1. INTRODUCTION. There’s nothing quite like a day at the races....The quickening of the pulse as the starter’s pistol sounds, the thrill when your favorite contestant speeds out into the lead (or the distress if another contestant dashes out ahead of yours), and the accompanying fear (or hope) that the leader might change. And what if the race is a marathon? Maybe one of the contestants will be far stronger than the others, taking

To Benoît Mandelbrot, on the occasion of his jubilee. Abstract. We present an overview of a theory of complex dimensions of selfsimilar fractal strings, and compare this theory to the theory of varieties over a finite field from the geometric and the dynamical point of view. Then we combine the several strands to discuss a possible approach to establishing a cohomological interpretation of the complex dimensions. 1.

Abstract: We present the results of the numerical experiments in favor of the Baez-Duarte criterion for the Riemann Hypothesis. We give formulae allowing calculation of numerical values of the numbers ck appearing in this criterion for arbitrary large k. We present plots of ck 9 for k ∈ ( 1,10).

We study the concentration properties of high-energy eigenfunctions for the Laplace-Beltrami operator of the hyperbolic plane with special consideration of automorphic eigenfunctions. At the center of our investigation is the microlocal lift of an eigenfunction to SL(2; R) introduced by S. Zelditch. The microlocal lift is based on S. Helgason’s Fourier transform and has a straightforward description in terms of the Lie algebra sl(2; R). We begin with an elementary demonstration of Zelditch’s exact differential equation for the microlocal lift. Further, for a sequence of suitably bounded eigenfunctions with eigenvalues tending to infinity, we show that the limit of microlocal lifts is a geodesic-flow-invariant positive measure. Our main consideration is the microlocal lifts of the elementary eigenfunctions constructed from the Macdonald-Bessel functions. We find that with a scaling of auxiliary parameters the corresponding high-energy limit converges to the positive Dirac measure for the lift to SL(2; R) of a single geodesic on the upper half-plane. In particular, at high energy the microlocal lift of a Macdonald-Bessel function is

The main purpose of this survey is to introduce the inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic machinery used to study them, and try to give a representative sample of the directions of current research.

. 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 \Gamma5:895 \Delta 10 \Gamma9 ! : This is done with the help of a spectacularly close pair of consecutive zeros of the Riemann zeta function. Key words. Lehmer pairs of zeros, de Bruijn-Newman constant, Riemann Hypothesis. AMS subject classifications. 30D10, 30D15, 65E05. 1. Introduction. It is known (cf. Titchmarsh [9, p. 255]) that the Riemann ¸-function can be expressed in the form ¸ i x 2 j =8 = Z 1 0 \Phi(u) cos(xu)du (x 2 I C); (1.1) where \Phi(u) := 1 X n=1 \Gamma 2ß 2 n 4 e 9u \Gamma 3ßn 2 e 5u \Delta exp \Gamma \Gammaßn 2 e 4u \Delta (0 u ! 1); (1.2) and the Riemann Hypothesis is the statement that all zeros of ¸ are real. If we define H t (x) := Z 1 0 e tu 2 \Phi(u) cos(xu)du...

Abstract. The Chowla-Selberg formula is applied in approximating a given Epstein zeta function. Partial sums of the series derive from the Chowla-Selberg formula, and although these partial sums satisfy a functional equation as does in an Epstein zeta function, they do not possess an Euler product. What we call partial sums throughout this paper may be considered as special cases concerning a more general function satisfying a functional equation only. In this article we study the distribution of zeros of the function. We show that in any strip containing the critical line, all but finitely many zeros of the function are simple and on the critical line. For many Epstein zeta functions we show that all but finitely many nontrivial zeros of partial sums in the Chowla-Selberg formula are simple and on the critical line. 1.

Abstract. A strategy for proving Riemann hypothesis is suggested. The vanishing of the Rieman Zeta reduces to an orthogonality condition for the eigenfunctions of a non-Hermitian operator D + having the zeros of Riemann Zeta as its eigenvalues. The construction of D + is inspired by the conviction that Riemann Zeta is associated with a physical system allowing conformal transformations as its symmetries. The eigenfunctions of D + are analogous to the so called coherent states and in general not orthogonal to each other. The states orthogonal to a vacuum state (which has a negative norm squared) correspond to the zeros of the Riemann Zeta. The induced metric in the space of states which correspond to the zeros of the Riemann Zeta at the critical line Re[s] = 1/2 is hermitian. Riemann hypothesis follows both from the hermiticity of the induced metric and from the conformal gauge invariance in the subspace of states which correspond to the zeros of the Riemann Zeta. 1 1

Abstract. In this paper we obtain estimates for certain transcendence measures of an entire function f. Using these estimates, we prove Bernstein, doubling and Markov inequalities for a polynomial P(z, w) in C 2 along the graph of f. These inequalities provide, in turn, estimates for the number of zeros of the function P(z, f(z)) in the disk of radius r, in terms of the degree of P and of r. Our estimates hold for arbitrary entire functions f of finite order, and for a subsequence {nj} of degrees of polynomials. But for special classes of functions, including the Riemann ζ-function, they hold for all degrees and are asymptotically best possible. From this theory we derive lower estimates for a certain algebraic measure of a set of values f(E), in terms of the size of the set E. 1.

For integer partitions : n = a 1 + ::: + a k , where a 1 a 2 : : : a k 1, we study the sum a 1 + a 3 + : : : of the parts of odd index. We show that the average of this sum, over all partitions of n, is of the form n=2 + ( 6=(8)) n log n + c 2;1 n +O(log n): More generally, we study the sum a i + am+i + a 2m+i + : : : of the parts whose indices lie in a given arithmetic progression and we show that the average of this sum, over all partitions of n, is of the form n=m + b m;i n +O(log n), with explicitly given constants b m;i ; c m;i . Interestingly, for m odd and i = (m+ 1)=2 we have b m;i = 0, so in this case the error term is of lower order. The methods used involve asymptotic formulas for the behavior of Lambert series and the Zeta function of Hurwitz. We also show that if f(n; j) is the number of partitions of n the sum of whose parts of even index is j, then for every n, f(n; j) agrees with a certain universal sequence, Sloane's sequence #A000712, for j n=3 but not for any larger j.