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15
The Riemann Zeros and Eigenvalue Asymptotics
 SIAM Rev
, 1999
"... Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many feat ..."
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Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t n have a similar structure to those of the semiclassical En ; in particular, they display randommatrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n can be computed accurately from formulae with quantum analogues. The RiemannSiegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.
APPLICATIONS OF THE LFUNCTIONS RATIOS CONJECTURES
 PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY
, 2006
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2003 Random matrix theory and the zeros of ζ
 Preprint mathph/0207044
"... Abstract. We study the density of the roots of the derivative of the characteristic polynomial Z(U, z) of an N × N random unitary matrix with distribution given by Haar measure on the unitary group. Based on previous random matrix theory models of the Riemann zeta function ζ(s), this is expected to ..."
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Abstract. We study the density of the roots of the derivative of the characteristic polynomial Z(U, z) of an N × N random unitary matrix with distribution given by Haar measure on the unitary group. Based on previous random matrix theory models of the Riemann zeta function ζ(s), this is expected to be an accurate description for the horizontal distribution of the zeros of ζ ′ (s) to the right of the critical line. We show that as N → ∞ the fraction of roots of Z ′ (U, z) that lie in the region 1−x/(N −1) ≤ z  < 1 tends to a limit function. We derive asymptotic expressions for this function in the limits x → ∞ and x → 0 and compare them with numerical experiments. Mathematics Subject Classification: 15A52, 11M99Random matrix theory and the zeros of ζ ′ (s) 2 1.
Moments of the derivative of characteristic polynomials with an application to the Riemann zeta function
 Comm. Math. Phys
"... Abstract. We investigate the moments of the derivative, on the unit circle, of characteristic polynomials of random unitary matrices and use this to formulate a conjecture for the moments of the derivative of the Riemann ζ function on the critical line. We do the same for the analogue of Hardy’s Zf ..."
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Abstract. We investigate the moments of the derivative, on the unit circle, of characteristic polynomials of random unitary matrices and use this to formulate a conjecture for the moments of the derivative of the Riemann ζ function on the critical line. We do the same for the analogue of Hardy’s Zfunction, the characteristic polynomial multiplied by a suitable factor to make it real on the unit circle. Our formulae are expressed in terms of a determinant of a matrix whose entries involve the IBessel function and, alternately, by a combinatorial sum. 1.
MOMENTS OF THE DERIVATIVE OF THE RIEMANN ZETAFUNCTION AND OF CHARACTERISTIC POLYNOMIALS
"... Characteristic polynomials of unitary matrices are extremely useful models for the Riemann zetafunction ζ(s). The distribution of their eigenvalues give insight into the distribution of zeros of the Riemann zetafunction and the values of these characteristic polynomials give a model for the value ..."
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Characteristic polynomials of unitary matrices are extremely useful models for the Riemann zetafunction ζ(s). The distribution of their eigenvalues give insight into the distribution of zeros of the Riemann zetafunction and the values of these characteristic polynomials give a model for the value distribution of ζ(s). See the works [KS] and [CFKRS] for detailed
THE TWISTED FOURTH MOMENT OF THE RIEMANN ZETA FUNCTION
, 709
"... Abstract. We compute the asymptotics of the fourth moment of the Riemann zeta function times an arbitrary Dirichlet polynomial of length T 1 ..."
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Abstract. We compute the asymptotics of the fourth moment of the Riemann zeta function times an arbitrary Dirichlet polynomial of length T 1
Zeros of the Derivatives of the Riemann Zetafunction ∗
, 2009
"... We study properties of zeros of the derivatives of the Riemann zeta function ζ(s). Levinson and Montgomery [8] achieved several important theorems for the behavior of zeros of ζ (m)(s) (m = 1, 2, 3, · · ·). If we assume the Riemann hypothesis, ζ ′(s) has no nonreal zero in Re s < 1 ..."
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We study properties of zeros of the derivatives of the Riemann zeta function ζ(s). Levinson and Montgomery [8] achieved several important theorems for the behavior of zeros of ζ (m)(s) (m = 1, 2, 3, · · ·). If we assume the Riemann hypothesis, ζ ′(s) has no nonreal zero in Re s < 1
Quantum Knots and Riemann Hypothesis
, 2006
"... In this paper we propose a quantum gauge system from which we construct generalized Wilson loops which will be as quantum knots. From quantum knots we give a classification table of knots where knots are onetoone assigned with an integer such that prime knots are bijectively assigned with prime nu ..."
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In this paper we propose a quantum gauge system from which we construct generalized Wilson loops which will be as quantum knots. From quantum knots we give a classification table of knots where knots are onetoone assigned with an integer such that prime knots are bijectively assigned with prime numbers and the prime number 2 corresponds to the trefoil knot. Then by considering the quantum knots as periodic orbits of the quantum system and by the identity of knots with integers and an approach which is similar to the quantum chaos approach of Berry and Keating we derive a trace formula which may be called the von MangoldtSelbergGutzwiller trace formula. From this trace formula we then give a proof of the Riemann Hypothesis. For our proof of the Riemann Hypothesis we show that the HilbertPolya conjecture holds that there is a selfadjoint operator for the nontrivial zeros of the Riemann zeta function and this operator is the Virasoro energy operator with central charge c = 1 2. Our approach for proving the Riemann Hypothesis can also be extended to prove the Extended Riemann Hypothesis. We also investigate the relation of our approach for proving the Riemann Hypothesis with the Random Matrix Theory for Lfunctions.
THREE LECTURES ON THE RIEMANN ZETAFUNCTION
, 2004
"... These lectures were delivered at the “International Conference on Subjects Related to the Clay Problems” held at Chonbuk National University, Chonju, Korea in July, 2002. My aim was to give mathematicians and graduate students unfamiliar with analytic number theory an introduction to the theory of t ..."
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These lectures were delivered at the “International Conference on Subjects Related to the Clay Problems” held at Chonbuk National University, Chonju, Korea in July, 2002. My aim was to give mathematicians and graduate students unfamiliar with analytic number theory an introduction to the theory of the Riemann zeta–function focusing, in particular, on the distribution of its zeros. Professor Y. Yildirin of the University
THE RIEMANN HYPOTHESIS
, 2007
"... The Riemann Hypothesis is a conjecture made in 1859 by the great mathematician Riemann that all the complex zeros of the zeta function ζ(s) lie on the ‘critical line’ Rls = 1/2. Our analysis shows that the assumption of the truth of the Riemann Hypothesis leads to a contradiction. We are therefore l ..."
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The Riemann Hypothesis is a conjecture made in 1859 by the great mathematician Riemann that all the complex zeros of the zeta function ζ(s) lie on the ‘critical line’ Rls = 1/2. Our analysis shows that the assumption of the truth of the Riemann Hypothesis leads to a contradiction. We are therefore led to the conclusion that the Riemann Hypothesis is not true. 1. The Zeta function of Riemann 1 is the analytic function obtained by the analytic continuation of the sumfunction ζ(s) of the infinite series n=1