Related papers: The Collision Spectrum

Let $\pi$ be a Hecke-Maass cusp form for $SL(3,\mathbb{Z})$ and $\chi=\chi_1\chi_2$ a Dirichlet character with $\chi_i$ primitive modulo $M_i$. Suppose that $M_1$, $M_2$ are primes such that $\max\{(M|t|)^{1/3+2\delta/3},M^{2/5}|t|^{-9/20},…

A consistent treatment of $B\rightarrow \pi l \nu$ decay is given on the light-front. The $B$ to $\pi$ transition form factors are calculated in the entire physical range of momentum transfer for the first time. The valence-quark…

We prove that the spectrum of the Kirchhoff Laplacian H0 of a finite simple Barycentric refined graph and the spectrum of the connection Laplacian L of G determine each other: we prove that L-L^(-1) is similar to the Hodge Laplacian H of G…

The newform Dedekind sum $S_{\chi_1, \chi_2}$ associated to a pair of primitive Dirichlet characters $\chi_1$, $\chi_2$ of respective conductors $q_1$, $q_2$, is a group homomorphism from $\Gamma_1(q_1 q_2)$ into the number field…

We present the first analytic ${\cal O}(\alpha_s)$-computation at twist-$1$,$2$ of the $\bar B_{u,d,s} \to \gamma$ form factors within the framework of sum rules on the light-cone. These form factors describe the charged decay $\bar B_u \to…

Dirichlet's theorem guarantees infinitely many primes in each reduced residue class modulo q, but the analytic mechanism underlying this separation is often difficult to visualize directly. In this article we construct simplified…

We propose a method to factor numbers based on two interacting bosonic atoms in a central potential where the single-particle spectrum depends logarithmically on the radial quantum numbers of the zero angular momentum states. The bosons…

Let $\chi$ be a primitive Dirichlet character modulo $q$ and $L(s,\chi)$ be the Dirichlet L-function associated to $\chi$. Using a new two-piece mollifier we show that $L(\tfrac{1}{2},\chi)\ne0$ for at least 34% of the characters in the…

Let $L(s,\chi)$ be the Dirichlet $L$-function associated to a non trivial primitive Dirichlet character $\chi$ defined $\bmod\ q$, where $q$ is an odd prime. In this paper we introduce a fast method to compute $\vert L(1,\chi) \vert$ using…

We obtain (conditional and unconditional) results on large values of $L$-functions $L(s,\chi)$ in the critical strip $1/2 \leq \Re s \leq 1$ when the character $\chi$ runs through a thin subgroup of all characters modulo an integer $q$.…

We obtain an asymptotic formula for the first moment of quadratic Dirichlet $L$--functions over function fields at the central point $s=\tfrac{1}{2}$. Specifically, we compute the expected value of $L(\tfrac{1}{2},\chi)$ for an ensemble of…

When the fundamental invariant of $SLq(2)$ is expressed as $\epsilon_q = (\matrix{0 & \alpha_2 \cr -\alpha_1 & 0})$, then the deformation parameter, $q$, defining the knot algebra is $q = \frac{\alpha_1}{\alpha_2}$. We consider models in…

In this note we investigate the behavior at the central point of the symmetric square $L$-functions, the most frequently used $\rm{GL}(3)$ $L$-functions. We establish an asymptotic formula with arbitrary power saving for the first moment of…

Let G be any of the complex classical groups GL(n), SO(2n+1), Sp(2n), O(2n), let g denote the Lie algebra of G, and let Z(g) denote the subalgebra of G-invariants in the universal enveloping algebra U(g). We derive a Taylor-type expansion…

A rapid transformation is derived between spherical harmonic expansions and their analogues in a bivariate Fourier series. The change of basis is described in two steps: firstly, expansions in normalized associated Legendre functions of all…

An explicit formula for the mean value of $\vert L(1,\chi)\vert^2$ is known, where $\chi$ runs over all odd primitive Dirichlet characters of prime conductors $p$. Bounds on the relative class number of the cyclotomic field ${\mathbb…

Let $F$ be a number field, $\pi$ either a unitary cuspidal automorphic representation of $\mathrm{GL}(2)/F$ or a unitary Eisenstein series, and $\chi$ a unitary Hecke character of analytic conductor $C(\chi).$ We develop a regularized…

We define an SFT-type invariant for Legendrian knots in the standard contact $\mathbb{R}^3$. The invariant is a deformation of the Chekanov-Eliashberg differential graded algebra. The differential consists of a part that counts index zero…

Let $\chi_D$ be the Dirichlet character associated to $\mathbb{Q}(\sqrt{D})$ where $D < 0$ is a fundamental discriminant. Improving Granville-Stark [DOI:10.1007/s002229900036], we show that \[ \frac{L'}{L}(1,\chi_D) = \frac{1}{6}\,…

We use a nine-parameter expanding source model that includes special relativity, quantum statistics, resonance decays, and freeze-out on a realistic hypersurface in spacetime to analyze invariant pi+, K+, and K- one-particle multiplicity…