2021/01/17 by Bjoern S. Schmekel, Schmekel, Bjoern S.
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #General Mathematics (math.GM) #Mathematical functions and polynomials #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2101.07651
openalex publication_date 2021/01/17 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We describe a numerical algorithm for evaluating the numbers of roots minus\nthe number of poles contained in a region based on the argument principle with\nthe function of interest being written as a Mellin transformation of a usually\nsimpler function. Because the function to be transformed may be simpler than\nits Mellin transform whose roots are to be sought we express the final\nintegrals in terms of the former accepting higher dimensional integrals.\nNonlinear terms are expressed as convolutions approximating reciprocal values\nby exponential sums. As an example the final expression is applied to the\nRiemann Zeta function. The procedure is very inefficient numerically. However,\ndepending on the function to be investigated it may be possible to find\nanalytical estimates of the resulting integrals.\n