The main aim of this paper is to study an analogue of the generalized divisor function in a number field K, namely, ?K,?(n). The Dirichlet series associated to this function is ?K(s)?K(s ??). We give an expression for the ...

Let $ r_k(n)$ denote the number of representations of the positive integer $ n$ as the sum of $ k$ squares. In 1934, the Russian mathematician A. I. Popov stated, but did not rigorously prove, a beautiful series transformation ...

Let rk(n) denote the number of representations of the positive integer n as the sum of k squares. In 1934, the Russian mathematician A. I. Popov stated, but did not rigorously prove, a beautiful series transformation ...

Let G=H×A be a finite group, where H is a purely non-abelian subgroup of G and A is a non-trivial abelian factor of G. Then, for n≥2, we show that there exists an isomorphism ϕ:Autγn(G)Z(G)(G)→Autγn(H)Z(H)(H) such that ...

In this article, we prove that the Faltings theta functions on moduli of parabolic bundles over a smooth projective curve can be used to give an explicit scheme-theoretic projective embedding.

Let G be a finite group and M,N be two normal subgroups of G. Let AutMN(G) denote the group of all automorphisms of G which fix N element wise and act trivially on G/M. Let n be a positive integer. In this article we have ...

To study the su ciency of an optimization problem, one either imposes some convexity assumptions or consider second order optimality conditions. In this paper we establish second order optimality conditions for nonsmooth ...

In this article, a q-series examined by Kluyver and Uchimura is generalized. This allows us to find generalization of the identities in the random acyclic digraph studied by Simon, Crippa, and Collenberg in 1993. As one ...