An explicit formula for the square of the Riemann zeta-function in the critical strip

In this article, we prove an explicit formula for |ζ(σ + iT)|², where ζ(s) is the Riemann zeta-function and 1/2 < σ < 1, which is an analogue of Jutila’s formula. Our proof differs from that of Jutila. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 381–398, July–September, 2007.     

On the error term in the mean square formula for the Riemann zeta-function in the critical strip

For 1/2<<1 fixed, letE (T) denote the error term in the asymptotic formula for . We obtain some new bounds forE (T), and an _-result which is the analogue of the strongest _-result in the classical Dirichlet divisor problem.     

A critical case of Rallis inner product formula

Letπbe a genuine cuspidal representation of the metaplectic group of rank n.We consider the theta lifts to the orthogonal group associated to a quadratic space of dimension 2n+1.We show a case of regularised Rallis inner product formula that relates the pairing of theta lifts to the central value of the Langlands L-function ofπtwisted by a character.The bulk of this paper focuses on proving a case of regularised Siegel-Weil formula,on which the Rallis inner product formula is based and whose proof is missing in the literature.     

Robust convex optimization: A new perspective that unifies and extends

Mathematical Programming - Robust convex constraints are difficult to handle, since finding the worst-case scenario is equivalent to maximizing a convex function. In this paper, we propose a new...     

A semi-classical trace formula at a non-degenerate critical level

We study the semi-classical trace formula at a critical energy level for a h-pseudodifferential operator whose principal symbol has a unique non-degenerate critical point for that energy. This leads to the study of Hamiltonian systems near equilibrium and near the non-zero periods of the linearized flow. The contributions of these periods to the trace formula are expressed in terms of degenerate oscillatory integrals. The new results obtained are formulated in terms of the geometry of the energy surface and the classical dynamics on this surface.     

On a minimal complex norm that extends the real euclidean norm

A minimal extension of the real euclidean norm to ⁿ is introduced and some complex analytic properties of its unit ball are studied.Research supported partially by Fulbright Grant (1986–1987) at Universität Osnabrück-Abteilung Vechta.     

Zeros of Dedekind zeta functions in the critical strip

In this paper, we describe a computation which established the GRH to height (resp. ) for cubic number fields (resp. quartic number fields) with small discriminant. We use a method due to E. Friedman for computing values of Dedekind zeta functions, we take care of accumulated roundoff error to obtain results which are mathematically rigorous, and we generalize Turing's criterion to prove that there is no zero off the critical line. We finally give results concerning the GRH for cubic and quartic fields, tables of low zeros for number fields of degree and , and statistics about the smallest zero of a number field.

     

Mean value of Dirichlet L-functions in the critical strip

Vilnius University. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 31, No. 4, pp. 678–686, October–December, 1991.     

Analogues of the Gauss-Vinogradov formula on the critical line

An asymptotic behavior of the sum for X is studied in the critical strip, where L(s, X_p) is the Dirichlet series with the quadratic character X_p modulo p, where p is a prime number; v=1 or 3. With the help of large seive estimates a formula for this sum is obtained with two asymptotic terms on the critical line of the variable s. As a corollary the asymptotic expansion of this sum at the point s=1/2 is presented. The asymptotic formula for the sum, where d runs over discriminants of quadratic fields, is also obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 109, pp. 41–82, 1981.     

The mean square of the DirichletL-functions in the critical strip

We study an asymptotic formula of the DirichletL-functions in the critical strip. This is an analogy of the Atkinson-type formula for DirichletL-functions. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 201–213, April–June, 2000.     

A representation formula for the voltage perturbations caused by diametrically small conductivity inhomogeneities. Proof of uniform validity

We revisit the asymptotic formulas originally derived in [D.J. Cedio-Fengya, S. Moskow, M.S. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems 14 (1998) 553–595; A. Friedman, M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: A theorem on continuous dependence, Arch. Ration. Mech. Anal. 105 (1989) 299–326]. These formulas concern the perturbation in the voltage potential caused by the presence of diametrically small conductivity inhomogeneities. We significantly extend the validity of the previously derived formulas, by showing that they are asymptotically correct, uniformly with respect to the conductivity of the inhomogeneities. We also extend the earlier formulas by allowing the conductivities of the inhomogeneities to be completely arbitrary L^∞$L^{\infty }$, positive definite, symmetric matrix-valued functions. We briefly discuss the relevance of the uniform asymptotic validity, and the admission of arbitrary anisotropically conducting inhomogeneities, as far as applications of the perturbation formulas to “approximate cloaking” are concerned.     

Contributions of non-extremum critical points to the semi-classical trace formula

We study the semi-classical trace formula at a critical energy level for a h-pseudo-differential operator on whose principal symbol has a totally degenerate critical point for that energy. We compute the contribution to the trace formula of isolated non-extremum critical points under a condition of “real principal type”. The new contribution to the trace formula is valid for all time in a compact subset of but the result is modest since we have restrictions on the dimension.