A sequence associated with the zeros of the Rimann zeta function

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

“…For instance, in principle, only improved estimation of the Stieltjes constants prevents the verification of the Riemann hypothesis by way of either the Li criterion (Li 1997) or criterion (c) of Bombieri & Lagarias (1999). Based upon the work of Matsuoka (1985aMatsuoka ( , 1986, we have been able to, yet again, derive an arithmetic formula for the Li/Keiper constants, without recourse to the Guinand-Weil explicit formula. where g is the Euler constant, and for p2C …”

“…This bound is not of direct use to us because it gives an exponentially large upper bound on l n . With the use of equation (3.15) and arguing as in Matsuoka (1985aMatsuoka ( , 1986, we are able to slightly improve the upper bound on js n j and jh nK1 j to O½ð1C 1=pÞ n =n. It may be of interest to interchange the two finite sums over j and h in equation (3.11),…”

“…In an earlier work (Coffey 2004), we showed low-order examples of how S 2 , and thus l k , could be written in terms of the Stieltjes constants g j . Here, we show how this can be generally done, based upon the work of Matsuoka (1985aMatsuoka ( , 1986. We then make some observations on the contributions of g j to these quantities.…”

“…The constants h j enter the expansion around sZ1 of the logarithmic derivative of the zeta function Matsuoka (1985aMatsuoka ( , 1986. We then make some observations on the contributions of g j to these quantities.…”

“…In this paper, we develop a new representation of the Li (Keiper 1992) constants l k (Li 1997). It is an explicit expression in terms of the Stieltjes constants g k and is based upon the work of Matsuoka (1985aMatsuoka ( , 1986. Correspondingly, 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.…”