An Explicit Formula and its Application to the Selberg Trace Formula

We prove that the explicit formula in a symmetric case for a triple (Z, [(Z)\tilde]\tilde{Z} , Φ) in Jorgenson-Lang’s fundamental class of functions holds for a larger class of (not necessarily differentiable or even continuous) test functions. As one of the most important applications, we show that the Selberg trace formula for a strictly hyperbolic cocompact Fuchsian group Γ is valid for a larger class of test functions. Further applications to growth estimates for the logarithmic derivative of the Selberg zeta function are considered.     

(Almost) primitivity of Hecke <Emphasis Type="Italic">L</Emphasis>-functions

Let f be a cusp form of the Hecke space and let L _f be the normalized L-function associated to f. Recently it has been proved that L _f belongs to an axiomatically defined class of functions . We prove that when λ ≤ 2, L _f is always almost primitive, i.e., that if L _f is written as product of functions in , then one factor, at least, has degree zeros and hence is a Dirichlet polynomial. Moreover, we prove that if then L _f is also primitive, i.e., that if L _f = F ₁ F ₂ then F ₁ (or F ₂) is constant; for the factorization of non-primitive functions is studied and examples of non-primitive functions are given. At last, the subset of functions f for which L _f belongs to the more familiar extended Selberg class is characterized and for these functions we obtain analogous conclusions about their (almost) primitivity in .     

Algebraic Independence Results Related to 〈q, r〉-Number Systems

We define 〈q, r〉-linear arithmetical functions attached to the 〈q, r〉-number systems and give a necessary and sufficient condition for their generating power series to be algebraically independent over . We also deduce algebraic independence of the functions values at a nonzero algebraic number in the circle of convergence.     

Systems of quadratic Diophantine inequalities and the value distribution of quadratic forms

Let Q ₁,…,Q _r be quadratic forms with real coefficients. We prove that the set is dense in , provided that the system Q ₁(x) = 0,…,Q _r (x) = 0 has a nonsingular real solution and all forms in the real pencil generated by Q ₁,…,Q _r are irrational and have rank larger than 8r. Moreover, we give a quantitative version of the above assertion. As an application we study higher correlation functions of the value distribution of a positive definite irrational quadratic form. Author’s address: Institut für Statistik, Technische Universit?t Graz, A-8010 Graz, Austria     

On Li's criterion for the Riemann hypothesis for the Selberg class

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In this paper, we shall prove a generalization of Li's positivity criterion for the Riemann hypothesis for the extended Selberg class with an Euler sum. We shall also obtain two arithmetic expressions for Li's constants , where the sum is taken over all non-trivial zeros of the function F and the indicates that the sum is taken in the sense of the limit as T→∞ of the sum over ρ with |Imρ|?T. The first expression of λF(n), for functions in the extended Selberg class, having an Euler sum is given terms of analogues of Stieltjes constants (up to some gamma factors). The second expression, for functions in the Selberg class, non-vanishing on the line , is given in terms of a certain limit of the sum over primes.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=EwDtXrkuwxA.     

On the irrationality measure for certain series

Let K be either the rational number field or an imaginary quadratic field. We give irrationality results for the number , where q (∣q∣ > 1) is an integer in K, r∈ K ^× (∣r∣ < ∣q∣), and with q ⁿ ≠ r ^l (n ≥ 1). Authors’ addresses: Kenji Amano, NS solutions Corporation, 2-27-1 Shinkawa, Chuo-ku, Tokyo 104-0033, Japan; Yohei Tachiya, Department of Mathematics, Keio University, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan     

Une Remarque sur une Formule Sommatoire liée à la Somme des Chiffres

Let g ≥ 2 be an integer, and let s(n) be the sum of the digits of n in basis g. Let f(n) be a complex valued function defined on positive integers, such that . We propose sufficient conditions on the function f to deduce the equality . Applications are given, for instance, on the equidistribution mod 1 of the sequence (s(n))^α, where α is a positive real number.     

