Abstract: Preface 1 Elements of the probability theory 2 Dirichlet series and Dirichlet polynomials 3 Limit theorems for the modulus of the Riemann Zeta-function 4 Limit theorems for the Riemann Zeta-function on the complex plane 5 Limit theorems for the Riemann Zeta-function in the space of analytic functions 6 Universality theorem for the Riemann Zeta-function 7 Limit theorem for the Riemann Zeta-function in the space of continuous functions 8 Limit theorems for Dirichlet L-functions 9 Limit theorem for the Dirichlet series with multiplicative coefficients References Notation Subject index

Abstract: In conformal geometry, the Sobolev inequality at a critical exponent has received much attention. In particular, the determination of the best constants has played a crucial role in the Yamabe problem. In dimension two the analogous problem deals with the Moser-Trudinger inequality: on a compact Riemann surface M2, there exists a constant c = c(M) so that f e47w2 < c(M) if f IVw12 < 1 and f w = 0. The connection of this inequality with geometry comes through the zeta functional determinant of the Laplacian as defined by Ray-Singer: for a Riemannian metric g, let 0 < A1 < A2 < ... be the spectrum

Abstract: Let $N(L)$ be the number of eigenvalues, in an interval of length $L$, of a matrix chosen at random from the Gaussian orthogonal, unitary, or symplectic ensembles of $N$ by $N$ matrices, in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$. We prove that $[N(L)\ensuremath{-}〈N(L)〉]/\sqrt{\mathrm{ln}L}$ has a Gaussian distribution when $L\ensuremath{\rightarrow}\ensuremath{\infty}$. This theorem, which requires control of all the higher moments of the distribution, elucidates numerical and exact results on chaotic quantum systems and on the statistics of zeros of the Riemann zeta function.

Abstract: An intriguing log-cosine integral is fully analyzed and shown to have value a rational multiple of rC(4), C being the Riemann zeta function. From this we deduce by means of generating functions and Parseval's identity the sums of certain series previously established by a completely different method.

Abstract: Using the Weyl quantization we formulate one-dimensional adelic quantum mechanics, which unifies and treats ordinary and p-adic quantum mechanics on an equal footing. As an illustration the corresponding harmonic oscillator is considered. It is a simple, exact and instructive adelic model. Eigenstates are Schwartz-Bruhat functions. The Mellin transform of the simplest vacuum state leads to the well-known functional relation for the Riemann zeta function. Some expectation values are calculated. The existence of adelic matter at very high energies is suggested.

Abstract: Tof ∈ℂ[x 1…,x n ] one associates thetopological zeta function which is an invariant of (the germ of)f at 0, defined in terms of an embedded resolution of (the germ of)f −1{0} inf −1{0}. By definition the topological zeta function is a rational function in one variable, and it is related to Igusa’s local zeta function. A major problem is the study of its poles. In this paper we exactly determine all poles of the topological zeta function forn=2 and anyf ∈ℂ[x 1,x 2]. In particular there exists at most one pole of order two, and in this case it is the pole closest to the origin. Our proofs rely on a new geometrical result which makes the embedded resolution graph of the germ off into an ‘ordered tree’ with respect to the so-callednumerical data of the resolution.

Abstract: L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Abstract: From the perspective of the well-known identity 2F (a, b; c; 1)= rc(cc a b) we clarify the connections between the Stirling numbers Sk and the Riemann zeta function C(n) . As a consequence, certain series and integrals can be evaluated in terms of C,(n) and Sn.

Abstract: © Annales de l’institut Fourier, 1995, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Abstract: JUNESANG CHOI, H.M. SRIVASTAVA AND J.R. QUINELots of formulas for series of zeta function have been developed in many ways.We show how we can apply the theory of the double gamma function, which hasrecently been revived according to the study of determinants of Laplacians, toevaluate some series involving the Riemann zeta function.1. INTRODUCTION

Abstract: In this paper, the problem of computing the exact value of the asymptotic efficiency of maximum likelihood estimators of a discontinuous signal in a Gaussian white noise is considered. A method based on constructing difference equations for the appropriate moments is presented and used to show that the exact variance of the Pitman estimator is $16\zeta(3)$, where $\zeta$ is the Riemann zeta function.

