Bilateral Lerch theta and theta star function and Quadrilateral Lerch zeta and zeta star functions
DOI:
https://doi.org/10.51094/jxiv.872Keywords:
functional equation, modular relation, theta functions, zeta functionsAbstract
In the present paper, we construct theta functions with two parameters $a,b \in {\mathbb{R}}$ which satisfy Jacobi's modular relation.
Moreover, we give zeta functions with two parameters $a,b \in {\mathbb{R}}$ which satisfy Riemann's functional equation by the theta functions with two parameters.
Conflicts of Interest Disclosure
The authors have no conflicts of interest directly relevant to the content of this article.Downloads *Displays the aggregated results up to the previous day.
References
T. M. Apostol, On the Lerch zeta-function, Pacific J. Math. 1, (1951), 161–167.
T. M. Apostol, Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics, Springer, New York, 1976.
H. Hamburger, ¨ Uber die Riemannsche Funktionalgleichung der ζ-Funktion. (German) Math. Z. 10 (1921), no. 3-4, 240–254.
A. A. Karatsuba and S. M. Voronin, The Riemann zeta-function. Translated from the Russian by Neal Koblitz. De Gruyter Expositions in Mathematics, 5. Walter de Gruyter & Co., Berlin, 1992.
N. Koblitz, Introduction to elliptic curves and modular forms. Second edition. Graduate Texts in Mathematics, 97. Springer-Verlag, New York, 1993.
A. Laurinˇcikas and R. Garunkˇstis, The Lerch zeta-function. Kluwer Academic Publishers, Dordrecht, 2002.
J. Lagarias and W. W-C Li, The Lerch zeta function I. Zeta integrals. Forum Math. 24 (2012), no 1, 1–48.
T. Nakamura, The functional equation and zeros on the critical line of the quadrilateral zeta function. J. Number Theory 233 (2022), 432–455.
T. Nakamura, On Lerch’s formula and zeros of the quadrilateral zeta function. preprint, arXiv:2001.01981.
T. Nakamura, On zeros of bilateral Hurwitz and periodic zeta and zeta star functions, Rocky Mountain Journal of Mathematics 53 (2023), no.1, 157–176 (arXiv:1910.10430).
E. C. Titchmarsh, The theory of the Riemann zeta-function, Second edition. Edited and with a preface by D. R. Heath-Brown. The Clarendon Press, Oxford University Press, New York, 1986.
Downloads
Posted
Submitted: 2024-09-02 06:54:32 UTC
Published: 2024-09-04 00:39:18 UTC
License
Copyright (c) 2024
Takashi Nakamura
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
