R Balasubramanian ; K Ramachandra - Mean square of the Hurwitz zeta-function and other remarks

hrj:149 - Hardy-Ramanujan Journal, January 1, 2004, Volume 27 - 2004 - https://doi.org/10.46298/hrj.2004.149
Mean square of the Hurwitz zeta-function and other remarksArticle

Authors: R Balasubramanian 1; K Ramachandra 2

The Hurwitz zeta-function associated with the parameter a(0<a1) 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)sn+1ndu(u+a)s)+a1ss1
gives the analytic continuation to (σ>0). A repetition of this several times shows that ζa1ss1
can be continued as an entire function to the whole plane. In Re(s)1,t2,ζ(s,a)as=O(t3) and by the functional equation (see \S2) it is O((|s|2π)12Re(s))
in Re(s)1,t2. 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 T13T+T13T|ζ(12+it)a12it|2dt<<(logT)3
uniformly in a(0<a1). 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 nj=2|bjbj1|+|bn|nε for every ε>0 and every nn0(ε). Then we prove T13T+T13T|n=1anbn(n+a)12+it|2dtTε
for every ε>0 and every TT0(ε). Here, as usual, 0<a1 and T0(ε) is independent of a.


Volume: Volume 27 - 2004
Published on: January 1, 2004
Imported on: March 3, 2015
Keywords: Hurwitz zeta-function,approximate functional equation,periodic sequence of complex numbers,[MATH]Mathematics [math]

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