Showing papers on "Riemann zeta function published in 1993"

Basic Analytic Number Theory

Abstract: I. Integer Points.- 1. Statement of the Problem, Auxiliary Remarks, and the Simplest Results.- 2. The Connection Between Problems in the Theory of Integer Points and Trigonometric Sums.- 3. Theorems on Trigonometric Sums.- 4. Integer Points in a Circle and Under a Hyperbola.- Exercises.- II. Entire Functions of Finite Order.- 1. Infinite Products. Weierstrass's Formula.- 2. Entire Functions of Finite Order.- Exercises.- III. The Euler Gamma Function.- 1. Definition and Simplest Properties.- 2. Stirling's Formula.- 3. The Euler Beta Function and Dirichlet's Integral.- Exercises.- IV. The Riemann Zeta Function.- 1. Definition and Simplest Properties.- 2. Simplest Theorems on the Zeros.- 3. Approximation by a Finite Sum.- Exercises.- V. The Connection Between the Sum of the Coefficients of a Dirichlet Series and the Function Defined by this Series.- 1. A General Theorem.- 2. The Prime Number Theorem.- 3. Representation of the Chebyshev Functions as Sums Over the Zeros of the Zeta Function.- Exercises.- VI. The Method of I.M. Vinogradov in the Theory of the Zeta Function.- 1. Theorem on the Mean Value of the Modulus of a Trigonometric Sum.- 2. Estimate of a Zeta Sum.- 3. Estimate for the Zeta Function Close to the Line ? = 1.- 4. A Function-Theoretic Lemma.- 5. A New Boundary for the Zeros of the Zeta Function.- 6. A New Remainder Term in the Prime Number Theorem.- Exercises.- VII. The Density of the Zeros of the Zeta Function and the Problem of the Distribution of Prime Numbers in Short Intervals.- 1. The Simplest Density Theorem.- 2. Prime Numbers in Short Intervals.- Exercises.- VIII. Dirichlet L-Functions.- 1. Characters and their Properties.- 2. Definition of L-Functions and their Simplest Properties.- 3. The Functional Equation.- 4. Non-trivial Zeros Expansion of the Logarithmic Derivative as a Series in the Zeros.- 5. Simplest Theorems on the Zeros.- Exercises.- IX. Prime Numbers in Arithmetic Progressions.- 1. An Explicit Formula.- 2. Theorems on the Boundary of the Zeros.- 3. The Prime Number Theorem for Arithmetic Progressions.- Exercises.- X. The Goldbach Conjecture.- 1. Auxiliary Statements.- 2. The Circle Method for Goldbach's Problem.- 3. Linear Trigonometric Sums with Prime Numbers.- 4. An Effective Theorem.- Exercises.- XI. Waring's Problem.- 1. The Circle Method for Waring's Problem.- 2. An Estimate for Weyl Sums and the Asymptotic Formula for Waring's Problem.- 3. An Estimate for G(n).- Exercises.- Hints for the Solution of the Exercises.- Table of Prime Numbers < 4070 and their Smallest Primitive Roots.

Exponential sums and the Riemann zeta function v

Abstract: A Van der Corput exponential sum is $S = \Sigma \exp (2 \pi i f(m))$, where $m$ has size $M$, the function $f(x)$ has size $T$ and $\alpha = (\log M) / \log T < 1$. There are different bounds for $S$ in different ranges for $\alpha $. In the middle range where $\alpha $ is near ${1\over 2}$, $S = O(\sqrt{M} T^{\theta + \epsilon })$. This $\theta $ bounds the exponent of growth of the Riemann zeta function on its critical line ${\rm Re} s = {1\over 2}$. Van der Corput used an iteration which changed $\alpha$ at each step. The Bombieri?Iwaniec method, whilst still based on mean squares, introduces number-theoretic ideas and problems. The Second Spacing Problem is to count the number of resonances between short intervals of the sum, when two arcs of the graph of $y = f'(x)$ coincide approximately after an automorphism of the integer lattice. In the previous paper in this series [Proc. London Math. Soc. (3) 66 (1993) 1?40] and the monograph Area, lattice points, and exponential sums we saw that coincidence implies that there is an integer point close to some ?resonance curve?, one of a family of curves in some dual space, now calculated accurately in the paper ?Resonance curves in the Bombieri?Iwaniec method?, which is to appear in Funct. Approx. Comment. Math. We turn the whole Bombieri?Iwaniec method into an axiomatised step: an upper bound for the number of integer points close to a plane curve gives a bound in the Second Spacing Problem, and a small improvement in the bound for $S$. Ends and cusps of resonance curves are treated separately. Bounds for sums of type $S$ lead to bounds for integer points close to curves, and another branching iteration. Luckily Swinnerton-Dyer's method is stronger. We improve $\theta $ from 0.156140... in the previous paper and monograph to 0.156098.... In fact $(32/205 + \epsilon , 269/410 + \epsilon)$ is an exponent pair for every $\epsilon > 0$.

