2007/11/30 by Ghaith A. Hiary, Hiary, Ghaith Ayesh
Mathematics · #11Y16 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.0711.5002
openalex publication_date 2007/11/30 · openalex created_date 2022/09/27 · openalex updated_date 2026/07/28
A poly-log time method to compute the truncated theta function, its\nderivatives, and integrals is presented. The method is elementary, rigorous,\nexplicit, and suited for computer implementation. We repeatedly apply the\nPoisson summation formula to the truncated theta function while suitably\nnormalizing the linear and quadratic arguments after each repetition. The\nmethod relies on the periodicity of the complex exponential, which enables the\nsuitable normalization of the arguments, and on the self-similarity of the\nGaussian, which ensures that we still obtain a truncated theta function after\neach application of the Poisson summation. In other words, our method relies on\nmodular properties of the theta function. Applications to the numerical\ncomputation of the Riemann zeta function and to finding the number of solutions\nof Waring type Diophantine equations are discussed.\n