Sums of squares and products of Bessel functions

Abstract

Letrk(n) denote the number of representations of the positive integernas thesum ofksquares. We rigorously prove for the first time a Vorono ?? summation formula forrk(n), k?2,proved incorrectly by A. I. Popov and later rediscovered by A. P. Guinand,but without proof and without conditions on the functions associated in the transformation.Using this summation formula we establish a new transformation between a series consistingofrk(n) and a product of two Bessel functions, and a series involvingrk(n) and the Gaussianhypergeometric function. This transformation can be considered as a massive generalizationof well-known results of G. H. Hardy, and of A. L. Dixon and W. L. Ferrar, as well as ofa classical result of A. I. Popov that was completely forgotten. An analytic continuationof this transformation yields further useful results that generalize those obtained earlier byDixon and Ferrar.1.IntroductionInfinite series involving arithmetic functions and Bessel functions are instrumental in study-ing some notoriously difficult problems in analytic number theory, for example, the circle andthe divisor problems. As mentioned by G. H. Hardy [19, p. 266], S. Wigert [43] was the firstmathematician to recognize the importance of series of Bessel functions in analytic numbertheory. Since then, several mathematicians have studied, and continue to study, such series,for example, with the point of view of understanding and improving the order of magnitudeof error terms associated with the summatory functions of certain arithmetic functions. Aprime tool in making the connection between a summatory function and certain series ofBessel functions is the Vorono ?? summation formula associated with the corresponding arith-metic function.Letrk(n) denote the number of representations of a positive integernas the sum ofksquares, where different signs and different orders of the summands give distinct representa-tions. The ordinary Bessel functionJ?(z) of order?is defined by [42, p. 40]J?(z) :=?Xm=0(?1)m(z/2)2m+?m!?(m+ 1 +?),|z|<?.(1.1)We record the Vorono ?? summation formula associated withr2(n), sometimes known as the Hardy Landau summation formula, in the form given in [26, p.274] (or [13, Thm. A]).Theorem 1.1.If0?? < ?andh(y)is real and of bounded variation in(?,?), thenX??n??r2(n)12(h(n?0) +h(n+ 0)) =??Xn=0r2(n)Z??h(y)J0(2??ny)dy,(1.2)2010Mathematics Subject Classification.Primary: 11E25; Secondary: 33C10, 30B40.Key words and phrases.sums of squares, Bessel functions, Vorono ?? summation formula, analyticcontinuation.1