# Volume 9 -1986

### 1. On the frequency of Titchmarsh's phenomenon for $\zeta(s)$ IV.

The inequality in the main theorem of this paper improves on an earlier weaker version by a certain shaper constant.

### 2. Hybrid mean square of $L$-functions.

This paper discusses the Dirichlet's $L$-function $L(s,\chi)$ for a character $\chi$ mod $k$, and we prove a refinement of the error term using the result on Hurwitz Zeta function, proved by the author in an earlier paper.

### 3. The number of finite non-isomorphic abelian groups in mean square.

Let $\Delta(x)=\sum_{n\leq x}a(n)-\sum_{j=1}^6 c_jx^{1/j}$ denote the error term in the abelian group problem. Using zeta-function methods it is proved that $\int_1^X\Delta^2(x)\,dx~<\!\!<~ X^{39/29} \log^2X$ where the exponent $39/29=1.344827\ldots$ is close to the best possible exponent $4/3$ in this problem.