New results concerning power series expansions of the Riemann xi function and the Li/Keiper constants

Abstract

The Riemann hypothesis is equivalent to the Li criterion governing a sequence of real constants fl k g N kZ1 that are certain logarithmic derivatives of the Riemann xi function evaluated at unity. A new representation of l k is developed in terms of the Stieltjes constants g j and the subcomponent sums are discussed and analysed. Accompanying this decomposition, we find a new representation of the constants h j entering the Laurent expansion of the logarithmic derivative of the Riemann zeta function about sZ1. We also demonstrate that the h j coefficients are expressible in terms of the Bernoulli numbers and certain other constants. We determine new properties of h j and s j , where s j Z P r r Kj are the sums of reciprocal powers of the non-trivial zeros of the Riemann zeta function.

The theta-Laguerre calculus formulation of the Li/Keiper constants

“…What remains for a rigorous confirmation of the Riemann hypothesis is a verification that the subdominant contribution of the Li/Keiper constants is at most linear in n. In fact, we have conjectured that aside from explicitly linear terms in n , the remaining terms are O(n 1/2+ ) where > 0 is otherwise arbitrary [8][9][10]. Numerical computations to date are in agreement with this conjecture [27].…”

“…In our research program of the last few years, we have shown among other results that the leading order of the Li/Keiper constants n is O(n ln n), effectively verifying a conjecture of Keiper [7][8][9]. What remains for a rigorous confirmation of the Riemann hypothesis is a verification that the subdominant contribution of the Li/Keiper constants is at most linear in n. In fact, we have conjectured that aside from explicitly linear terms in n , the remaining terms are O(n 1/2+ ) where > 0 is otherwise arbitrary [8][9][10].…”

New results on the Stieltjes constants: Asymptotic and exact evaluation

“…Such knowledge would be very influential in deciding the nature of a sum denoted S 2 (n) [7][8][9] and thereby the resulting Li/Keiper constants λ k [19,22]. As a byproduct of this work we obtain interesting infinite series for fundamental constants such as the Euler constant and ln π .…”

“…The integrals appearing in Eqs. (9) can be written in several different ways via integration by parts, but we do not pursue this here.…”

On certain sums over the non-trivial zeta zeroes

“…For further discussion of the Li criterion, its application, and results on series expansion of the x function, see, for example, Bombieri & Lagarias (1999) and Coffey (2004Coffey ( , 2005Coffey ( , 2007aCoffey ( , 2008). …”