arXiv subjects
Publications and source records attributed to Suraj Singh Khurana.
In this article, we show that the Riemann hypothesis for an $L$-function $F$ belonging to the Selberg class implies that all the derivatives of $F$ can have at most finitely many zeros on the left of the critical line with imaginary part greater than a certain constant. This was shown for the Riemann zeta function by Levinson and Montgomery in 1974.
The purpose of this article is twofold. First, we introduce the constants $ζ_k(α,r,q)$ where $α\in (0,1)$ and study them along the lines of work done on Euler constant in arithmetic progression $γ(r,q)$ by Briggs, Dilcher, Knopfmacher, Lehmer and some other authors. These constants are used for evaluation of certain integrals involving error term for Dirichlet divisor problem with congruence conditions and also to provide a closed form expression for the value of a class of Dirichlet L-series at any real critical point. In the second half of this paper, we consider the behaviour of the Laurent Stieltjes constants $γ_k(χ)$ for a principal character $χ.$ In particular, we study a generalization of the "Generalized Euler constants" introduced by Diamond and Ford in 2008. We conclude with a short proof for a closed form expression for the first generalized Stieltjes constant $γ_1(r/q)$ which was given by Blagouchine in 2015.