Abstract: A very simple class of algorithms for the computation of the Riemann-zeta function to arbitrary precision in arbitrary domains is proposed. These algorithms out perform the standard methods based on Euler-Maclaurin summation, are easier to implement and are easier to analyse.

Abstract: Article history: Received 2 March 2010 Revised 28 April 2010 Communicated by David Goss MSC: 11M06

Abstract: We discuss various results about weighted dynamical zeta functions for real and complex hyperbolic dynamical systems, mainly their relationship with transfer operators.

Abstract: To a polynomial f over a p-adic field K and a character X of the group of units of the valuation ring of K one associates Igusa's local zeta function Z(s, f, X), which is a meromorphic function on C. Several theorems and conjectures relate the poles of Z(s, f, X) to the monodromy of f; the so-called holomorphy conjecture states roughly that if the order of X does not divide the order of any eigenvalue of monodromy of f, then Z(s, f, X) is holomorphic on C. We prove mainly that if the holomorphy conjecture is true for f(xI . xn-) , then it is true for f(xl . .. x ) + xk with k > 3, and we give some applications.

Abstract: The problem of numerical evaluation of the classical trigonometric series ... (formule)... where ν > 1 in the case of S 2n (α) and C 2n+1 (α) with n = 1, 2, 3,... has been recently addressed by Dempsey, Liu, and Dempsey; Boersma and Dempsey; and by Gautschi. We show that, when α is equal to a rational multiple of 2π, these series can in the general case be summed in closed form in terms of known constants and special functions. General formulae giving C ν (α) and S ν (α) in terms of the generalized Riemann zeta function and the cosine and sine functions, respectively, are derived. Some simpler variants of these formulae are obtained, and various special results are established

Abstract: Explicit formulae for the zeta functions of the zeros of Hahn-Exton and Jackson's q-Bessel functions are derived. They can be regarded as spectral sum rules for some discrete quantum billiards.

