Abstract: 1. Introduction.- 2. Preliminaries.- 3. Zeta Functions of the Geodesic Flow of Compact Locally Symmetric Manifolds.- 4. Operators and Complexes.- 5. The Verma Complexes on SY and SX.- 6. Harmonic Currents and Canonical Complexes.- 7. Divisors and Harmonic Currents.- 8. Further Developments and Open Problems.- 9. A Summary of Important Formulas.- Index of Equations.

Abstract: The theory of geometric zeta functions for locally symmetric spaces is generalized to the case of higher rank spaces. We show that the zeta functions can be continued to meromorphic functions on the plane, describe the divisor in terms of tangential cohomology and in terms of group cohomology which generalizes a conjecture of Patterson. We also extend the range of zeta functions in considering higher dimensional flats.

Abstract: The Riemann hypothesis is equivalent to the conjecture that the de Bruijn–Newman constant Λ satisfies Λ≤0 However, so far all the bounds that have been proved for Λ go in the other direction, and provide support for the conjecture of Newman that Λ≥0 This paper shows how to improve previous lower bounds and prove that −27⋅10−9<Λ This can be done using a pair of zeros of the Riemann zeta function near zero number 1020 that are unusually close together The new bound provides yet more evidence that the Riemann hypothesis, if true, is just barely true

Abstract: Euler discovered a recursion formula for the Riemann zeta function evaluated at the even integers. He also evaluated special Dirichlet series whose coefficients are the partial sums of the harmonic series. This paper introduces a new method for deducing Euler's formulas as well as a host of new relations, not only for the zeta function but for several allied functions.

Abstract: This paper gives an explicit zero-free region for the Riemann zeta-function derived from the VinogradovKorobov method. We prove that the Riemann zeta-function does not vanish in the region σ ≥ 1 − .00105 log−2/3 |t| (log log |t|)−1/3 and |t| ≥ 3.

Abstract: Many interesting families of rapidly convergent series representations for the Riemann Zeta function $\zeta (2n+1)$ $(n\in {\Bbb N})$ were considered recently by various authors In this survey-cum-expository paper, the author presents a systematic (and historical) investigation of these series representations Relevant connections of the results presented here with several other known series representations for $\zeta (2n+1)$ $(n\in {\Bbb N})$ are also pointed out In one of many computationally useful special cases presented here, it is observed that $\zeta (3)$ can be represented by means of a series which converges much faster than that in Euler's celebrated formula as well as the series used recently by Ap\'{e}ry in his proof of the irrationality of $\zeta (3)$ Symbolic and numerical computations using {\em Mathematica} (Version 40) for Linux show, among other things, that only 50 terms of this series are capable of producing an accuracy of seven decimal places

Abstract: The Laplace transform of $|\zeta(1/2+it)|$ is investigated, for which a precise expression is obtained, valid in a certain region in the complex plane. The method of proof is based on complex integration and spectral theory of the non-Euclidean Laplacian.

Abstract: Let G(z) be an entire function of order less than 2 that is real for real z with only real zeros Then we classify certain distribution functions F such that the convolution (G ∗ dF )(z) = ∫∞ −∞G(z − is) dF (s) of G with the measure dF has only real zeros all of which are simple This generalizes a method used by Polya to study the Riemann zeta function

Abstract: A vanishing theorem is proved for l-adic cohomology with compact support on an affine (singular) complete intersection As an application, it is shown that for an affine complete intersection defined over a finite field of q elements, the reciprocal "poles" of the zeta function are always divisible by q as algebraic integers A p-adic proof is also given, which leads to further q-divisibility of the poles or equivalently an improvement of the polar part of the Ax-Katz theorem for an affine complete intersection Similar results hold for a projective complete intersection

Abstract: Let K be the simplest cubic field defined by the irreducible polynomial where m is a nonnegative rational integer such that m 2 + 3 m + 9 is square-free. We estimate the value of the Dedekind zeta function ζ K (s ) at s = −1 and get class number 1 criterion for the simplest cubic fields.

Abstract: In this paper, we introduce new non-abelian zeta functions for number fields and study their basic properties. Recall that for number fields, we have the classical Dedekind zeta functions. These functions are usually called abelian, since, following Artin, they are associated to one dimensional representations of Galois groups; moreover, following Tate and Iwasawa, they may be constructed as integrations over abelian spaces, i.e., GL1 over adelic space AF for F . Thus to define non-abelian versions of zeta functions for number fields, naturally, mathematicians use higher dimensional representations of Galois groups and/or algebraic groups. This turns to be extremely important and very fruitful. As a result, now we have the so-called Artin L-functions, automorphic Lfunctions, etc.. However in this paper, we are not going to touch any part of such a fascinating representation oriented number theoretical theory. Instead, we do it more geometrically. It consists of two aspects, i.e., the one for integrands and the one for integration domains, along with the pioneer works of Tata and Iwasawa. To construct quite satisfied integrands, we need a completed cohomology theory, form which RiemannRoch theorem holds. For this purpose, in Part I of this paper, for a number field F with KF a canonical element of degree log |∆F |, we first introduce an adelic version of vector bundles E over number fields; then, we define the 0-th cohomology h(F,E) and the 1-st cohomology h(F,E) for these vector bundles which satisfy the standard duality h(F,E) = h(F,E ⊗ KF ); and finally, we prove the following Riemann-Roch theorem for them: h(F,E) − h(F,E) = deg(E) − rank(E) 2 · log |∆F |.

Abstract: Let X be a regular scheme projective and flat over Spec(Z), equidimensional of relative dimension d. Consider the Hasse-Weil zeta function of X, ζ(X, s) = ∏ x(1 −N(x) −s)−1 where x ranges over the closed points of X and N(x) is the order of the residue field of x. Denote by L(X, s) the zeta function with Γ-factors L(X, s) = ζ(X, s)Γ(X, s). The L-function conjecturally satisfies a functional equation

Abstract: Let be the Hurwitz zeta function. Furthermore, let p > 1 and α ≠ 0 be real numbers and n ≥ 2 be an integer. We determine the best possible constants a(p, α, n), A(p, α, n), b(p, n) and B(p, n) such that the inequalities and hold for all positive real numbers x1,…,xn.

Abstract: In this paper, we introduce a geometrically stylized arithmetic cohomology for number fields. Based on such a cohomology, we define and study new yet genuine non-abelian zeta functions for number fields, using an intersection stability.

Abstract: We consider a zeta function of the classical dynamics in the exterior of several convex bodies. The main result is that the poles of the zeta function cannot converge to the line of absolute convergence if the abscissa of absolute convergence of the zeta function is positive.

Abstract: Let n ≥ 2 be an integer. Let P be the set of all integers in [1, n + 1] and let σ be a cyclic permutation on P . Assume that f is the linearisation of σ on P . Then we show that f has rational Artin-Mazur zeta function which is closely related to the characteristic polynomial of some n × n matrix with entries either zero or one. Some examples of non-conjugate maps with the same Artin-Mazur zeta function are also given.