This article deals with the value distribution of multiplicative prime-independent arithmetic functions (α(n)) with α(n) = 1 if n is N-free (N ≥ 2 a fixed integer), α(n) > 1 else, and α(2) → ∞. An asymptotic result is established with an error term probably definitive on the basis of the present knowledge about the zeros of the zetafunction. Applications to the enumerative functions of Abelian ...

The classification of complex wave functions of infinite variables is an important problem since it is related to the classification of possible quantum states of matter. In this paper, we propose a way to classify symmetric polynomials of infinite variables using the pattern of zeros of the polynomials. Such a classification leads to a construction of a class of simple non-Abelian quantum Hall...

We study topology change in (2+1)D gravity coupling with non-Abelian SO(2, 1) Higgs field from the point of view of Morse theory. It is shown that the Higgs potential can be identified as a Morse function. The critical points of the latter (i.e. loci of change of the spacetime topology) coincide with zeros of the Higgs field. In these critical points twodimensional metric becomes degenerate, bu...

in which the functions r, p and q are continuous on (0, °o) while r is positive there. The point x = 0 is singular for the functional in the sense that the conditions on r, p, and q may not hold for an interval of the form [O, b]. Finally, all integrals which appear are Lebesgue integrals. We denote by: F[0, b], the class of all functions y such that y is absolutely continuous and y'ÇzL2 on eve...

The cylindrical Bessel differential equation and the spherical Bessel differential equation in the interval [Formula: see text] with Neumann boundary conditions are considered. The eigenfunctions are linear combinations of the Bessel function [Formula: see text] or linear combinations of the spherical Bessel functions [Formula: see text]. The orthogonality relations with analytical expressions ...

Criterion for the Riemann hypothesis found by Báez-Duarte involves certain real coefficients ck defined as alternating binomial sums. These coefficients can be effectively investigated using Nörlund-Rice’s integrals. Their behavior exhibits characteristic trend, due to trivial zeros of zeta, and fading oscillations, due to complex zeros. This method enables to calculate numerical values of ck f...

The quantum gravity path integral involves a sum over topologies that invites comparisons to worldsheet string theory and Feynman diagrams of field theory. However, the latter are naturally associated with non-abelian algebra fields, while former has been argued define an abelian superselected observables partition-function-like quantities at asymptotic boundary. We resolve this apparent tensio...

This paper generalizes Bismut’s equivariant Chern character to the setting of abelian gerbes. In particular, associated to an abelian gerbe with connection, an equivariantly closed differential form is constructed on the space of maps of a torus into the manifold. These constructions are made explicit using a new local version of the higher Hochschild complex, resulting in differential forms gi...

In the present paper we consider Fk(x) = x k − ∑ k−1 t=0 x, the characteristic polynomial of the k-th order Fibonacci sequence, the latter denoted G(k, l). We determine the limits of the real roots of certain odd and even degree polynomials related to the derivatives and integrals of Fk(x), that form infinite sequences of polynomials, of increasing degree. In particular, as k → ∞, the limiting ...

This paper studies the non-holomorphic Eisenstein series E(z, s) for the modular surface PSL(2,Z)\H, and shows that integration with respect to certain non-negative measures μ(z) gives meromorphic functions Fμ(s) that have all their zeros on the line R(s) = 1 2 . For the constant term a0(y, s) of the Eisenstein series the Riemann hypothesis holds for all values y ≥ 1, with at most two exception...