Abstract: In this paper we consider L-functions satisfying certain standard conditions, their approximation by Dirichlet polynomials and, especially, lower bounds for the lengths of the polynomials that provide good approximations. 1. Introduction For ~(s) the Riemann zeta-function one has the Dirichlet series representation valid for where s = u + it. By the absolute convergence of this series one sees that, even for x not very large, the Dirichlet polynomial L n-s gives a rather good n2 approximation to ~(s), with a remainder which is o( 1 ) as x -~ oo. This is a nice property, since one would expect the finite sum to be easier to work with for purposes of estimation. However, one is of course more interested in estimating ~(s) in the critical strip 0 u 1. Here the above polynomial still provides [T, § 4.11 ] (at least away from the pole) a useful approximation to ~(s), moreover the smoothed polynomials * Supported in part by NSERC Grant A5123. Pervenuto alla Redazione il 2 Luglio 1994. 518 do an even better job, but all of these only for x > (1 + 0 ( 1 )) l!l, and this limits their usefulness for application. 27r Thus the question arises whether shorter approximations of the same quality to ~(s), or for that matter to L-series and general zeta functions, exist. In this paper we investigate approximations by Dirichlet polynomials to L-functions of a fairly general type, and show in many cases that it is not possible to achieve a very good level of approximation by means of polynomials essentially shorter than the known approximations. Thus we may view such a result as a first step toward understanding the analytic complexity of a zeta function. We shall consider L-functions L(s) having the following properties (compare, for example, [S]): (HI) L(s) is given by an absolutely convergent Dirichlet series in the half-plane u > 1, with coefficients an satisfying a, = 1 and an « n°(1). (H2) L(s) is meromorphic of finite order in the whole complex plane, has only finitely many poles and satisfies a functional equation 1 where with constants satisfying From the fact that L(s) is of finite order with finitely many poles and satisfies a functional equation of the above type and from the Phragmen-Lindelof principle, it follows that L(s) has, away from the poles, polynomial growth in any fixed vertical strip. Moreover L(s), for a 1 has order not less than 2 1 and for a 0 has order precisely where It now follows by a well-known argument (cf. [T, § 9.4]) that the number N(T; L) of non-trivial zeros (that is, those not located at the poles of the r factors) of L(s) satisfying 0 t T, is given asymptotically by 1 For a function f (s) we define 7(s)=f(-~). 519 where cL is a constant depending on L. Since we assume a 1 = 1 we may compute the constants explicitly and write this in the form where One should remark that the choice of the parameter Q and the Gamma factors in the above decomposition of are not uniquely determined due to the multiplication formula for the Gamma function. However the key quantities A and CL used in this paper are uniquely determined by L(s). The next assumption that we make about our L-function is that it satisfy a weak zero-density estimate. Let N(u, T; L) denote the number of non-trivial zeros p = {3 + iq of L with 0 q T and {3 > ~ . Then we assume: (H3) For any fixed 6 > 0, we have Our two main results place a limitation on the length of the Dirichlet polynomial (actually, may be replaced by any fixed positive constant) if it is to be a 2 useful approximation to L(s). Specifically, we prove THEOREM 1. Let L(s) satisfy assumptions (Hl)-(H3), and let e,,-’ > 0. Suppose that we have on the segment Our basic strategy is to use a well known lemma of Littlewood to compare, in a suitable rectangle, the number of zeros of the function L(s) with that of the approximating polynomial These should be nearly equal if the approximation is sufficiently good. On the other hand we shall be able to estimate the former using (1.1). This will give a contradiction provided that we can give a smaller upper bound to the number of zeros of D2(s) in case x is not too large. Such a result is provided by the following: 520 PROPOSITION 1. Let Dx(s) given by (1.2) satisfy Then, uniformly for a a oo we have Let also N(a, T, T + H; Dx) denote the number of zeros of D2(s) satisfying u > a, T t T + H, where H T. Then, uniformly for -H a l, we have where the implied constant is absolute. The exponent 2A given in Theorem 1 is sharp, as will be seen in the next section. Nevertheless, a slightly different argument using Rouche’s theorem shows that the bound can be made still more precise if one is willing to strengthen the assumptions to some extent. Specifically, we have: THEOREM 2. Let L(s) satisfy assumptions (HI), (H2), and also, for every 6 > 0, the strengthened zero density bound Suppose that we have on the segment (u = -c’, T t (I + Then x > (1 + o(I»CLT2A, with CL as in ( 1.1’). In Theorem 2, not only the exponent, but even the constant CL is the best possible. We remark that the assumption of (1.3) on the segment with u = -g’ is stronger than the assumption on a corresponding segment with a = 1 -s’; see Proposition 3. In the event that one assumes a stronger version as in Theorem 2, but is willing to settle for the weaker conclusion of Theorem 1, then it is possible to give a somewhat simpler proof which combines the principle of the argument with the result of Proposition 1. Throughout the paper, implied constants may depend on L(s) which is considered to be fixed. It would be of interest to have analogous results that are uniform in the parameter Q. The paper is organized as follows. In Section 2 we give a number of examples illustrating the sharpness of our results. In Section 3 we give an alternative argument that is considerably shorter than the proof of the main 521 theorems, but which gives only weaker bounds except in the case A 1. The remaining sections are devoted to the proof of the Theorems. In Section 4, we give the proof of Proposition 1, bounding the number of zeros of Dirichlet polynomials. In Section 5 we prove Proposition 3 which shows that, given an approximation of the type hypothesized by the theorems, that approximation continues to hold for all larger values of u. In Section 6 we prove a number of consequences of our hypothesis of a zero-density bound. We find, with good localization, thin horizontal strips on which there holds a Lindelof strength bound for L, and within each of these, a horizontal line on which holds a similar bound for L-1. These bounds, which are needed for our application of the Littlewood lemma and the Rouche theorem, improve earlier results which would not have sufficed. Finally, in Section 7, we combine the above preparations to complete the proofs of our results. 2. Some Examples and Remarks EXAMPLE 1. Zeros of Dirichlet polynomials The finite Euler product has length by the prime number theorem and has, for T t 2T, zeros on the imaginary axis at t = 2n7r/ log p. These number again by the prime number theorem. Thus the bound given in Proposition 1 is asymptotically sharp. EXAMPLE 2. Approximate functional equation As is well known, it is possible to approximate the Riemann zeta function, using two Dirichlet polynomials rather than one, in a way which allows shorter polynomials, namely: 522 for x y : , and s in any fixed vertical strip away from the pole at Here appears in the functional equation It is likely that there should be an analogue to our Theorem for approximate functional equations of this type stating that such an approximate functional equation can only hold for L(s) in the range T t 2T provided that xy > At first we hoped that our method, based on counting zeros, would lead to this result, but were stopped by the following example which shows that the analogue for Proposition 1 (at least in its obvious form) does not hold. Take L(s) to be ~(s); take x = y = 1. Then the \"approximation\" is 1 + x(s) and, despite the fact that x and y are bounded, this has asymptotically (in fact, with an error term only O(log T)) as many zeros as ~(s) itself inside the rectangle 0 Q 1, T t 2T. EXAMPLE 3. Existence of approximations It is well-known that smoothed truncations of a Dirichlet series can provide very good approximations. Let u(x) be a C°° function with compact support in (0, 1], such that -

