A product identity for Dirichlet series satisfying Hecke's functional equation

Year 2025, Volume: 54 Issue: 5 , 1792 - 1805 , 29.10.2025

Efe Gürel

https://doi.org/10.15672/hujms.1595265

https://izlik.org/JA52PN88PS

Abstract

In this paper, we give an analogue of Wilton’s product formula for Dirichlet series that satisfy Hecke’s functional equation. We apply our results to obtain identities for Hecke series, L-functions associated to modular forms, Ramanujan’s L-function, Epstein zeta functions, Dedekind zeta functions of imaginary quadratic fields and Dirichlet L-functions. A $4$-term product identity for Riemann zeta function is also given.

Keywords

Wilton’s formula , Dirichlet series , Hecke correspondence theorem , Hecke series

References

• [1] T.M. Apostol and A. Sklar, The approximate functional equation of Heckes Dirichlet series. Trans. Am. Math. Soc. 86 (2), 446-462, 1957.
• [2] T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory, volume 41 of Graduate Texts in Mathematics. Springer-Verlag, Inc., New York, 9, 125-141, 1997.
• [3] D. Banerjee and J. Mehta, Linearized product of two Riemann zeta functions, Proc. Jpn. Acad. Ser. A Math. Sci. 90 (8), 123 126, 2014.
• [4] S. Banerjee, K. Chakraborty and A. Hoque, An analogue of Wiltons formula and special values of Dedekind zeta functions, arXiv preprint arXiv:1611.08693, 2016.
• [5] S. Banerjee, A. Hoque and K. Chakraborty, On the product of Dedekind zeta functions, arXiv preprint arXiv:1611.08693, 2016.
• [6] S. Banerjee, K. Chakraborty and A. Hoque, An analogue of Wilton’s formula and values of Dedekind zeta functions, J. Math. Anal. Appl. 495 (1), 124675, 2021.
• [7] H. Bateman and A. Erdélyi, Higher transcendental functions, volume II, Mc Graw- Hill Book Company, 1953.
• [8] B.C. Berndt and M.I. Knopp, Hecke’s theory of modular forms and Dirichlet series, volume 5, World Scientific, 2008.
• [9] K. Chandrasekharan and R. Narasimhan, The approximate functional equation for a class of zeta-functions, Math. Ann. 152 (1), 3064, 1963.
• [10] H. Davenport, Multiplicative Number Theory, volume 74, Springer Science & Business Media, 2013.
• [11] P. Epstein, Zur theorie allgemeiner zetafunctionen, Math. Ann. 56 (4), 615644, 1903.
• [12] G. Hardy and J. Littlewood, The approximate functional equations for $\zeta(s)$ and $\zeta^2(s)$, Proc. Lond. Math. Soc. 2 (1), 8197, 1929.
• [13] G. H. Hardy and M. Riesz, The general theory of Dirichlet’s series, Courier Corporation, 2013.
• [14] A.F. Lavrik, Approximate functional equations for Dirichlet functions, Izv. Math. 2 (1), 129, 1968.
• [15] M. Nakajima, A new expression for the product of the two Dirichlet series I, Proc. Jpn. Acad. Ser. A Math. Sci. 79 (2), 19 22, 2003.
• [16] J. Wilton, An Approximate Functional Equation for the Product of Two $\zeta$-Functions, Proc. Lond. Math. Soc. 2 (1), 1117, 1930.