Newton polygons of zeta functions and L functions

Abstract: In this article we give a systematic treatment of Newton polygons of exponential sums. The Newton polygon is a nice way to describe p-adic values of the zeroes or poles of zeta functions and L functions. Our main objective is to show that the Adolphson-Sperber conjecture 12], which asserts that under a simple condition the generic Newton polygon of L functions coincides with its lower bound, is false in its full form, but true in a slightly weaker form. We also show that the full form is true in various important special cases. For example, we show that for a generic projective hypersurface of degree d, the Newton polygon of the interesting part of the zeta function coincides with its lower bound (the Hodge polygon). This gives a p-adic proof of a recent theorem of Illusie, conjectured by Dwork and Mazur. For more examples, let us consider the family of affine hypersurfaces of degree d or the family of affine hypersurfaces defined by polynomials f(xi, . . ., x7n) of degree di with respect to xi (1 < i < n), where the di are fixed positive integers. Then, for all large prime numbers p, the generic Newton polygon for the zeta functions of each of the two families of hypersurfaces coincides with its lower bound. We obtain our main results, namely several decomposition theorems, using certain maximizing functions from linear programming. Our work suggests a possible connection between Newton polygons and the resolution of singularities of toric varieties. Let p be a prime, q = pa, and let Fq be the finite field of q elements and Fqm its extension of degree m. Fix a nontrivial additive character qP of Fp. For any Laurent polynomial f(xi, . . ., xn) E Fq[xl, xj1,.. . ., x, xi1] we form

Symmetry decomposition of chaotic dynamics

Abstract: Discrete symmetries of dynamical flows give rise to relations between periodic orbits, reduce the dynamics to a fundamental domain, and lead to factorizations of zeta functions. These factorizations in turn reduce the labour and improve the convergence of cycle expansions for classical and quantum spectra associated with the flow. The general formalism is developed, with the N-disk pinball model used as a concrete example and a series of physically interesting cases worked out in detail.

Zeta regularized products

Abstract: If λ k is a sequence of nonzero complex numbers, then we define the zeta regularized product of these numbers, Π k l k , to be exp(−Z'(0)) where Z(s) = Σ k=0 ∞l k −s . We assume that Z(s) has analytic continuation to a neighborhood of the origin. If λ k is the sequence of positive eigenvalues of the Laplacian on a manifold, then the zeta regularized product is known as det' Δ, the determinant of the Laplacian, and Π k (λ k − λ) is known as the functional determinant. The purpose of this paper is to discuss properties of the determinant and functional determinant for general sequences of complex numbers. We discuss asymptotic expansions of the functional determinant as λ→−∞ and its relationship to the Weierstrass product

Arbitrary-spin effective potentials in anti-de Sitter spacetime.

Abstract: The one-loop effective potentials and stress-energy tensors for fields of arbitrary spin in anti-de Sitter (AdS) spacetime are calculated using the $\ensuremath{\zeta}$-function technique. The arbitrary-spin $\ensuremath{\zeta}$ function is computed exactly on the four-dimensional hyperbolic space ${H}^{4}$, which is the (noncompact) Euclidean section appropriate to AdS spacetime. The finite parts of the effective potentials are given explicitly in terms of integrals containing $\ensuremath{\psi}$ functions. Applications to AdS supergravity are pointed out.

On a ferromagnetic spin chain

Abstract: The quotient ξ(s-1)/ξ(s) of Riemann zeta functions is shown to be the partition function of a ferromagnetic spin chain for inverse temperatures.

An example of a two-term asymptotics for the “counting function” of a fractal drum

Abstract: In this paper we study the spectrum of the Dirichlet Laplacian in a bounded domain Ω ⊂ R n with fractal boundary ∂Ω We construct an open set Q for which we can effectively compute the second term of the asymptotics of the «counting function» N(λ, Q), the number of eigenvalues less than λ In this example, contrary to the M V Berry conjecture, the second asymptotic term is proportional to a periodic function of Inλ, not to a constant We also establish some properties of the ζ-function of this problem We obtain asymptotic inequalities for more general domains and in particular for a connected open set O derived from Q Analogous periodic functions still appear in our inequalities These results have been announced in [FV]

Lectures on Mean Values of the Riemann Zeta Function

Abstract: This is an advanced text on the Riemann zeta function, a continuation of the author's earlier book. It presents the most recent results on mean values. An especially detailed discussion is given of the second and the fourth moment, and the latter is studied by the use of spectral theory, one of the most powerful methods used in analytic number theory. The book presupposes a reasonable knowledge of zeta function theory and complex analysis. It will be of use to researchers in the field, and to all those who have the need for application of zeta function theory.

