Suraj Singh, Research Scholar

Area of Interest

My research area of interest is mainly analytic number theory. During my PhD I was interested in the study of certain generalizations of the Euler constant. Euler constant has been studied by many mathematicians since the 18th century. For a nice survey of results related to Euler's constant see here. Currently I am interested in special values of L-functions, L-functions in Selberg Class, application of Euler constant to Divisor problems and Euler-Kronecker constants.

List of publications

Suraj Singh Khurana; A closed-form expression for the Euler–Kronecker constant of a quadratic field, The Ramanujan Journal, Volume 63, 2024, pp. 507-526.
[Link] [Summary]
Given a number field, the Euler–Kronecker constant is defined as the constant term in the Laurent series expansion of the logarithmic derivative of the Dedekind zeta function at the point the point s=1. In the case of real and imaginary quadratic fields, a closed-form expression for the Euler–Kronecker constants can be obtained with the help of suitable Kronecker limit formulas. In this article, we avoid the use of Kronecker limit formulas and derive an explicit series representation of these constants with the help of an asymptotic series representation of the coefficients appearing in the Laurent series expansion of the Dedekind zeta function at s=1. As a result, the expressions obtained do not require evaluation of the special functions appearing in the Kronecker limit formulas.
Suraj Singh Khurana; On irrationality criteria for the Ramanujan summation of certain series, International Journal of Number Theory, Volume 19, Number 07, 2023, pp. 1571-1587.
[Link] [Summary]
In this paper, we prove a criterion for the irrationality of certain constants which arise from the Ramanujan summation of a family of infinite divergent sums. As an application, we provide a sufficient criterion for the irrationality of the values of the Riemann zeta function in the interval (0,1). We further see that our discussion leads to a natural generalization of a result of Sondow on the irrationality criterion for the Euler–Mascheroni constant.
Sneha Chaubey, Suraj Singh Khurana, Ade Irma Suriajaya; Zeros of derivatives of L-functions in Selberg Class on Re(s)<1/2, Proc. Amer. Math. Soc., Volume 151, Number 5, May 2023, pp. 1855-1866.
[Link] [Summary]
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 Riemannzeta function by Levinson and Montgomery in 1974.
Tapas Chatterjee, Suraj Singh Khurana; A series representation of Euler–Stieltjes constants and an identity of Ramanujan, Rocky Mountain J. Math. Volume 52, Issue 1, February 2022 pp. 49-64.
[Link] [Summary]
In this article, we derive a series representation of the generalized Stieltjes constants which arise in the Laurent series expansion of partial zeta function at the point s=1. In the process, we introduce a generalized gamma function and deduce its properties such as functional equation, Weierstrass product and reflection formulas along the lines of the study of a generalized gamma function introduced by Dilcher in 1994. These properties are used to obtain a series representation for the k-th derivative of Dirichlet series with periodic coefficients at the point s=1. Another application involves evaluation of a class of infinite products whose special case includes an identity of Ramanujan.
Tapas Chatterjee, Suraj Singh Khurana; Shifted Euler constants and a generalization of Euler-Stieltjes constants, J. Number Theory, Volume 204, November 2019, pp. 185-210.
[Link] [Summary]
The purpose of this article is twofold. First, we introduce the constants ζ_k(α,r,q) where α ∈ (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 γ₁(r/q) which was given by Blagouchine in 2015.
Tapas Chatterjee, Suraj Singh Khurana; Erdősian functions and an identity of Gauss, Proc. Japan Acad. Ser. A Math. Sci. 95 (2019), no. 6, pp. 58-63.
[Link] [Summary]
A famous identity of Gauss gives a closed form expression for the values of the digamma function ψ(x) at rational arguments x in terms of elementary functions. Linear combinations of such values are intimately connected with a conjecture of Erdős which asserts non vanishing of an infinite series associated to a certain class of periodic arithmetic functions. In this note we give a different proof for the identity of Gauss using an orthogonality like relation satisfied by these functions. As a by product we are able to give a new interpretation for nth Catalan number in terms of these functions.
Tapas Chatterjee, Suraj Singh Khurana; A note on generalizations of Stieltjes constants, J. Ramanujan Math. Soc. Volume 34, Issue 4, December 2019 pp. 457–468.
[Link] [Summary]
In this article we consider a generalization of Stieltjes constants and study its relation with special values of certain Dirichlet series. Further we show a connection of these constants with a generalization of Digamma function. Some of the results obtained are a natural generalization of the identities of Gauss, Lehmer, Dilcher and many other authors.