.3

Ghaith Ayesh Hiary 1

1 Pure Mathematics<br/> University of Waterloo<br/> Waterloo, Ontario<br/> Canada N2L 3G1 
@article{10_4007_annals_2011_174_2_3,
     author = {Ghaith Ayesh Hiary},
     title = {A nearly-optimal method to compute the truncated theta function, its derivatives, and integrals},
     journal = {Annals of mathematics},
     pages = {859--889},
     publisher = {mathdoc},
     volume = {174},
     number = {2},
     year = {2011},
     doi = {10.4007/annals.2011.174.2.3},
     mrnumber = {2831109},
     zbl = {05960718},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2011.174.2.3/}
}
TY  - JOUR
AU  - Ghaith Ayesh Hiary
TI  - A nearly-optimal method to compute the truncated theta function, its derivatives, and integrals
JO  - Annals of mathematics
PY  - 2011
SP  - 859
EP  - 889
VL  - 174
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.4007/annals.2011.174.2.3/
DO  - 10.4007/annals.2011.174.2.3
LA  - en
ID  - 10_4007_annals_2011_174_2_3
ER  - 
%0 Journal Article
%A Ghaith Ayesh Hiary
%T A nearly-optimal method to compute the truncated theta function, its derivatives, and integrals
%J Annals of mathematics
%D 2011
%P 859-889
%V 174
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.4007/annals.2011.174.2.3/
%R 10.4007/annals.2011.174.2.3
%G en
%F 10_4007_annals_2011_174_2_3
Ghaith Ayesh Hiary. A nearly-optimal method to compute the truncated theta function, its derivatives, and integrals. Annals of mathematics, Tome 174 (2011) no. 2, pp. 859-889. doi : 10.4007/annals.2011.174.2.3. http://geodesic.mathdoc.fr/articles/10.4007/annals.2011.174.2.3/

Cité par Sources :