Long mollifiers of the Riemann Zeta-function

Abstract: The best current bounds for the proportion of zeros of ζ( s ) on the critical line are due to Conrey [C], using Levinson's method [Lev]. This method can also be used to detect simple zeros on the critical line. To apply Levinson's method one first needs an asymptotic formula for the meansquare from 0 to T of ζ( s ) M ( s ) near the -line, where where μ ( n ) is the Mobius function, h ( x ) is a real polynomial with h (0) = 0, and y = T θ for some θ > 0. It turns out that the parameter θ is critical to the method: having an asymptotic formula valid for large values of θ is necessary in order to obtain good results. For example, if we let κ denote the proportion of nontrivial zeros of ζ( s ) which are simple and on the critical line, then having the formula valid for 0 yields κ > 0·3562, having 0 gives κ > 0·40219, and it is necessary to have θ > 0·165 in order to obtain a positive lower bound for κ. At present, it is known that the asymptotic formula remains valid for 0 , this is due to Conrey. Without assuming the Riemann Hypothesis, Levinson's method provides the only known way of obtaining a positive lower bound for κ.

Special values of the Lerch zeta function and the evaluation of certain integrals

Abstract: The Lerch zeta function 0(x, a, s) is defined by the series coe2nxix 1D(x, a, s) = S + a)s' n=O (n +as where x is real, 0 1 if x is an integer and a > 0 otherwise. In this paper we study the function J(s , a) = ( 2, a, s) . We use its integral representation as -s (tI y\ e'ly dy J(s, a) = + 2 (a2 + y2)/2 sin s tan } e2KY -1 to obtain the values of certain definite integrals; for example, we show that [00 coshxlogx dx JO cosh 2x cos 2ra = 7r flo7((l+ a)/2) +log 27tcot 7ra 2snra log IF(a/2-) 2 \ 2)1 < <

Entire Fredholm determinants for evaluation of semiclassical and thermodynamical spectra.

Abstract: Proofs that Fredholm determinants of transfer operators for hyperbolic flows are entire can be extended to a large new class of multiplicative evolution operators. We construct such operators both for the Gutzwiller semiclassical quantum mechanics and for classical thermodynamic formalism, and introduce a new functional determinant which is expected to be entire for Axiom A flows, and whose zeros coincide with the zeros of the Gutzwiller-Voros zeta function.

Shintani zeta functions

Abstract: The theory of prehomogeneous vector spaces is a relatively new subject although its origin can be traced back through the works of Siegel to Gauss. The study of the zeta functions related to prehomogeneous vector spaces can yield interesting information on the asymptotic properties of associated objects, such as field extensions and ideal classes. This is amongst the first books on this topic, and represents the author's deep study of prehomogeneous vector spaces. Here the author's aim is to generalise Shintani's approach from the viewpoint of geometric invariant theory, and in some special cases he also determines not only the pole structure but also the principal part of the zeta function. This book will be of great interest to all serious workers in analytic number theory.

Zero point energy and analytic regularizations.

Abstract: We discuss analytic regularization methods used to obtain the renormalized vacuum energy of quantum fields in an arbitrary ultrastatic spacetime. After proving that the $\ensuremath{\zeta}$-function method is equivalent to the cutoff method with the subtraction of the polar terms, we present two examples where the analytic extension method gives a finite result, but in disagreement with the cutoff and $\ensuremath{\zeta}$-function methods.

Periodic-Orbit Theory

Abstract: [[ RM: A review paper on cycle expansions. I quote the introduction: in section (2) ]] I will summarize Gutzwiller's theory for the spectrum of eigenenergies and extend it to diagonal matrix elements as well. The derivation of the associated zeta function is given (2.2) and the identification of suitable scaling variables discussed (2.3). In section 3 tools necessary for the organization of chaos will be discussed: symbolic dynamics (3.1), the connectivity matrix (3.3), the topological zeta function (3.4) and general transfer matrices and zeta functions (3.5). Although illustrated for the case of hard collisions in a billiard, the symbolic dynamics can be extended to `smooth collisions' in smooth potentials (3.2). In systems with discrete symmetries, zeta functions factorize into zeta functions on invariant subspaces. This symmetry factorization and the associated reduction in symbolics is discussed in section 4. The ideas developed here are illustrated for the example of a free particle reflected elastically off three disks in section 5. Methods to find periodic orbits (5.1), the convergence of the trace formula (5.2), the semiclassical computation of scattering resonances (5.3), the convergence of the cycle expansion (5.4) and methods to obtain eigenvalues of the bounded billiard (5.5) are discussed. The relevant parts of a classical periodic orbit theory are developed in section 6.1, including a discussion of escape rates and the Hannay-Ozorio de Almeida sum rule (6.2). Finally, the issue of semiclassical matrix elements is taken up again and applications to experiments are discussed.

