Volume 1 - 1978


1. Some remarks on the mean value of the Riemann zeta-function and other Dirichlet series 1

K Ramachandra.
The present paper is concerned with Ω-estimates of the quantity (1/H)T+HT|(dm/dsm)ζk(12+it)|dt
where k is a positive number (not necessarily an integer), m a nonnegative integer, and (logT)δHT, where δ is a small positive constant. The main theorems are stated for Dirichlet series satisfying certain conditions and the corollaries concerning the zeta function illustrate quite well the scope and interest of the results. %It is proved that if 2k1 and TT0(δ), then (1/H)T+HT|ζ(12+it)|2kdt>(logH)k2(loglogH)C
and (1/H)T+HT|ζ(12+it)|dt>(logH)5/4(loglogH)C,
where C is a constant depending only on δ.