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For Anosov flows on compact Riemann manifolds we study the rate of decay along the flow of diameters of balls Bs(x, ) on local stable manifolds at Lyapunov regular points x. We prove that this decay rate is similar for all sufficiently small values of > 0. From this and the main result in an earlier paper, we derive strong spectral estimates for Ruelle transfer operators for contact Anosov flows with Lipschitz local stable holonomy maps. These apply in particular to geodesic flows on compact locally symmetric manifolds of strictly negative curvature. As is now well known, such spectral estimates have deep implications in some related areas, e.g. in studying analytic properties of Ruelle zeta functions and partial differential operators, asymptotics of closed orbit counting functions, etc. [ABSTRACT FROM AUTHOR]

The holomorphy conjecture predicts that the topological zeta function associated to a polynomial $ f \in \mathbb{C}[x_1,\ldots,x_n]$ is holomorphic unless $ d$. In this paper, we generalise the holomorphy conjecture to the setting of arbitrary ideals in $ \mathbb{C}[x_1,\ldots,x_n]$. [ABSTRACT FROM AUTHOR]

In this paper we shall establish the counterpart of Szmidt, Urbanowicz and Zagier's formula in the sense of the Hecker correspondence. The motivation is the derivation of the values of the Riemann zeta-function at positive even integral arguments from the partial fraction expansion for the hyperbolic cotangent function (or the cotangent function). Since the last is equivalent to the functional equation, we may view their elegant formula as one for the Lambert series, and comparing the Laurent coefficients, we may give a functional equational approach to the short-interval character sums with polynomial weight. In view of the importance of these short-interval character sums, we assemble some handy formulations for them that are derived from Szmidt, Urbanowicz and Zagier's formula and Yamamoto's method, which also gives the conjugate sums. We shall also state the formula for the values of the Dirichlet $ L$ [ABSTRACT FROM AUTHOR]

Given a circle-valued Morse function of a closed oriented manifold, we prove that Reidemeister torsion over a non-commutative formal Laurent polynomial ring equals the product of a certain non-commutative Lefschetz-type zeta function and the algebraic torsion of the Novikov complex over the ring. This paper gives a generalization of the result of Hutchings and Lee on abelian coefficients to the case of skew fields. As a consequence we obtain a Morse theoretical and dynamical description of the higher-order Reidemeister torsion. [ABSTRACT FROM AUTHOR]

In this paper we shall prove that the combination of the general distribution property and the functional equation for the Lipschitz-Lerch transcendent capture the whole spectrum of deeper results on the relations between the values at rational arguments of functions of a class of zeta-functions. By Theorem 1 and its corollaries, we can cover all the previous results in a rather simple and lucid way. By considering the limiting cases, we can also deduce new striking identities for Milnor's gamma functions, among which is the Gauss second formula for the digamma function. [ABSTRACT FROM AUTHOR]

It is known that the Lerch (or periodic) zeta function of non-positive integer order, $ell _{-n}(xi )$, $nin mathbb {N}_{0}:= {0, 1, 2, 3, ldots }$, is a polynomial in $cot (pi xi )$ of degree $n + 1$. In this paper, a very simple explicit closed-form formula for this polynomial valid for any degree is derived. In addition, novel analogous explicit closed-form formulae for the Legendre chi function, the alternating Lerch zeta function and the alternating Legendre chi function are established. The obtained formulae involve the Carlitz-Scoville tangent and secant numbers of higher order, and the derivative polynomials for tangent and secant are used in their derivation. Several special cases and consequences are pointed out, and some examples are also given. [ABSTRACT FROM AUTHOR]

In this paper, we compute the scattering matrix for the Hilbert modular group over any number field $K$. We then compute the determinant of the scattering matrix and show it is a ratio of completed Dedekind zeta functions associated to the Hilbert class field of $K$. This generalizes work of Efrat and Sarnak in the imaginary quadratic case. [ABSTRACT FROM AUTHOR]

We study a $q$-logarithm which was introduced by Euler and give some of its properties. This $q$-logarithm has not received much attention in the recent literature. We derive basic properties, some of which were already given by Euler in a 1751 paper and in a 1734 letter to Daniel Bernoulli. The corresponding $q$-analogue of the dilogarithm is introduced. The relation to the values at $1$ and $2$ of a $q$-analogue of the zeta function is given. We briefly describe some other $q$-logarithms that have appeared in the recent literature. [ABSTRACT FROM AUTHOR]

In this paper we give a short and simple proof of the remarkable evaluations of multiple zeta values established by D. Bowman and D. M. Bradley. [ABSTRACT FROM AUTHOR]

In a recent paper, A. Libgober showed that the multiplicative sequence $\{Q_i(c_1,\dots,c_i)\}$ of Chern classes corresponding to the power series $Q(z)=\Gamma(1+z)^{-1}$ appears in a relation between the Chern classes of certain Calabi-Yau manifolds and the periods of their mirrors. We show that the polynomials $Q_i$ can be expressed in terms of multiple zeta values. [ABSTRACT FROM AUTHOR]

In this paper we shall define the analytic continuation of the multiple (Euler-Riemann-Zagier) zeta functions of depth $d$: \[\zeta(s_1,\dots,s_d):= \sum_{01$ and $\sum_{j=1}^d\re(s_j)>d$. We shall also study their behavior near the poles and pose some open problems concerning their zeros and functional equations at the end. [ABSTRACT FROM AUTHOR]

We prove, subject to certain hypotheses, that a positive proportion of the a-points of the Riemann zeta-function and Dirichlet L-functions with primitive characters are simple and discuss corresponding results for other functions in the Selberg class. We also prove an unconditional result of this type for the a-points in fixed strips to the right of the line ℜs = 1/2. [ABSTRACT FROM AUTHOR]

Let Λ be the von Mangoldt function and R(n)=Σh+k=nΛ(h)Λ(k) be the counting function for the Goldbach numbers. Let N ≥ 2 and assume that the Riemann Hypothesis holds. We prove that [Multiple line equation(s) cannot be represented in ASCII text] where p = 1/2+iγ runs over the non-trivial zeros of the Riemann zeta-function ζ(s). This improves a recent result by Bhowmik and Schlage-Puchta [ABSTRACT FROM AUTHOR]