# Volume 27 - 2004

### 1. Hardy-Littlewood first approximation theorem for quasi $L$-functions

The object of the present note (which is an addendum to [R. Balasubramanian and K. Ramachandra, Indian J. Pure Appl. Math. 18 (1987), no. 9, 790--793]) is to make a few remarks on the results and prove that a certain quasi $L$-function $L(s,\chi)$ is uniformly convergent in any compact subset, and that it can be continued as an entire function.

### 2. Mean square of the Hurwitz zeta-function and other remarks

The Hurwitz zeta-function associated with the parameter $a\,(0< a\leq1)$ is a generalisation of the Riemann zeta-function namely the case $a=1$. It is defined by $\zeta(s,a)=\sum_{n=0}^{\infty}(n+a)^{-s},\,(s=\sigma+it,\,\sigma>1)$ and its analytic continuation. %In fact $\zeta(s,a)=\sum_{n=0}^{\infty}\left((n+a)^{-s}-\int_{n}^{n+1}\frac{du}{(u+a)^s} \right)+\frac{a^{1-s}}{s-1}$ gives the analytic continuation to $(\sigma>0)$. A repetition of this several times shows that $\zeta-\frac{a^{1-s}}{s-1}$ can be continued as an entire function to the whole plane. In $Re(s)\geq-1,\,t\geq2,\,\zeta(s,a)-a^{-s}=O(t^3)$ and by the functional equation (see \S2) it is $O\left(\left(\frac{\vert s\vert}{2\pi}\right)^{\frac{1}{2}-Re(s)}\right)$ in $Re(s)\leq-1,\,t\geq2$. From these facts In this paper, we deduce an `Approximate function equation' (see \S3), which is a generalisation of the approximate functional equation for $\zeta(s)$. Combining this with an important theorem due to van-der-Corput, we prove $T^{-\frac{1}{3}}\int_{T}^{T+T^{\frac{1}{3}}} \vert\zeta(\frac{1}{2}+it)-a^{-\frac{1}{2}-it}\vert^2 dt <\!\!\!< (\log T)^3$ uniformly in $a(0< a\leq1)$. From this we deduce similar results for quasi $L$-functions and more general functions. %Let $a_1, a_2,\ldots$, be any periodic sequence of complex numbers for which the sum over a period is zero. Let $b_1, b_2,\ldots$ be any sequence of complex numbers for which \$\sum_{j=2}^{n}\vert b_j-b_{j-1}\vert+\vert […]

### 3. A generalization of Bochner's formula

In this note we expound our general hierarchy theorems by the example of a Ramified-Type Functional Equarion H, which gives all possbile forms, in terms of se-ries with H-function coefficients, of the functional equation of higher hierarchy arising from the original ramified one satisfied by the Dirichlet series. Then by sepcifying the parameters, we shall deduce a few concrete examples scattered in the literature in the most natural way.