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 γ 1 (r/q) which was given by Blagouchine in 2015.

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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 [Formula: see text]. 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.

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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.