Related papers: Quadratic relations between Bessel moments

Recently Conrey, Farmer, and Zirnbauer developed the L-functions Ratios conjecture, which gives a recipe that predicts a wealth of statistics, from moments to spacings between adjacent zeros and values of L-functions. The problem with this…

In this paper, the integral $\pmatrix{\lambda_1 &\lambda_2 &\lambda_3\cr 0 &0 &0\cr}\, \int_0^\infty \, r^{\lambda_3+2}\, \exp{(-\alpha r^2)}\, j_{\lambda_1}(k_1r) \,j_{\lambda_2}(k_2r) \,dr$, where $k_1$, $k_2$ and $\alpha$ are positive,…

This paper will focus on the proof of local integrability of Bessel functions for GL(n) (p-adic case) by using the relations between Bessel functions and local Kloosterman (orbital) integrals proved in several papers of E. M. Baruch [Ba03]…

We investigate the moments of a smooth counting function of the zeros near the central point of L-functions of weight k cuspidal newforms of prime level N. We split by the sign of the functional equations and show that for test functions…

Multivariate Bessel processes $(X_{t,k})_{t\ge0}$ describe interacting particle systems of Calogero-Moser-Sutherland type and are related with $\beta$-Hermite and $\beta$-Laguerre ensembles. They depend on a root system and a multiplicity…

By elaborating on the recent progress made in the area of Feynman integrals, we apply the intersection theory for twisted de Rham cohomologies to simple integrals involving orthogonal polynomials, matrix elements of operators in Quantum…

This is a survey of recent work on values of Rankin-Selberg $L$-functions of pairs of cohomological automorphic representations that are {\it critical} in Deligne's sense. The base field is assumed to be a CM field. Deligne's conjecture is…

We derive explicit expressions for a family of radially symmetric, non-differentiable, Spartan covariance functions in $\mathbb{R}^2$ that involve the modified Bessel function of the second kind. In addition to the characteristic length and…

We obtain the $n$th centered moments of one level densities of a large orthogonal family of $L$-functions associated with holomorphic Hecke newforms of level $q$, averaged over $q\sim Q$. We verify the Katz-Sarnak conjecture for these…

Bloch theorem for a periodic operator is being revisited here, and we notice extra orthogonality relationships. It is shown that solutions are bi-periodic, in the sense that eigenfunctions are periodic with respect to one argument, and…

Integral transforms arising from the separable solutions to the Helmholtz differential equation are presented. Pairs of these integral transforms are related via Plancherel theorem and, ultimately, any of these integral transforms may be…

We present a straightforward discretization of the Bessel functions $J_n(x)$ to discrete counterparts $B^{(N)}_n(x_m)$, of $N$ integer orders $n$ on $N$ integer points $x_m \equiv m$, that we call discrete Bessel functions. These are built…

For vector-valued Maass cusp forms for~$SL_2(\mathbb{Z})$ with real weight~$k\in\mathbb{R}$ and spectral parameter $s\in\mathbb{C}$, $\mathrm{Re} s\in (0,1)$, $s\not\equiv \pm k/2$ mod $1$, we propose a notion of vector-valued period…

Quasisymmetry (QS) is a hidden symmetry of the magnetic field strength, B, that effectively confines charged particles in a three-dimensional toroidal plasma equilibrium. Here, we show that QS has a deep connection to the underlying…

The QCD-coupling is a necessary input in the computation of many observables, and the parametric error on input parameters can be a dominant source of uncertainty. The coupling can be extracted by comparing high order perturbative…

This paper studies the first moment of symmetric-square $L$-functions at the critical point in the weight aspect. Asymptotics with the best known error term $O(k^{-1/2})$ were obtained independently by Fomenko in 2005 and by Sun in 2013. We…

We obtain for the Kempner series (i.e. harmonic series where certain digits are excluded from all denominators, for example the digit 9 in base 10) new representations as geometrically convergent series. The coefficients for these…

We establish improved bounds for bilinear forms with Kloosterman fractions of the form ${\sum\sum}_{m,n} \alpha_m \beta_n e(a\overline{m}/(bn))$ with $M<m\le 2M$, $N < n \le 2N$ and $(m,n)=1$. Our approach works directly with arbitrary…

It is an open question whether the fractional parts of nonlinear polynomials at integers have the same fine-scale statistics as a Poisson point process. Most results towards an affirmative answer have so far been restricted to almost sure…

Commensurate scale relations are perturbative QCD predictions which relate observable to observable at fixed relative scale, such as the "generalized Crewther relation", which connects the Bjorken and Gross-Llewellyn Smith deep inelastic…