The divisor problem for \(d_4(n)\) in short intervals
This paper is concerned with the average order in short intervals of the divisor function \(d_k(n)\) which, for fixed \(k\), counts the number of ways to write \(n\) as the product of \(k\) integers. The authors investigate the exceptional case \(k=4\) and show that \[ \sum_{x < n\leq x+y} d_4(n)=\frac{1}{6}y\log ^3x\left(1+O\left(\left(\frac{x^{1/2}\log x}{y}\right)^{2/3}\right)\right) \] for \(x^{1/2}\log x <y<x^{1/2}\log ^{5/2} x\). Their method is analytic based on Perron's inversion formula applied to the Dirichlet series \(\zeta^4(s)=\sum_{n\geq 1}d_4(n)n^{-s}\).
scientific article; zbMATH DE number 5013986