The Riemann zeta function and exact exponential sum identities of divisor functions (Q6635221)

Let \(s=\sigma+it\) with \(\sigma, t\in{\mathbb{R}}\) and let \N\[\N\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}\N\]\Nfor \(\Re (s)>1\) denote the Riemann zeta function. The divisor function \(\sigma_{a}(n)\) is defined by \N\[\N\sigma_{a}(n)=\sum_{d\mid n}d^{a},\N\]\Nwith \(n\in{\mathbb{N}},\ a\in{\mathbb{C}}\). Its generating function is given by \N\[\N\zeta(s)\zeta(s-a)=\sum_{n=1}^{\infty}\frac{\sigma_{a}(n)}{n^{s}}\N\]\Nfor \(\Re (s)>1\) and \(\Re (s)>\Re(a)+1\). Let \(\xi\) be the completed Riemann zeta function \N\[\N\xi(s)=\frac{1}{2}s(s-1)\pi^{-1/2}\Gamma(s/2)\zeta(s).\N\]\NIn this paper under review, the authors approximate \(\zeta(s)\zeta(s-a)\) when \(a=1,3,5\) with high precision and express \(\zeta(s_{0})\zeta(s_{0}-a)\) for any \(s_{0}\in{\mathbb{C}}\) in terms of a vertical line integral (see formula (1.3)). As a consequence, they deduce relations involving the divisor function \(\sigma_{a}(n)\) for \(a=1,3,5\) which can be obtained differently by the theory of modular forms. In the end of the paper, they state some conjectures on the values of the divisor functions \(\sigma_{a=1,3,5}(n)\) and show that it can be obtained by inverting certain matrices, without employing divisibility techniques. In particular, each of these conjectures can be used as a primality test.