On a Relation Between Sums of Arithmetical Functions and Dirichlet Series
We give a simple proof that critical values of any Artin -function attached to a representation with character are stable under twisting by a totally even character , up to the -th power of the Gauss sum related to and an element in the field generated by the values of and over . This extends a result of Coates and Lichtenbaum as well as the previous work of Ward.
Let be a nonprincipal Dirichlet character modulo a prime number and let . Define the mean value We give an identity for which, in particular, shows that for fixed and .
We obtain a series of new conditional lower bounds for the modulus and the argument of the Riemann zeta function on very short segments of the critical line, based on the Riemann hypothesis. In particular, we prove that for any large fixed constant A > 1 there exist(non-effective) constants T₀(A) > 0 and c₀(A) > 0 such that the maximum of |ζ (0.5+it)| on the interval (T-h,T+h) is greater than A for any T > T₀ and h = (1/π)lnlnln{T}+c₀.