On Functional Determinants of Laplacians in Polygons and Simplices

Abstract: The functional determinant of an elliptic operator with positive, discrete spectrum may be defined as $e^{-Z'(0)}$, where $Z(s)$, the zeta function, is the sum $\sum_n^{\infty} \lambda_n^{-s}$ analytically continued to $s$ around the origin. In this paper $Z'(0)$ is calculated for the Laplace operator with Dirichlet boundary conditions inside polygons and simplices with the topology of a disc in the Euclidean plane. The domains we consider are hence piece--wise flat with corners on the boundary and in the interior. Our results are complementary to earlier investigations of the determinants on smooth surfaces with smooth boundaries. We have explicit closed integrated expressions for triangles and regular polygons.

Anomalous diffusion in dynamical systems: Transport coefficients of all order.

Abstract: The theory of Ruelle's zeta function [Thermodynamic Formalism (Addison-Wesley, Reading, MA, 1978)] is extended to describe anomalous transport induced by dynamical chaos. It is shown that P(q) for the generating function of the displacement may not exist for supradiffusive processes, and that the difficulty may be overcome by the introduction of a two-parameter function P (β,q). We present two exactly solvable examples of anomalous diffusion induced by intermittency, to which our method is applied

Anomalous diffusion in dynamical systems : transport coefficients of all order

Abstract: The theory of Ruelle's zeta function [Thermodynamic Formalism (Addison-Wesley, Reading, MA, 1978)] is extended to describe anomalous transport induced by dynamical chaos. It is shown that P(q) for the generating function of the displacement may not exist for supradiffusive processes, and that the difficulty may be overcome by the introduction of a two-parameter function P (β,q). We present two exactly solvable examples of anomalous diffusion induced by intermittency, to which our method is applied

A new lehmer pair of zeros and a new lower bound for the de bruijn-newman constant

Abstract: The de Bruijn-Newman constant has been investigated extensively because the truth of the Riemann Hypothesis is equivalent to the assertion that 0 On the other hand, C M Newman conjectured that 0 This paper improves previous lower bounds by showing that 5:895 10 9 < : This is done with the help of a spectacularly close pair of consecutive zeros of the Riemann zeta function

Semiclassical computations of energy levels

Abstract: Different methods of semiclassical calculations of energy levels of two-dimensional ergodic models are discussed and compared. Special attention is given to the calculation of the dynamical zeta function via the Rieman-Siegel relations.

Zeros of the Riemann Zeta-function on the critical line

Abstract: It was shown by Selberg [3] that the Riemann Zeta-function has at least cT log T zeros on the critical line up to height T, for some positive absolute constant c. Indeed Selberg’s method counts only zeros of odd order, and counts each such zero once only, regardless of its multiplicity. With this in mind we shall write γi for the distinct ordinates of zeros of ζ(s) on the critical line of odd multiplicity. We shall number the points γi so that 0 < γ1 < γ2 < . . . . The purpose of the present note is to extract a little more from Selberg’s argument, by obtaining further information on the distribution of the γi. This is given in the following result.

Resummation of classical and semiclassical periodic-orbit formulas.

Abstract: The convergence properties of cycle expanded periodic orbit expressions for the spectra of classical and semiclassical time evolution operators have been studied for the open three disk billiard. We present evidence that both the classical and the semiclassical Selberg zeta function have poles. Applying a Pad\'{e} approximation on the expansions of the full Euler products, as well as on the individual dynamical zeta functions in the products, we calculate the leading poles and the zeros of the improved expansions with the first few poles removed. The removal of poles tends to change the simple linear exponential convergence of the Selberg zeta functions to an $\exp\{-n^{3/2}\}$ decay in the classical case and to an $\exp\{-n^2\}$ decay in the semiclassical case. The leading poles of the $j$th dynamical zeta function are found to equal the leading zeros of the $j+1$th one: However, in contrast to the zeros, which are all simple, the poles seem without exception to be {\em double}\/. The poles are therefore in general {\em not}\/ completely cancelled by zeros, which has earlier been suggested. The only complete cancellations occur in the classical Selberg zeta function between the poles (double) of the first and the zeros (squared) of the second dynamical zeta function. Furthermore, we find strong indications that poles are responsible for the presence of spurious zeros in periodic orbit quantized spectra and that these spectra can be greatly improved by removing the leading poles, e.g.\ by using the Pad\'{e} technique.

Zeta function for the Lyapunov exponent of a product of random matrices

Abstract: A cycle expansion for the Lyapunov exponent of a product of random matrices is derived. The formula is non-perturbative and numerically effective, which allows the Lyapunov exponent to be computed to high accuracy. In particular, the free energy and the heat capacity are computed for the one-dimensional Ising model with quenched disorder. The formula is derived by using a Bernoulli dynamical system to mimic the randomness.