Weighted divisor sums and Bessel function series. IV
The paper is devoted to an identity published in Ramanujan's Lost Notebook (p. 335) involving a finite trigonometric sum and a doubly infinite series of Bessel functions, connected with the classical divisor and circle problems. First the result from the notebook is generalized to an identity for Riesz sums (taking \(\nu=1\) below, the original identity is recovered): Theorem 2.1. Let \(x>0,0<\theta<1\) and \(\nu\) be a positive integer, then \[ \begin{align*}{\sideset\and {'}\to\sum_{n\leq x}(x-n)^{\nu-1}&\sum_{r|n}\,\sin{(2\pi r\theta)}= {\pi x^{\nu}\over\nu}\left({1\over 2}-\theta\right) -{x^{\nu-1}\over 4}\cot{(\pi\theta)}+\cr &\qquad +{(\nu-1)!x^{\nu/2}\over 2(2\pi)^{\nu-1}}\sum_{m=1}^{\infty}\sum_{n=0}^{\infty} \left\{{J_{\nu}(4\pi\sqrt{m(n+\theta)x}\over (m(n+\theta))^{\nu/2}}-{J_{\nu}(4\pi\sqrt{m(n+1-\theta)x}\over (m(n+1-\theta))^{\nu/2}}\right\}. \cr}\end{align*} \] (\(\sideset\and {'}\to\sum\): if \(x\) is an integer, the term with \(n=x\) is given a factor \(1/2\) in the sum) Furthermore, if \(\chi\) is a Dirichlet character, denote by \(d_{\chi}\) \[ d_{\chi}(n)=\sum_{d\mid n}^{}\chi(d), \] then the authors prove that Theorem 2.1 is equivalent to Theorem 2.2. Let \(q\) be a positive integer and let \(\chi\) be an odd primitive character modulo \(q\), then for any \(x>0\) \[ \begin{align*}{\sideset\and {'}\to\sum_{n\leq x} d_{\chi}(n)(x-n)^{\nu-1}=&{x^{\nu}\over\nu}L(1,\chi)+{i\tau(\chi)x^{\nu-1}\over 2\pi} L(1, \overline{\chi})+\cr &\qquad +{iq^{\nu/2}(\nu-1)!\over (2\pi)^{\nu-1}\tau(\overline{\chi})}\sum_{n=1}^{\infty}\,d_{\overline{\chi}} \left({x\over n}\right)^{\nu/2} J_{\nu}(4\pi\sqrt{nx/q}).\cr}\end{align*} \] (here \(\tau(\chi):=\sum_{k=1}^{q-1}\,\chi(k)\exp{(2\pi ik/q)}\), the Gauss sum) Moreover, the authors generalize the formula to a trigonometric analogon: Theorem 4.1. Let \(J_1(x)\) denote the ordinary Bessel function of order \(1\); if \(0<\theta,\sigma<1\) and \(x>0\), then \[ \begin{align*}{\sideset\and {'}\to\sum_{mn\leq x} \cos{(2\pi m\sigma)}&\sin{(2\pi n\theta)}=-{\cot{(\pi\theta)}\over 4}+\cr &\qquad+{\sqrt{x}\over 4}\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\left\{{J_1(4\pi\sqrt{(m+\sigma)(n+\theta)x}\over \sqrt{(m+\sigma)(n+\theta)}}+{J_1(4\pi\sqrt{(m+1-\sigma)(n+\theta)x}\over\sqrt{(m+1-\sigma)(n+\theta)}}\right.\cr &\qquad\left.-{J_1(4\pi\sqrt{(m+\sigma)(n+1-\theta)x}\over\sqrt{(m+\sigma)(n+1-\theta)}} -{J_1(4\pi\sqrt{(m+1-\sigma)(n+1-\theta)x}\over\sqrt{(m+1-\sigma)(n+1-\theta)}}\right\}.\cr}\end{align*} \] Finally, Theorem 4.1 is proved to hold with the summation order in the double sum on the right hand side reversed (Theorem 4.2).
scientific article; zbMATH DE number 6120463