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,χ) is uniformly convergent in any compact subset, and that it can be continued as an entire function.
The Hurwitz zeta-function associated with the parameter a(0<a≤1) is a generalisation of the Riemann zeta-function namely the case a=1. It is defined by ζ(s,a)=∞∑n=0(n+a)−s,(s=σ+it,σ>1)
and its analytic continuation. %In fact ζ(s,a)=∞∑n=0((n+a)−s−∫n+1ndu(u+a)s)+a1−ss−1
gives the analytic continuation to (σ>0). A repetition of this several times shows that ζ−a1−ss−1
can be continued as an entire function to the whole plane. In Re(s)≥−1,t≥2,ζ(s,a)−a−s=O(t3) and by the functional equation (see \S2) it is O((|s|2π)12−Re(s))
in Re(s)≤−1,t≥2. From these facts In this paper, we deduce an `Approximate function equation' (see \S3), which is a generalisation of the approximate functional equation for ζ(s). Combining this with an important theorem due to van-der-Corput, we prove T−13∫T+T13T|ζ(12+it)−a−12−it|2dt<<(logT)3
uniformly in a(0<a≤1). From this we deduce similar results for quasi L-functions and more general functions.%Let a1,a2,…, be any periodic sequence of complex numbers for which the sum over a period is zero. Let b1,b2,… be any sequence of complex numbers for which $\sum_{j=2}^{n}\vert b_j-b_{j-1}\vert+\vert […]
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.