On the sum of divisors function
A classical identity of Ramanujan states that, for \(\sigma_ z (n)= \sum_{d\mid n} d^ z\) and \(\text{Re } s>1+ \max (0, \text{Re } a\), \(\text{Re } b,\text{Re} (a+b))\) we have \[ \sum_{n=1}^ \infty \sigma_ a (n) \sigma_ b (n) n^{-s}= {{\zeta (s) \zeta (s-a) \zeta (s-b) \zeta (s-a-b)} \over {\zeta (2s-a -b)}}. \] The reviewer asked in 1990 for an extension of this identity in the case of the generating function of \(\sigma_ a (n) \sigma_ b (n) \sigma_ c (n)\). The author successfully solves this problem in the general case by finding a meromorphic analytic continuation (the formula is too complicated to be reproduced here), in a certain region, of the generating function of \(\sigma_{a_ 1} (n) \ldots \sigma_{a_ k} (n)\) for arbitrary complex numbers \(a_ 1, \ldots, a_ k\). He also proves that, for a fixed natural number \(k\geq 2\), we have \((\sigma (n)= \sigma_ 1 (n))\) \[ \sum_{n\leq x} \sigma^ k (n)= C_ k x^{k+1}+ E_ k (x), \] where \(C_ k\) \((>0)\) is explicitly given, and \(E_ k (x)= O_ k (x^ k \log^{k- 1/3})\). He also obtains an asymptotic formula for the summatory function of \(E_ k (n)\).
asymptotic formula for the summatory function
scientific article; zbMATH DE number 751309