Mathematika最新文献

A regularized Petersson inner product on the space of Jacobi forms is defined and the regularized Petersson norms of Jacobi–Eisenstein series are computed. We use this result to establish Gross–Kohnen–Zagier's formula for Eisenstein series. In addition, we give an answer to the question raised by Böcherer and Das asking whether the regularized norm of Jacobi–Eisenstein series defined by them is non-zero. In the Supporting Information, we compute the Fourier coefficients of a suitable “new” basis of the space of Jacobi–Eisenstein series and give a remark on the proportional constant of the inner product formula in the theory of Jacobi forms.

Conrey, Ghosh and Gonek studied the first moment of the derivative of the Riemann zeta function evaluated at the non-trivial zeros of the zeta function, resolving a problem known as Shanks' conjecture. Conrey and Ghosh studied the second moment of the Riemann zeta function evaluated at its local extrema along the critical line to leading order. In this paper, we combine the two results, evaluating the first moment of the zeta function and its derivatives at the local extrema of zeta along the critical line, giving a full asymptotic. We also consider the factor from the functional equation for the zeta function at these extrema.

It has recently been conjectured by Bogosel, Henrot, and Michetti that the second positive eigenvalue of the Neumann Laplacian is maximized, among all planar convex domains of fixed perimeter, by the rectangle with one edge length equal to twice the other. In this note, we prove that this conjecture is true within the class of parallelogram domains.

In this paper, we establish different sharp forms of Mahler's conjecture for $$-concave even functions in dimensions $$, for $$ and 2, for $$, thus generalizing our previous results in Fradelizi and Nakhle (Int. Math. Res. Not. 12 (2023), 10067–10097) on log-concave even functions in dimension 2, which corresponds to the case $$. The functional volume product of an even $$-concave function $$ is

Let $$ be the $$ matrix algebra over $$ and $$ be the invertible elements in $$. Inspired by Kaplansky–Lv́ov conjecture, we explore the image of polynomials with constants, namely polynomials from the free algebra $$. In this article, we compute the images of the polynomial maps given by (a) generalized sum of powers $$ and (b) generalized commutator map $$, where $$, $$ are nonzero elements of $$ when $$ is an algebraically closed field. We show that the images of these maps are vector spaces. For the polynomial in (a), we compute the images by fixing a simultaneous conjugate pair for $$, and it turns out that it is surjective in most cases.

The $$ Theorem for $$ asserts that, if $$ are finite, nonempty subsets with $$ and $$, then there exist arithmetic progressions $$ and $$ of common difference such that $$ and $$ for all $$. These are instances of Freiman's theorem with precise bounds. There is much partial progress extending this result to nonempty subsets $$ with $$ prime, $$ and $$. The ideal conjectured density restriction under which such a version of the $$ Theorem modulo $$ is expected is $$. Under this ideal density constraint, we show that there are arithmetic progressions $$, $$, and $$ of common difference with $$ and $$ for all $$, where $$, provided $$. This generalizes a result of Serra and Zémor [33] by extending their work from the special case $$ to that of general sumsets $$, removes all unnecessary sufficiently large $$ restrictions, and improves (even in the case $$) their constant 100-fold, from 0.0001 to 0.01. As part of the proof, we additionally obtain a yet better 1000-fold improvement of their constants at the cost of a near optimal density restriction of the form $$ (Theorem 3.5 and Corollary 3.7). These give rare high-density versions of the $$ Theorem for general sumsets $$ modulo $$ and are the first instances with tangible (rather than effectively existential) values for the constants for general sumsets $$ with high density, or indeed for any density without added constraints on the relative sizes of $$ and $$.

Kuperberg and Lalín stated some conjectures on the variance of certain sums of the divisor function $$ over number fields, which were inspired by analogous results over function fields proven by the authors. These problems are related to certain symplectic matrix integrals. While the function field results can be directly related to the random matrix integrals, the connection between the random matrix integrals and the number field results is less direct and involves arithmetic factors. The goal of this article is to give heuristic arguments for the formulas of these arithmetic factors.

Recently, we showed that global root numbers of modular forms are biased toward $$. Together with Pharis, we also showed an initial bias of Fourier coefficients toward the sign of the root number. First, we prove analogous results with respect to local root numbers. Second, a subtle correlation between Fourier coefficients and global root numbers, termed murmurations, was recently discovered for elliptic curves and modular forms. We conjecture murmurations in a more general context of different (possibly empty) combinations of local root numbers. Last, the Appendix corrects a sign error in our joint paper with Pharis.

We construct maximal $$-subsets on a large class of curved manifolds, in an optimal range of Lebesgue exponents $$. Our arguments combine restriction estimates and decoupling with old and new probabilistic estimates.

Let $$ denote the integer part of $$. In 1947, Mills constructed a real number $$ such that $$ is always a prime number for every positive integer $$. We define Mills' constant as the smallest real number $$ satisfying this property. Determining whether this number is irrational has been a long-standing problem. In this paper, we show that Mills' constant is irrational. Furthermore, we obtain partial results on the transcendency of this number.