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Probability laws related to the Jacobi theta and Riemann zeta functions, and the Brownian excursions
 Bulletin (New series) of the American Mathematical Society
"... Abstract. This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to onedimensional ..."
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Cited by 57 (11 self)
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Abstract. This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to onedimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann’s zeta function which are related to these laws. Contents
A normal form for elliptic curves
 Bulletin of the American Mathematical Society
"... Abstract. The normal form x2 +y2 = a2 +a2x2y 2 for elliptic curves simplifies formulas in the theory of elliptic curves and functions. Its principal advantage is that it allows the addition law, the group law on the elliptic curve, to be stated explicitly ..."
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Cited by 55 (0 self)
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Abstract. The normal form x2 +y2 = a2 +a2x2y 2 for elliptic curves simplifies formulas in the theory of elliptic curves and functions. Its principal advantage is that it allows the addition law, the group law on the elliptic curve, to be stated explicitly
Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
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Quantum Spin Chains and Riemann Zeta Function with Odd Arguments
 J. Phys.A: Math. Gen
"... Riemann zeta function is an important object of number theory. We argue that it is related to Heisenberg spin 1/2 antiferromagnet. In the XXX spin chain we study the probability of formation of a ferromagnetic string in the antiferromagnetic ground state. We prove that for short strings the pro ..."
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Cited by 29 (9 self)
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Riemann zeta function is an important object of number theory. We argue that it is related to Heisenberg spin 1/2 antiferromagnet. In the XXX spin chain we study the probability of formation of a ferromagnetic string in the antiferromagnetic ground state. We prove that for short strings the probability can be expressed in terms of Riemann zeta function with odd arguments.
An elementary problem equivalent to the Riemann hypothesis
 Amer. Math. Monthly
"... ABSTRACT. The problem is: Let Hn = n∑ n ≥ 1, that with equality only for n = 1. j=1 1 j d ≤ Hn + exp(Hn)log(Hn), ..."
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Cited by 21 (2 self)
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ABSTRACT. The problem is: Let Hn = n∑ n ≥ 1, that with equality only for n = 1. j=1 1 j d ≤ Hn + exp(Hn)log(Hn),
Evaluation of Integrals Representing Correlations in XXX Heisenberg Spin Chain
 in Progress in Mathematics
, 2001
"... We study XXX Heisenberg spin 1/2 antiferromagnet. We evaluate a probability of formation of a ferromagnetic string in the antiferromagnetic ground state in thermodynamics limit. We prove that for short strings the probability can be expressed in terms of Riemann zeta function with odd arguments. ..."
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Cited by 17 (6 self)
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We study XXX Heisenberg spin 1/2 antiferromagnet. We evaluate a probability of formation of a ferromagnetic string in the antiferromagnetic ground state in thermodynamics limit. We prove that for short strings the probability can be expressed in terms of Riemann zeta function with odd arguments.
On the distribution of ranked heights of excursions of a Brownian bridge
 In preparation
, 1999
"... The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge (B br t ; 0 t 1) is described. The height M br+ j of the jth highest maximum over a positive excursion of the bridge has the same distribution as M br+ 1 =j, where th ..."
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Cited by 11 (6 self)
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The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge (B br t ; 0 t 1) is described. The height M br+ j of the jth highest maximum over a positive excursion of the bridge has the same distribution as M br+ 1 =j, where the distribution of M br+ 1 = sup 0t1 B br t is given by L'evy's formula P (M br+ 1 ? x) = e \Gamma2x 2 . The probability density of the height M br j of the jth highest maximum of excursions of the reflecting Brownian bridge (jB br t j; 0 t 1) is given by a modification of the known `function series for the density of M br 1 = sup 0t1 jB br t j. These results are obtained from a more general description of the distribution of ranked values of a homogeneous functional of excursions of the standardized bridge of a selfsimilar recurrent Markov process. Keywords: Brownian bridge, Brownian excursion, Brownian scaling, local time, selfsimilar recurrent Markov process, Bessel p...
Zeros of Dirichlet LFunctions near the Real Axis and Chebyshev's Bias
 JOURNAL OF NUMBER THEORY
, 2001
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New results on the Stieltjes constants: Asymptotic and exact evaluation
 J. Math. Anal. Appl
, 2006
"... The Stieltjes constants γk(a) are the expansion coefficients in the Laurent series for the Hurwitz zeta function about s = 1. We present new asymptotic, summatory, and other exact expressions for these and related constants. Key words and phrases Stieltjes constants, Riemann zeta function, Hurwitz z ..."
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Cited by 10 (6 self)
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The Stieltjes constants γk(a) are the expansion coefficients in the Laurent series for the Hurwitz zeta function about s = 1. We present new asymptotic, summatory, and other exact expressions for these and related constants. Key words and phrases Stieltjes constants, Riemann zeta function, Hurwitz zeta function, Laurent expansion, integrals of periodic Bernoulli polynomials, functional equation, Kreminski
Finding Meaning in Error Terms
, 2007
"... (In memory of Serge Lang) Four decades ago, Mikio Sato and John Tate predicted the shape of probability distributions to which certain “error terms ” in number theory conform. Their prediction—known as the SatoTate ..."
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Cited by 9 (1 self)
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(In memory of Serge Lang) Four decades ago, Mikio Sato and John Tate predicted the shape of probability distributions to which certain “error terms ” in number theory conform. Their prediction—known as the SatoTate