Search Results (5)

In this paper, the authors derive some believed-to-be new recursion and explicit formulas for the generalized Ramanujan numbers ${\tau }_s\left(n\right)$$(s\in \mathbb{N}\cup \{-1\left\}\right)$, where, as usual, $\mathbb{N}$ is the set of positive integers. The authors consider the associated partition functions and derive connections of the Eisenstein series with the numbers ${\tau }_s\left(n\right)$. Several related corollaries and consequences of each of the presented results are also given. The paper concludes by presenting an open problem that is related to one of these results. Full article

Zeta-functions play a fundamental role in many fields where there is a norm or a means to measure distance. They are usually given in the forms of Dirichlet series (additive), and they sometimes possess the Euler product (multiplicative) when the domain in question has a unique factorization property. In applied disciplines, those zeta-functions which satisfy the functional equation but do not have Euler products often appear as a lattice zeta-function or an Epstein zeta-function. In this paper, we shall manifest the underlying principle that automorphy (which is a modular relation, an equivalent to the functional equation) is intrinsic to lattice (or Epstein) zeta-functions by considering some generalizations of the Eisenstein series of level $2\varkappa$ to the complex variable level s. Naturally, generalized Eisenstein series and Barnes multiple zeta-functions arise, which have affinity to dissections, as they are (semi-) lattice functions. The method of Lewittes (and Chapman) and Kurokawa leads to some limit formulas without absolute value due to Tsukada and others. On the other hand, Komori, Matsumoto and Tsumura make use of the Barnes multiple zeta-functions, proving their modular relation, and they give rise to generalizations of Ramanujan’s formula as the generating zeta-function of ${\sigma }_s\left(n\right)$, the sum-of-divisors function. Lewittes proves similar results for the 2-dimensional case, which holds for all values of s. This in turn implies the eta-transformation formula as the extreme case, and most of the results of Chapman. We shall unify most of these as a tapestry of ideas arising from the merging of additive entity (Dirichlet series) and multiplicative entity (Euler product), especially in the case of limit formulas. Full article

This paper presents an approach to summing a few families of infinite series involving hyperbolic functions, some of which were first studied by Ramanujan. The key idea is based on their contour integral representations and residue computations with the help of some well-known results of Eisenstein series given by Ramanujan, Berndt, et al. As our main results, several series involving hyperbolic functions are evaluated and expressed in terms of $z={}_{2}F_1(1/2,1/2;1;x)$ and $z^{\prime }=dz/dx$. When a certain parameter in these series is equal to $\pi$, the series are expressed in closed forms in terms of some special values of the Gamma function. Moreover, many new illustrative examples are presented. Full article

In this paper, the desymmetrized PSL(2, Z) group is studied. The Fourier coefficients of the non-holomorphic one-cusp Eisenstein series expansion are summed, and as a further result, a new dependence on the Euler’s $\gamma$ constant is found. The congruence subgroups of the desymmetrized $PSL(2,\mathbf{Z})$ are scrutinized, and the related structures are investigated. A ‘square-box’ one-cusp congruence subgroup is constructed. New leaky tori are constructed. Full article

In this study, we consider codes over Euclidean domains modulo their ideals. In the first half of the study, we deal with arbitrary Euclidean domains. We show that the product of generator matrices of codes over the rings mod a and mod b produces generator matrices of all codes over the ring mod $ab$ , i.e., this correspondence is onto. Moreover, we show that if a and b are coprime, then this correspondence is one-to-one, i.e., there exist unique codes over the rings mod a and mod b that produce any given code over the ring mod $ab$ through the product of their generator matrices. In the second half of the study, we focus on the typical Euclidean domains such as the rational integer ring, one-variable polynomial rings, rings of Gaussian and Eisenstein integers, p-adic integer rings and rings of one-variable formal power series. We define the reduced generator matrices of codes over Euclidean domains modulo their ideals and show their uniqueness. Finally, we apply our theory of reduced generator matrices to the Hecke rings of matrices over these Euclidean domains. Full article