On the limiting distribution of a generalized divisor problem for the case \(-1/2<a<0\) (Q2717628)

Let \(\sigma_a(n) = \sum_{d| n}d^a\), and set NEWLINE\[NEWLINE\sum_{n\leq t}{}'\sigma_a(n) =\zeta(1 - a)t + \frac{\zeta(1+a)}{1+a}\;t^{1+a}+\tfrac12\,\zeta(-a)+\Delta_a(t).NEWLINE\]NEWLINE This paper is concerned with the distribution of \(\Delta_a(t)\) in the case \(-1/2\leq a < 0\). Note that \(\sigma_a(n) = n^a\sigma_{-a}(n)\), whence it suffices to consider nonpositive values of \(a\). Let NEWLINE\[NEWLINED_{a,T}(u)=T^{-1}\text{meas}\{t\in [1,T]:\Delta_a(t)\leq u\}.NEWLINE\]NEWLINE Then, for \(-1/2<a < 0\) NEWLINE\[NEWLINED_{a,T}(u)-D_a(u)\ll_a\left(\frac{\log T}{\log \log T}\right)^{-(1+2a)/8}NEWLINE\]NEWLINE for the rate of convergence to the limiting distribution.