Dr. Jay Mehta

To study point groups, their irreducible characters are essential. The table of irreducible characters of the icosahedral group A_5 is usually obtained by using its duality to the dodecahedral group. It seems that there is no literature which gives a routine computational way to complete it. In the works of Harter and Allen, a computational method is given and the character table up to the tetrahedral group A_4 using the group algebra table and linear algebra. In this paper, we employ their method with the aid of computer programming to complete the table. The method is applicable to any other more complicated groups.

Summary: In this article, we present a very short proof of infinitude of primes. This generalized proof is based on the idea of factorization of the product of primes and it unifies three proofs, the famous proof due to Euclid, a proof of Métrod, and the one due to Stieltjes.

In this article we obtain the meromorphic continuation of multiple zeta functions, together with a complete list of their poles and residues, by means of an elementary and simple translation formula for these multiple zeta functions. The use of matrices to express this translation formula leads, in particular, to a succinct description of the residues of the multiple zeta functions. We conclude our paper by introducing certain interesting weighted variants of multiple zeta functions. They are shown to behave particularly nicely with respect to product formulas and location of poles.

We recall the well-known notion of the set of uniqueness for arithmetical functions, introduced by Kátai and several other mathematicians like Indlekofer, Elliot and Hoffman, independently. We define its analogue for completely additive complex-valued functions over the set of non-zero Gaussian integers with some examples. We show that the set of «Gaussian prime plus one’s» along with finitely many Gaussian primes of norm up to some constant K is a set of uniqueness with respect to Gaussian integers. This is analogous to Kátai’s result in the case of positive integers [I. Kátai, On sets characterizing number theoretical functions, II, Acta Arith.16 (1968) 1-14].