On products of <Emphasis Type="Italic">k</Emphasis> atoms

Let H be an atomic monoid. For let denote the set of all with the following property: There exist atoms (irreducible elements) u ₁, …, u _k , v ₁, …, v _m ∈ H with u ₁· … · u _k = v ₁ · … · v _m . We show that for a large class of noetherian domains satisfying some natural finiteness conditions, the sets are almost arithmetical progressions. Suppose that H is a Krull monoid with finite cyclic class group G such that every class contains a prime (this includes the multiplicative monoids of rings of integers of algebraic number fields). We show that, for every , max which settles Problem 38 in [4]. Authors’ addresses: W. Gao, Center for Combinatorics, Nankai University, Tianjin 300071, P.R. China; A. Geroldinger, Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universit?t Graz, Heinrichstra?e 36, 8010 Graz, Austria     

Arithmetic expressions of Selberg's zeta functions for congruence subgroups

In [P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory 15 (1982) 229-247], it was proved that the Selberg zeta function for SL₂(Z) is expressed in terms of the fundamental units and the class numbers of the primitive indefinite binary quadratic forms. The aim of this paper is to obtain similar arithmetic expressions of the logarithmic derivatives of the Selberg zeta functions for congruence subgroups of SL₂(Z). As applications, we study the Brun-Titchmarsh type prime geodesic theorem and the asymptotic formula of the sum of the class number.     

Entropy conditions for subsequences of random variables with applications to empirical processes

We introduce new entropy concepts measuring the size of a given class of increasing sequences of positive integers. Under the assumption that the entropy function of is not too large, many strong limit theorems will continue to hold uniformly over all sequences in . We demonstrate this fact by extending the Chung-Smirnov law of the iterated logarithm on empirical distribution functions for independent identically distributed random variables as well as for stationary strongly mixing sequences to hold uniformly over all sequences in . We prove a similar result for sequences (n _k ω) mod 1 where the sequence (n _k ) of real numbers satisfies a Hadamard gap condition. Authors’ addresses: István Berkes, Department of Statistics, Technical University Graz, Steyrergasse 17/IV, A-8010 Graz, Austria; Walter Philipp, Department of Statistics, University of Illinois, 725 S. Wright Street, Champaign, IL 61820, USA; Robert F. Tichy, Department of Analysis and Computational Number Theory, Technical University Graz, Steyrergasse 30, A-8010 Graz, Austria     

Geometric progressions in sumsets over finite fields

Given two sets , the set of d dimensional vectors over the finite field with q elements, we show that the sumset contains a geometric progression of length k of the form vΛ ^j , where j = 0,…, k − 1, with a nonzero vector and a nonsingular d × d matrix Λ whenever . We also consider some modifications of this problem including the question of the existence of elements of sumsets on algebraic varieties.     

Positive Harmonic Functions for Semi-Isotropic Random Walks on Trees, Lamplighter Groups, and DL-Graphs

We determine all positive harmonic functions for a large class of “semi-isotropic” random walks on the lamplighter group, i.e., the wreath product ℤ_q≀ℤ, where q≥2. This is possible via the geometric realization of a Cayley graph of that group as the Diestel–Leader graph . More generally, (q,r≥2) is the horocyclic product of two homogeneous trees with respective degrees q+1 and r+1, and our result applies to all -graphs. This is based on a careful study of the minimal harmonic functions for semi-isotropic walks on trees. Mathematics Subject Classifications (2000) 60J50, 05C25, 20E22, 31C05, 60G50. Supported by European Commission, Marie Curie Fellowship HPMF-CT-2002-02137.     

A relation between the zeros of an <Emphasis Type="Italic">L</Emphasis>-function belonging to the Selberg class and the zeros of an associated <Emphasis Type="Italic">L</Emphasis>-function twisted by a Dirichlet character

Let F (s) be a function belonging to the Selberg class. For a primitive Dirichlet character , we can define the -twist F(s) of F (s). If F(s) also belongs to the Selberg class and satisfies some other conditions then there is a relation between the zeros of F (s) and the zeros F(s). Further we give an operator theoretic interpretation of this relation according to A. Connes study.Received: 5 January 2004     

Existence of Solutions of a Vector Optimization Problem with a Generic Lower Semicontinuous Objective Function

We study a class of vector minimization problems on a complete metric space X which is identified with the corresponding complete metric space of lower semicontinuous bounded-from-below objective functions . We establish the existence of a G _δ everywhere dense subset ℱ of such that, for any objective function belonging to ℱ, the corresponding minimization problem possesses a solution.     

