On the summation formula of Voronoi

Nasim, C.

doi:10.1090/S0002-9947-1972-0284410-9

On the summation formula of Voronoi
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Trans. Amer. Math. Soc. 163 (1972), 35-45 Request permission

Abstract:

A formula involving sums of the form $\Sigma d(n)f(n)$ and $\Sigma d(n)g(n)$ is derived, where $d(n)$ is the number of divisors of $n$, and $f(x),g(x)$ are Hankel transforms of each other. Many forms of such a formula, generally known as Voronoi’s summation formula, are known, but we give a more symmetrical formula. Also, the reciprocal relation between $f(x)$ and $g(x)$ is expressed in terms of an elementary kernel, the cosine kernel, by introducing a function of the class ${L^2}(0,\infty )$. We use ${L^2}$-theory of Mellin and Fourier-Watson transformations.

References

A theory of general transforms for functions of the class

9

On the summation formulas of Voronoĭ and Poisson

8

Summation formulae and self-reciprocal functions

9

A. P. Guinand, General transformations and the Parseval theorem, Quart. J. Math. Oxford Ser. 12 (1941), 51–56. MR 4330, DOI 10.1093/qmath/os-12.1.51

Koshliakov, On Voronoĭi’s sum-formula

58

John Boris Miller, A symmetrical convergence theory for general transforms, Proc. London Math. Soc. (3) 8 (1958), 224–241. MR 96948, DOI 10.1112/plms/s3-8.2.224
J. B. Miller, A symmetical convergence theory for general transforms, Proc. London Math. Soc. (3) 9 (1959), 451–464. MR 109995, DOI 10.1112/plms/s3-9.3.451
E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford, at the Clarendon Press, 1951. MR 0046485

Introduction to the theory of Fourier integrals

Sur une fonction transcendante et ses applications à la sommation de quelques séries

21

General transforms

35

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746

Voronoĭ’s summation formula

3