AMS Subject Classification 2010: 11M26, 33C45, 42A38.Necessary and sufficient conditions for absence of zeros of ζ(s) in the half-plane σ ... Expansion of holomorphic functions in series of Hermite polynomials ...
We prove formulas for the k-higher Mahler measure of a family of rational functions with an arbitrary number of variables. Our formulas reveal relations with multiple polylogarithms evaluated at certain roots of unity.
We establish “higher depth” analogues of regularized determinants due to Milnor for zeros of cuspidal automorphic -functions of over a general number field. This is a generalization of the result of Deninger about the regularized determinant for zeros of the Riemann zeta function.
As usual, let s = σ + it. For any fixed value of t with |t| ≥ 8 and for σ < 0, we show that |ζ(s)| is strictly decreasing in σ, with the same result also holding for the related functions ξ of Riemann and η of Euler. The following inequality related to the monotonicity of all three functions is proved: ℜ (η'(s)/η(s)) < ℜ (ζ'(s)/ζ(s)) < ℜ (ξ'(s)/ξ(s)). It is also shown that extending the above monotonicity result for |ζ(s)|, |ξ(s)|, or |η(s)| from σ <...
For a cocompact group of we fix a real non-zero harmonic -form . We study the asymptotics of the hyperbolic lattice-counting problem for under restrictions imposed by the modular symbols . We prove that the normalized values of the modular symbols, when ordered according to this counting, have a Gaussian distribution.