On the error term for the counting functions of finite Abelian groups (Q1207657)

Let \(a(n)\) denote the number of non-isomorphic Abelian groups with \(n\) elements and let \[ \Delta(x):=\sum_{n\leq x} a(n)-\sum^ 9_{j=1}\underset{s=1/j}{\text{Res}}F(s) x^ s s^{-1}, \] \[ \Delta_ 1(x):=\sum_{mn\leq x} a(m)a(n)-\sum^ 5_{j=1}\underset{s=1/j}{\text{Res}}F^ 2(s) x^ s s^{-1}, \] where \(F(s)=\zeta(s)\zeta(2s)\zeta(3s)\dots\). Thus \(\Delta_ 1(x)\) is the error term in the asymptotic formula for the sum \(\sum t(G)\), where \(t(G)\) denotes the number of direct factors of an Abelian group \(G\), and summation is extended over all Abelian groups whose orders do not exceed \(x\). By using the complex integration technique and power moments of \(\zeta(s)\) [see Chapter 8 of the author's monograph, ``The Riemann zeta- function'' (1985; Zbl 0556.10026)] it is proved that \[ \int^ X_ 1\Delta(x)\;dx\ll_ \varepsilon\;X^{11/10+\varepsilon},\quad\int^ X_ 1\Delta_ 1(x)\;dx\ll_ \varepsilon\;X^{7/6+\varepsilon}, \] \[ \int^ X_ 1\Delta^ 2_ 1(x)\;dx=\Omega(X^{3/2}\log^ 4 X),\quad\int^ X_ 1\Delta^ 2_ 1(x)\;dx\ll_ \varepsilon\;X^{8/5+\varepsilon}. \] Moreover, if \(\int^ T_ 1|\zeta(1/2+\text{it})|^ 8 dt\ll_ \varepsilon\;T^{1+\varepsilon}\) holds, then \(\int^ X_ 1\Delta^ 2_ 1(x)dx\ll_ \varepsilon\;X^{3/2+\varepsilon}\).