Narrow Results

Quantization of the Riemann zeta-function is proposed. We treat the Riemann zeta-function as a symbol of a pseudodifferential operator and study the corresponding classical and quantum field theories. This approach is motivated by the theory of p-adic strings and by recent works on stringy cosmological models. We show that the Lagrangian for the zeta-function field is equivalent to the sum of the Klein–Gordon Lagrangians with masses defined by the zeros of the Riemann zeta-function. Quantization of the mathematics of Fermat–Wiles and the Langlands program is indicated. The Beilinson conjectures on the values of L-functions of motives are interpreted as dealing with the cosmological constant problem. Possible cosmological applications of the zeta-function field theory are discussed.

Let . A result concerning analytic continuation of Z₁(s) to ℂ is proved, and also a result relating the order of to the order of .

Given a lattice in the plane, we consider zeta-functions encoding the number of well-rounded sublattices of a given index. We are particularly interested in the abscissa of convergence of this function and show that the quality of convergence is related to arithmeticity questions concerning the ambient lattice. In particular, we discover that there are infinitely many similarity classes of well-rounded sublattices in a plane lattice if there is at least one. This generalizes results about the rings of Gaussian and of Eisenstein integers by Fukshansky and his coauthors.

The zeta Mahler measure is the generating function of higher Mahler measures. In this article, explicit formulas of higher Mahler measures, and relations between higher Mahler measures and multiple zeta (star) values are showed by observing connections between zeta Mahler measures and the generating functions of multiple zeta (star) values. Additionally, connections between higher Mahler measures and Dirichlet L-values associated with primitive quadratic characters are discussed.

In this paper, we prove that the constant terms of powers of a Laurent polynomial satisfy certain congruences modulo prime powers. As a corollary, the generating series of these numbers considered as a function of a $p$-adic variable admits an analytic continuation in a non-trivial way, as it was shown previously by Dwork for a class of hypergeometric series.

The Casimir-Polder and van der Waals interaction energy of an atom with infinitely thin sphere with finite conductivity is investigated in the framework of the hydrodynamic approach. We put the sphere into spherical cavity inside the infinite dielectric media, then calculate the energy of vacuum fluctuations in the context of the zeta-function approach. The energy for a single atom is obtained by rarefying media. The Casimir-Polder expression for an atom and plate is recovered in the limit of the infinite radius of the sphere. Assuming a finite radius of the sphere, the interaction energy of an atom falls down monotonic as third power of distance between atom and sphere for short distance and as seventh power for large distance from the sphere.

An new invariant of a knot, called the zeta-function, is introduced. It is defined by counting the non-diagonalizable representations of the fundamental group into SL₂(𝔽_q), up to conjugacy, and forming a formal power series, similar to a zeta-function from algebraic geometry. The structure of this invariant in the general case is given. The special case of (2, n) torus knots is studied, and basic topological properties are noted.