On the limiting distribution of a generalized divisor problem for the case \(-1\leq a<-1/2\) (Q2717629)

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 \(-1 \leq a < -1/2\). 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 In [J. Number Theory 94, 359--374 (2002; Zbl 1014.11058)] the author has shown that \(D_{a,T}(u)\) converges to a limiting distribution \(D_a(u)\) as \(T\to\infty\), for \(1\leq a < -1/2\). The present paper shows firstly that \(D_a(u)\) satisfies a Lipschitz condition \(D_a(u + \varepsilon) - D_a(u)\ll_a \varepsilon^{1/2}\) for \(0 <\varepsilon< 1\), uniformly in \(u\). Moreover there is an estimate NEWLINE\[NEWLINED_{a,T}(u)-D_a(u)\ll_a\left(\frac{\log T}{\log \log T}\right)^{(1+2a)/6}NEWLINE\]NEWLINE for the rate of convergence to the limiting distribution. Finally it is shown that \(D_a(u)\) is symmetric, in the sense that \(D_a(-u) =1- D_a(u)\). This contrasts with the case \(a = 0\), which is the classical Dirichlet divisor problem, for which it is known that \(D_0(u)\) is not symmetric.