Abstract: Some preliminaries some fundamental theorems on Titchmarsh series and applications Titchmarsh's phenomenon some recent progress (statements without proofs of some difficult results mentioned in the introduction). Appendix: asymptotic formulae for the gamma function and so on.

Abstract: We compute the Lyapunov exponent, generalized Lyapunov exponents and the diffusion constant for a Lorentz gas on a square lattice, thus having infinite horizon. Approximate zeta functions, written in terms of probabilities rather than periodic orbits, a re used in order to avoid the convergence problems of cycle expansions. The emphasis is on the relation between the analytic structure of the zeta function, where a branch cut plays an important role, and the asymptotic dynamics of the system. We find a diverging diffusion constant $D(t) \sim \log t$ and a phase transition for the generalized Lyapunov exponents.

Abstract: We prove that any spectral sequence obeying a certain growth law is the quantum spectrum of an equivalence class of classically integrable nonlinear oscillators. This implies that exceptions to the Berry-Tabor rule for the distribution of quantum energy gaps of classically integrable systems, are far more numerous than previously believed. In particular, we show that for each finite dimension k, there are an infinite number of classically integrable k-dimensional nonlinear oscillators whose quantum spectrum reproduces the imaginary part of zeros on the critical line of the Riemann zeta function.

Abstract: We prove that any spectral sequence obeying a certain growth law is the quantum spectrum of an equivalence class of classically integrable non-linear oscillators. This implies that exceptions to the Berry-Tabor rule for the distribution of quantum energy gaps of classically integrable systems, are far more numerous than previously believed. In particular we show that for each finite dimension $k$, there are an infinite number of classically integrable $k$-dimensional non-linear oscillators whose quantum spectrum reproduces the imaginary part of zeros on the critical line of the Riemann zeta function.

Abstract: Continuing previous work, we discuss applications of our summation/integration procedure to some classes of complex slowly convergent series. Especially, we consider the series of the form $$\sum olimits_{k = 1}^{ + \infty } {( \pm 1)^k k^{v - 1} } R(k)$$ , where 0=2/((2m+1)π),m∈ℕ0, involving the hyperbolic weightw(t)=1/cosh2 t. Numerical results are included to illustrate the method.

Abstract: The zeta function associated with higher-spin fields on the Euclidean $4$-ball is investigated. The leading coefficients of the corresponding heat-kernel expansion are given explicitly and the zeta functional determinant is calculated. For fermionic fields the determinant is shown to differ for local and spectral boundary conditions.

Abstract: We bring together two apparently disconnected lines of research (of mathematical and of physical nature, respectively) which aim at the definition, through the corresponding zeta function, of the determinant of a differential operator possessing, in general, a complex spectrum. It is shown explicitly how the two lines have in fact converged to a meeting point at which the precise mathematical conditions for the definition of the zeta function and the associated determinant are easy to understand from the considerations coming up from the physical approach, which proceeds by stepwise generalization starting from the most simple cases of physical interest. An explicit formula that establishes the bridge between the two approaches is obtained.

Abstract: We prove known identities for the Khinchin constant and develop new identities for the more general Hoelder mean limits of continued fractions. Any of these constants can be developed as a rapidly converging series involving values of the Riemann zeta function and rational coefficients. Such identities allow for efficient numerical evaluation of the relevant constants. We present free-parameter, optimizable versions of the identities, and report numerical results.