Abstract
We provide a simple proof of a Voronoi type identity using the Laplace transform for a class of arithmetical functions whose generating Dirichlet series satisfy a functional equation with gamma factors. As an application, we derivea Voronoi-type identity for Gaussian integers.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. Atsushi and N. Templier, On the Voronoi formula for GL(n), Amer. J. Math., 35(2013), 65–101.
D. Banerjee, E. M. Baruch and D. Bump, Voronoi summation formula for Gaussian integers, Ramanujan J., 57 (2022), 253–274.
D. Banerjee, E. M. Baruch and E. Tenetov, A Voronoi–Oppenheim summation formula for totally real number fields, J. Number Theory, 199 (2019), 63–97.
J. Beineke and D. Bump, Moments of the Riemann Zeta function and Eisenstein series. I, J. Number Theory, 105 (2004), 150–174.
B. C. Berndt, Identities involving the coefficients of a class of Dirichlet series. I, Trans. Amer. Math. Soc., 137 (1969), 345–359.
K. Chandrasekharan and R. Narasimhan, Functional equations with multiple gamma factors and the average order of arithmetical functions, Ann. of Math., 76(1962), 93–136.
K. Chandrasekharan and R. Narasimhan, Hecke’s functional equation and arithmetical identities, Ann. of Math., 74 (1961), 1–23.
K. Chandrasekharan, Arithmetical Functions, Springer Verlag (Berlin etc., 1970).
G. Doetsch, Handbuch der Laplace-Transformation, Band I, Birkh¨auser Verlag (Basel, Stuttgart, 1950).
A. L. Dixon and W. L. Ferrar, Lattice point summation formulae, Quart. J. Math., 2 (1931), 310-54.
R. Gupta and S. Pandit, On a certain divisor function in number fields, arXiv:2106.05257 (2021).
J. L. Hafner, On the representation of the summatory functions a class of arithmetical functions, in: Analytic Number Theory (Philadelphia, Pa., 1980), Lecture Notes inMath., vol. 899, Springer-Verlag (Berlin, Heidelberg, New York, 1981),pp. 148–165.
J. L. Hafner, On the average order of a class of arithmetical functions, J. Number Theory, 15 (1982), 36–76.
G. H. Hardy and M. Riesz, The general theory of Dirichlet’s series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 18, Stechert-Hafner, Inc. (New York, 1964).
G. H. Hardy, On Dirichlet’s divisor problem, Proc. London Math. Soc. (2), 15 (1916), 1–25.
G. H. Hardy, On the expression of a number as the sum of two squares, Quart. J. Math., 46 (1915), 263–283.
G. H. Hardy, The average order of the arithmetical functions P(x) and Δ(x), Proc. London Math. Soc. (2), 15 (1916), 192–213.
M. N. Huxley, Exponential sums and lattice points. III, Proc. London Math. Soc.(3), 87 (2003), 591–609.
M. N. Huxley, Exponential sums and the Riemann zeta-function, Proc. London Math. Soc. (3), 90 (2005), 1–41.
A. Ivić, Voronoi identity via the Laplace transform, Ramanujan J., 2 (1998), 39–45.
A. Ivić, The Riemann Zeta-function, John Wiley and Sons (New York, 1985).
A. Ivić, Large values of certain number-theoretic error terms, Acta Arith.(2), 56 (1990), 135–159.
M. Jutila, A Method in the Theory of Exponential Sums, Tata Institute of Fundamental Research, Lecture Notes, vol. 80 (Bombay, 1987).
A. Laurincikas, Once again on the function σα(m), Liet. Mat. Rink., 32 (1992), 105–121 (in Russian); translation in Lithuanian Math., 32 (1992), 81–93.
T. Meurman, A simple proof of Voronoi’s identity, Ast´erisque, 209 (1992), 265–274.
S. D. Miller and W. Schmid, Automorphic distributions, L-functions, and Voronoi summation for GL(3), Ann. of Math. (2), 164 (2006), 423–488.
C. Nasim, On the summation formula of Voronoi, Trans. Amer. Math. Soc., 163 (1972), 35–45.
A. Oppenheim, Some identities in the theory of numbers, Proc. London Math. Soc., 26 (1927), 295–350.
W. Sierpinski, O pewnem zagadnieniu z rachunku funkcyj asymptotycznych, Prace Mat. Fiz., 17 (1906), 77–118.
M. Voronoϊ, Sur une fonction transcendante et ses applications à la sommation de quelques séries, Ann. Sci. École Norm. Sup. (3), 21 (1904), 459–533.
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press (1952).
J. R. Wilton, Voronoi’s summation formula, Proc. London Math. Soc. (2), 3 (1932), 26–32.
J. R. Wilton, A note on Ramanujan’s arithmetical function τ (n), Proc. Camb. Philos. Soc., 25 (1929), 121–129.
F. Zhou, Voronoi summation formulae on GL(n), J. Number Theory, 162 (2016),483–495.
Acknowledgements
The author would like to thank Prof. Aleksandar Ivić for his valuable suggestions and comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
The author acknowledges the support of the Department of Science and Technology, Government of India (grant No. NPDF-PDF/2017/000435).
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Banerjee, D. Voronoi-type identity for a class of arithmetical functions via the Laplace transform. Acta Math. Hungar. 168, 50–62 (2022). https://doi.org/10.1007/s10474-022-01261-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-022-01261-y