Let k ≥ 1 denote any positive rational integer. We give formulae for the sums (where χ ranges over the ϕ(f)/2 odd Dirichlet characters modulo f > 2) whenever k ≥ 1 is odd, and for the sums (where χ ranges over the ϕ(f)/2 even Dirichlet characters modulo f>2) whenever k ≥ 1 is even.
It is shown that the multiplicative independence of Dedekind zeta functions of abelian fields is equivalent to their functional independence. We also give all the possible multiplicative dependence relations for any set of Dedekind zeta functions of abelian fields.
Let F/k be a finite abelian extension of global function fields, totally split at a distinguished place ∞ of k. We show that a complex Gras conjecture holds for Stark units, and we derive a refined analytic class number formula.
In this paper we solve three open problems and a conjecture related to the calculations of some classes of multiple series posed by Furdui in [1].
We survey some of the universality properties of the Riemann zeta function and then explain how to obtain a natural quantization of Voronin’s universality theorem (and of its various extensions). Our work builds on the theory of complex fractal dimensions for fractal strings developed by the second author and M. van Frankenhuijsen in [60]. It also makes an essential use of the functional analytic framework developed by the authors in [25] for rigorously studying the spectral operator (mapping...
We link together three themes which had remained separated so far: the Hilbert space properties of the Riemann zeros, the “dual Poisson formula” of Duffin-Weinberger (also named by us co-Poisson formula), and the “Sonine spaces” of entire functions defined and studied by de Branges. We determine in which (extended) Sonine spaces the zeros define a complete, or minimal, system. We obtain some general results dealing with the distribution of the zeros of the de-Branges-Sonine entire functions. We...