An explicit formula and estimations for Hecke L-functions: Applying the Li criterion

Abstract

We advance the use of a criterion for the validity of the generalized Riemann hypothesis. We rewrite and then estimate various terms of an explicit formula for Hecke L-functions very recently obtained by Li. We demonstrate how additional information on a coefficient function appearing in the logarithmic derivative of the L function would lead directly to verification of a generalized Riemann hypothesis. More generally, we mention a direct link between the generalized Ramanujan conjectures and the validity of generalized Riemann hypotheses. Mathematics Subject Classification: 11M26, 11M36

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

“…The corresponding l constants have been defined for Dirichlet & Hecke L-functions and other zeta functions (Li 2004), and the same leading behaviour O( j ln j ) has been found (Coffey 2005a(Coffey , 2007a. Our attention here is only with the classical zeta function.…”

The Li criterion and the Riemann hypothesis for the Selberg class II

“…Bombieri and Lagarias [1], Coffey [2,3] and Li [8] obtained an arithmetic expression for the Li coefficients λ n and gave an asymptotic formula as n → ∞. More recently, Maslanka [9] computed λ n for 1 n 3300 and empirically studied the growth behavior of the Li coefficients.…”

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). …”

“…For n > 1, l n is absolutely convergent, while for l 1 the sum should be taken over complex conjugate pairs of zeroes of increasing imaginary part. 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.…”