Selberg and Ruelle zeta functions for non-unitary twists

In this paper we study the dynamical zeta functions of Ruelle and Selberg associated with the geodesic flow of a compact hyperbolic odd-dimensional manifold. These are functions of a complex variable s in some right half-plane of \(\mathbb {C}\). Using the Selberg trace formula for arbitrary finite dimensional representations of the fundamental group of the manifold, we establish the meromorphic continuation of the dynamical zeta functions to the whole complex plane. We explicitly describe the singularities of the Selberg zeta function in terms of the spectrum of certain twisted Laplace and Dirac operators.     

Regularities of the Distribution of β-adic van der Corput Sequences

For Pisot numbers β with irreducible β-polynomial, we prove that the discrepancy function D(N, [0,y)) of the β-adic van der Corput sequence is bounded if and only if the β-expansion of y is finite or its tail is the same as that of the expansion of 1. If β is a Parry number, then we can show that the discrepancy function is unbounded for all intervals of length . We give explicit formulae for the discrepancy function in terms of lengths of iterates of a reverse β-substitution.     

Sums of the error term function in the mean square for ζ(s)

Sums of the form are investigated, where is the error term in the mean square formula for . The emphasis is on the case k = 1, which is more difficult than the corresponding sum for the divisor problem. The analysis requires bounds for the irrationality measure of e^2πm and for the partial quotients in its continued fraction expansion. Authors’ addresses: Y. Bugeaud, Université Louis Pasteur, Mathématiques, 7 rue René Descartes, F-67084 Strasbourg cedex, France; A. Ivić, Katedra Matematike RGF-a, Universitet u Beogradu, Đušina 7, 11000 Beograd, Serbia     

A mean value theorem on the binary Goldbach problem and its application

We study the binary Goldbach problem with one prime number in a given residue class, and obtain a mean value theorem. As an application, we prove that for almost all sufficiently large even integers n satisfying n ≢ 2(mod 6), the equation p ₁ + p ₂ = n is solvable in prime variables p ₁, p ₂ such that p ₁ + 2 = P ₃, and for every sufficiently large odd integer satisfying ≢ 1(mod 6), the equation p ₁ + p ₂ + p ₃ = is solvable in prime variables p ₁, p ₂, p ₃ such that p ₁ + 2 = P ₂, p ₂ + 2 = P ₃. Here P _k denotes any integer with no more than k prime factors, counted according to multiplicity.     

Regulator constants and the parity conjecture

The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p ^∞-Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/ℚ is semistable at 2 and 3, K/ℚ is abelian and K ^∞ is its maximal pro-p extension, then the p-parity conjecture holds for twists of E by all orthogonal Artin representations of . We also give analogous results when K/ℚ is non-abelian, the base field is not ℚ and E is replaced by an abelian variety. The heart of the paper is a study of relations between permutation representations of finite groups, their “regulator constants”, and compatibility between local root numbers and local Tamagawa numbers of abelian varieties in such relations. T. Dokchitser is supported by a Royal Society University Research Fellowship.     

Freeknot splines approximation of Sobolev-type classes of <Emphasis Type="Italic">s</Emphasis>-monotone functions

Let I be a finite interval, s ∈ ℕ₀, and r,ν,n ∈ ℕ. Given a set M, of functions defined on I, denote by M the subset of all functions y ∈ M such that the s-difference is nonnegative on I, ∀τ > 0. Further, denote by the Sobolev class of functions x on I with the seminorm . Also denote by Σ _ν,n , the manifold of all piecewise polynomials of order ν and with n – 1 knots in I. If ν ≥ max {r,s}, 1 ≤ p,q ≤ ∞, and (r,p,q) ≠ (1,1,∞), then we give exact orders of the best unconstrained approximation and of the best s-monotonicity preserving approximation . Part of this work was done while the first author visited Tel Aviv University in May 2003 and in March 2004.