# Volume 13 - 1990

### 1. Proof of some conjectures on the mean-value of Titchmarsh series I.

Let $F(s)$ be a Titchmarsh series, i.e. a kind of Dirichlet series, the coefficients of which are suitably bounded. This notion was introduced by us in an earlier paper where we stated a conjecture and proved certain theorems on the lower bound for $\int_0^H \vert F(it)\vert^k\,dt$, where $k=1$ or $2$. In this paper, better results and a proof of the conjecture are obtained.

### 2. Proof of some conjectures on the mean-value of titchmarsh series with applications to Titchmarsh's phenomenon

In the earlier paper written jointly with R. Balasubramanian, we proved certain lower bounds for mean values of Titchmarsh series. In this paper, we obtain analogous results for weak Titchmarsh series'', which are a kind of Dirichlet series $\sum a_n\lambda^{-s}_n$ such that $\sum_{n\leq X}\vert a_n\vert$ is suitably bounded above.

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

In the previous paper in this series, we proved a lower bound for $f(H)=\min_{T\geq1}\max_{T\leq t\leq T+H}\vert(\zeta(1+it))^z\vert,$ where $z=\exp(i\theta)$ and $0\leq\theta<2\pi$. In this paper, we prove an upper bound for $f(H)$ and present some applications.