Abstract
The classical Voronoi identity
is proved in a relatively simple way by the use of the Laplace transform. Here Δ(x) denotes the error term in the Dirichlet divisor problem, d(n) is the number of divisors of n and K_1, Y_1 are the Bessel functions. The method of proof may be used to yield other identities similar to Voronoi's.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B.C. Berndt, “Identities involving the coefficients of a class of Dirichlet series I,” Trans. Amer. Math. Soc. 137 (1969), 345–359.
K. Chandrasekharan, Arithmetical Functions, Springer-Verlag, Berlin, 1970.
A.L. Dixon and W.L. Ferrar, “Lattice point summation formulae,” Quart. J. Math. (Oxford) 2 (1931), 31–54.
G. Doetsch, Handbuch der Laplace-Transformation, Band I, Birkhäuser Verlag, Basel und Stuttgart, 1950.
G.H. Hardy, “The average order of the arithmetical functions P(x) and Δ(x),” Proc. London Math. Soc. 15(2) (1916), 192–213.
A. Ivić, The Riemann Zeta-Function, John Wiley & Sons, New York, 1985.
M. Jutila, “A method in the theory of exponential sums,” Lecture Notes, Tata Institute of Fundamental Research, vol. 80, Bombay, 1987 (distr. by Springer-Verlag, Berlin).
T. Meurman, “A simple proof of Voronoi's identity,” Astérisque 209 (1992), 265–274.
G.F. Voronoi, “Sur une fonction transcendante et ses applications à la sommation de quelques séries,” Ann. École Normale 21(3) (1904), 207–267, 459–534.
G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd edition, Cambridge University Press, Cambridge, 1944.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ivić, A. The Voronoi Identity via the Laplace Transform. The Ramanujan Journal 2, 39–45 (1998). https://doi.org/10.1023/A:1009753723245
Issue Date:
DOI: https://doi.org/10.1023/A:1009753723245