The mean value theorem of the divisor problem for short intervals (Q1289268)

Let \(-1< a\leq 0\) and let \(\sigma_a(n)\) denote the sum of the \(a\)th powers of positive divisors of \(n\). It is considered the error term \(\Delta_a(x;r)\) in the asymptotic representation of \(\sum_{n\leq x}\sigma_a(n) e^{2\pi irn}\), where \(r= h/k\) with coprime \(h,k\). It is proved a complicated asymptotic expansion of the mean value \[ \int_X^{X+H}| \Delta_a(x+U;r)- \Delta_a(x;r)|^2 dx \] for \(X\geq 2\), \(1\leq U\ll x^{1/2}\ll H\leq X\), \(4kU\leq X\).