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.
Nous montrons que pour tout rationnel de , l’ensemble des valeurs des polylogarithmes contient une infinité de nombres -linéairement indépendants.
AMS Subj. Classification: MSC2010: 11F72, 11M36, 58J37We point out the importance of the integral representations of the logarithmic derivative of the Selberg zeta function valid up to the critical line, i.e. in the region that includes the right half of the critical strip, where the Euler product definition of the Selberg zeta function does not hold. Most recent applications to the behavior of the Selberg zeta functions associated to a degenerating sequence of finite volume, hyperbolic manifolds of...
Integrals of logarithmic and hypergeometric functions are intrinsically connected with Euler sums. In this paper we explore many relations and explicitly derive closed form representations of integrals of logarithmic, hypergeometric functions and the Lerch phi transcendent in terms of zeta functions and sums of alternating harmonic numbers.