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Abstract: Conrey, Farmer, Keating, Rubinstein, and Snaith, recently conjectured formulas for the full asymptotics of the moments of L-functions. In the case of the Riemann zeta function, their conjecture states that the 2k-th absolute moment of zeta on the critical line is asymptotically given by a certain 2k-fold residue integral. This residue integral can be expressed as a polynomial of degree , whose coefficients are given in exact form by elaborate and complicated formulas. In this article, uniform asymptotics for roughly the first k coefficients of the moment polynomial are derived. Numerical data to support our asymptotic formula are presented. An application to bounding the maximal size of the zeta function is considered. [Copyright &y& Elsevier]

Stephen D. Miller showed that, assuming the Generalized Riemann Hypothesis, every entire L -function of real archimedean type has a zero in the interval 1 2 + i t with − t 0 < t < t 0 , where t 0 ≈ 14.13 corresponds to the first zero of the Riemann zeta function. We give a numerical example of a self-dual degree-4 L -function whose first positive imaginary zero is at t 1 ≈ 14.496 . In particular, Miller's result does not hold for general L -functions. We show that all L -functions satisfying some additional (conjecturally true) conditions have a zero in the interval ( − t 2 , t 2 ) with t 2 ≈ 22.661 . [ABSTRACT FROM AUTHOR]

Abstract: We derive formulas for the terms in the conjectured asymptotic expansions of the moments, at the central point, of quadratic Dirichlet L-functions, , and also of the L-functions associated to quadratic twists of an elliptic curve over . In so doing, we are led to study determinants of binomial coefficients of the form . [Copyright &y& Elsevier]

Abstract: We report a fully general technique addressing a long standing challenge of calculating conformational free energy differences between various states of a polymer chain from simulations using explicit solvent force fields. The main feature of our method is a special mapping variable, a path coordinate, which continuously connects two conformations. The path variable has been designed to preserve locality in the phase space near the path endpoints. We avoid the problem of sampling the unfolded states by creating an artificial confinement “tube” in the phase space that prevents the molecule from unfolding without affecting the calculation of the desired free energy difference. We applied our technique to compute the free energy difference between two native-like conformations of the small protein Trp-cage using the CHARMM force field with explicit solvent. We verified this result by comparing it with an independent, significantly more expensive calculation. Overall, the present study suggests that the new method of computing free energy differences between polymer chain conformations is accurate and highly computationally efficient. [Copyright &y& Elsevier]