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

Coffey, Mark W.

doi:10.1016/j.jat.2006.10.006

Cited by 10 publications

(9 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Besides the indications given in Ref. [11] that the Laguerre calculus is pervasive within the Li/Keiper formulation of the Riemann hypothesis, we have very recently systematically presented the structural origins of this framework [12]. The Li/Keiper constants arise as a sum over complex zeta zeros of a Laplace transform of the associated Laguerre polynomial L 1 n−1 (x).…”

Section: Discussionmentioning

confidence: 95%

The Stieltjes constants, their relation to the ηj coefficients, and representation of the Hurwitz zeta function

Coffey¹

2010

Analysis

Self Cite

View full text Add to dashboard Cite

The Stieltjes constants γ k (a) are the expansion coefficients in the Laurent series for the Hurwitz zeta function about its only pole at s = 1. We present the relation of γ k (1) to the η j coefficients that appear in the Laurent expansion of the logarithmic derivative of the Riemann zeta function about its pole at s = 1. We obtain novel integral representations of the Stieltjes constants and new decompositions such as S 2 (n) = S γ (n) + S Λ (n) for the crucial oscillatory subsum of the Li criterion for the Riemann hypothesis. The sum S γ (n) is O(n) and we present various integral representations for it. We present novel series representations of S 2 (n). We additionally present a rapidly convergent expression for γ k = γ k (1) and a variety of results pertinent to a parameterized representation of the Riemann and Hurwitz zeta functions.

show abstract

Section: Discussionmentioning

confidence: 95%

The Stieltjes constants, their relation to the ηj coefficients, and representation of the Hurwitz zeta function

Coffey¹

2010

Analysis

Self Cite

View full text Add to dashboard Cite

show abstract

“…The Laguerre polynomials are pervasive in formulating the Li criterion [7]. They provide certain test functions for a Weil inner product whose nonnegativity is equivalent to the Li criterion.…”

Section: Introduction Let ζ(S) Be the Riemann Zeta Function And The mentioning

confidence: 99%

An asymptotic form for the Stieltjes constants 𝛾_{𝑘}(𝑎) and for a sum 𝑆ᵧ(𝑛) appearing under the Li criterion

Knessl

Coffey

2011

Math. Comp.

View full text Add to dashboard Cite

Abstract. We present several asymptotic analyses for quantities associated with the Riemann and Hurwitz zeta functions. We first determine the leading asymptotic behavior of the Stieltjes constants γ k (a). These constants appear in the regular part of the Laurent expansion of the Hurwitz zeta function. We then use asymptotic results for the Laguerre polynomials L α n to investigate a certain sum S γ (n) involving the constants γ k (1) that appears in application of the Li criterion for the Riemann hypothesis. We confirm the sublinear growth of S γ (n) + n, which is consistent with the validity of the Riemann hypothesis.

show abstract

“…We have recently shown how the framework of the Li/Keiper constants may be written in terms of the Laguerre calculus [10].…”

Section: Summary and Brief Discussionmentioning

confidence: 99%

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

Coffey¹

2007

ijcms

View full text Add to dashboard Cite

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

show abstract

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

Cited by 10 publications

References 25 publications

The Stieltjes constants, their relation to the ηj coefficients, and representation of the Hurwitz zeta function

The Stieltjes constants, their relation to the ηj coefficients, and representation of the Hurwitz zeta function

An asymptotic form for the Stieltjes constants 𝛾_{𝑘}(𝑎) and for a sum 𝑆ᵧ(𝑛) appearing under the Li criterion

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

Contact Info

Product

Resources

About