A Generalization of the Riemann Functional Equation with Three Variables
Abstract
The discovery that the generalized Riemann functional equation from the literature can be extended to real b ≥ 0, by exploiting symmetries of a new formula for the Hurwitz zeta function with respect to the shift parameter (b), led to the search for a similar relation for the Lerch Φ function, Φ(ez,k,b), where a new formula is used as a starting point and the symmetries with respect to b are exploited again. This search resulted in a generalization of the Riemann functional equation with three variables, whose domain is then extended to b>1, with the aid of a new self-contained Lerch Φ formula, making the relation hold for all real b. As a secondary objective, we address the problem of obtaining the analytic continuation of a finite summation that appears in a formula for the Lerch Φ function. Above all, this paper demonstrates, with clear elementary mathematics, how a functional equation can be derived.
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BibTeX
@misc{sousa2026generalization,
title = {{A Generalization of the Riemann Functional Equation with Three Variables}},
author = {José Risomar Sousa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10609},
eprint = {2610.10609},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10609},
}
RIS
TY - UNPB
TI - A Generalization of the Riemann Functional Equation with Three Variables
AU - José Risomar Sousa
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10609
UR - https://arxiv.org/abs/2610.10609
AB - The discovery that the generalized Riemann functional equation from the literature can be extended to real b ≥ 0, by exploiting symmetries of a new formula for the Hurwitz zeta function with respect to the shift parameter (b), led to the search for a similar relation for the Lerch Φ function, Φ(e^z,k,b), where a new formula is used as a starting point and the symmetries with respect to b are exploited again. This search resulted in a generalization of the Riemann functional equation with three variables, whose domain is then extended to b>1, with the aid of a new self-contained Lerch Φ formula, making the relation hold for all real b. As a secondary objective, we address the problem of obtaining the analytic continuation of a finite summation that appears in a formula for the Lerch Φ function. Above all, this paper demonstrates, with clear elementary mathematics, how a functional equation can be derived.
ER -