Volume 13 - 1990

---

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

R Balasubramanian ; K Ramachandra.

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

K Ramachandra.

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.

K Ramachandra.

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.

---