Related papers: An Elementary Proof of the Transformation Formula …

We derive hypergeometric formulas for Euler's constant, gamma. A "by-product" of Thomae's transformation is an infinite product for e^gamma involving the binomial coefficients. Alternate, non-hypergeometric proofs use a double integral for…

We show that the generating series of Euler characteristics of Hilbert schemes of points on any algebraic surface with at worst $A_n$-type singularities is described by the theta series determined by integer valued positive definite…

We demonstrate that the decay eta -> 3 pi represents a sensitive probe for the breaking of chiral symmetry by the quark masses. The transition amplitude is proportional to the mass ratio (m_d^2-m_u^2)/(m_s^2-m^2). The factor of…

Certain objects of conformal field theory, for example partition functions on the rectangle and the torus, and one-point functions on the torus, are either invariant or transform simply under the modular group, properties which should be…

This paper is an annotated list of transformation properties and identities satisfied by the four theta functions $\theta _1$, $\theta _2$, $\theta _3$, $\theta _4$ of one complex variable, presented in a ready-to-use form. An attempt is…

In this paper, we take into account the intrinsic charm and gluonic contents into the $\eta-\eta^\prime$ mixing scheme and formulate the tetramixing $\eta-\eta^\prime-G-\eta_c$ to study the mixing properties of $\eta^{(\prime)}$ mesons.…

We show that the eighth power of the Jacobi triple product is a Jacobi--Eisenstein series of weight $4$ and index $4$ and we calculate its Fourier coefficients. As applications we obtain explicit formulas for the eighth powers of…

We examine the moduli space E=T* of complex tori T(t)=C/L(t) where L(t)=cost.n(t)Lt. We find that the Dedekind eta function furnishes a bridge between the euclidean and hyperbolic structures on T*=C-L/L as well as between the doubly…

The integral representation of the Hadamard product of two functions is used to prove several Euler-type series transformation formulas. As applications we obtain three binomial identities involving harmonic numbers and an identity for the…

We study the transition form factors of the pesudoscalar mesons ($\pi,\eta$ and $\eta^{\prime}$) as functions of the momentum transfer $Q^2$ within the light-front quark model. We compare our results with the recent experimental data by…

In this work we show that the Riemann hypothesis for the Dedekind zeta--function $\zeta_{\mathrm{K}}(s)$ of an algebraic number field $\mathrm{K}$ is equivalent to a problem of the rate of convergence of certain discrete measures defined…

We establish new explicit zero-free regions for the Dedekind zeta-function. Two key elements of our proof are a non-negative, even, trigonometric polynomial and explicit upper bounds for the explicit formula of the so-called differenced…

We study very smooth functions on the real line, namely Schwartz functions, that satisfy a finite identity relating their translates and a single modulation. Concretely, we assume there is a nontrivial linear combination of translates of…

In this paper, we calculate the absolute tensor square of the Dirichlet $L$-functions and show that it is expressed as an Euler product over pairs of primes. The method is to construct an equation to link primes to a series which has the…

Sequence transformations are valuable numerical tools that have been used with considerable success for the acceleration of convergence and the summation of diverging series. However, our understanding of their theoretical properties is far…

By employing the assessment of the asymptotic size of various sums of G\'{a}l studied by La Bret\`eche and Tenenbaum, we provide an improvement on the recent result of A. Bondarenko, P. Darbar, M. V. Hagen, W. Heap, and K. Seip regarding…

We present a new proof of Euler's formulas for $\zeta(2k)$, where $k = 1,2,3,...$, which uses only the defining properties of the Bernoulli polynomials, obtaining the value of $\zeta(2k)$ by summing a telescoping series. Only basic…

The classical modular equations involve bivariate polynomials that can be seen to be univariate with coefficients in the modular invariant $j$. Kiepert found modular equations relating some $\eta$-quotients and the Weber functions…

We study the form factors of the $\eta_c$ meson in the light-front quark model. We explicitly show that the transition form factor of $\eta_c \to \gamma^* \gamma$ as a function of the momentum transfer is consistent with the experimental…

In this paper we give a generalization of Iseki's formula and use it to prove the transformation law of $\theta_1(z, \tau)$.