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54
Prime Number Races
 Amer. Math. Monthly
"... 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 ..."
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Cited by 13 (1 self)
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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
Evidence in Favor of the BaezDuarte Criterion for the Riemann Hypothesis
, 2008
"... Abstract: We present the results of the numerical experiments in favor of the BaezDuarte 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). ..."
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Cited by 10 (3 self)
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Abstract: We present the results of the numerical experiments in favor of the BaezDuarte 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).
Fractality, selfsimilarity and complex dimensions
 ZETA FUNCTIONS OF FRACTALS AND POLYNOMIALS 25 MANDELBROT, PART 1. PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS 72
, 2004
"... 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 co ..."
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Cited by 7 (1 self)
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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.
Semiclassical limits for the hyperbolic plane
 Duke Math. J
"... We study the concentration properties of highenergy eigenfunctions for the LaplaceBeltrami 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. ..."
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Cited by 6 (0 self)
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We study the concentration properties of highenergy eigenfunctions for the LaplaceBeltrami 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 geodesicflowinvariant positive measure. Our main consideration is the microlocal lifts of the elementary eigenfunctions constructed from the MacdonaldBessel functions. We find that with a scaling of auxiliary parameters the corresponding highenergy limit converges to the positive Dirac measure for the lift to SL(2; R) of a single geodesic on the upper halfplane. In particular, at high energy the microlocal lift of a MacdonaldBessel function is
An invitation to additive prime number theory
, 2004
"... 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 sam ..."
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Cited by 4 (0 self)
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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.
A New Lehmer Pair Of Zeros And A New Lower Bound For The De BruijnNewman Constant Delta
, 1993
"... . The de BruijnNewman 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 ! : ..."
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Cited by 4 (2 self)
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. The de BruijnNewman 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 BruijnNewman 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...
Toward verification of the Riemann hypothesis: Application of the Li criterion, to appear in
 Math. Phys., Analysis and Geometry (2005). 14 M. W. Coffey, New
, 2004
"... We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence {λk}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we determine new bounds for relevant Riemann zeta function sums and the seque ..."
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Cited by 4 (2 self)
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We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence {λk}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we determine new bounds for relevant Riemann zeta function sums and the sequence itself. We find that the Riemann hypothesis holds if certain conjectured properties of a sequence ηj are valid. The constants ηj enter the Laurent expansion of the logarithmic derivative of the zeta function about s = 1 and appear to have remarkable characteristics. On our conjecture, not only does the Riemann hypothesis follow, but an inequality governing the values λn and inequalities for the sums of reciprocal powers of the nontrivial zeros of the zeta function. Key words and phrases Riemann zeta function, Riemann xi function, logarithmic derivatives, Riemann hypothesis,
The Stieltjes constants, their relation to the ηj coefficients, and representation of the Hurwitz zeta function
, 2009
"... ..."
TRANSCENDENCE MEASURES AND ALGEBRAIC GROWTH OF ENTIRE FUNCTIONS
, 2004
"... 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 z ..."
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Cited by 3 (1 self)
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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 ζfunction, 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.
ALL BUT FINITELY MANY NONTRIVIAL ZEROS OF THE APPROXIMATIONS OF THE EPSTEIN ZETA FUNCTION ARE SIMPLE AND ON THE CRITICAL LINE
"... Abstract. The ChowlaSelberg formula is applied in approximating a given Epstein zeta function. Partial sums of the series derive from the ChowlaSelberg formula, and although these partial sums satisfy a functional equation as does in an Epstein zeta function, they do not possess an Euler product. ..."
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Cited by 2 (2 self)
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Abstract. The ChowlaSelberg formula is applied in approximating a given Epstein zeta function. Partial sums of the series derive from the ChowlaSelberg 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 ChowlaSelberg formula are simple and on the critical line. 1.