New results on the Stieltjes constants: Asymptotic and exact evaluation

2005

Math Phys Anal Geom

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We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence {λ k }, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we determine new bounds for relevant Riemann zeta function sums and the sequence itself. We find that the Riemann hypothesis holds if certain conjectured properties of a sequence η j are valid. The constants η j enter the Laurent expansion of the logarithmic derivative of the zeta function about s = 1 and appear to have remarkable characteristics. On our conjecture, not only does the Riemann hypothesis follow, but an inequality governing the values λ n and inequalities for the sums of reciprocal powers of the nontrivial zeros of the zeta function.

Section: We Havementioning

confidence: 84%

“…This conjecture is partially motivated by our discussion elsewhere [11] of the possible connection between the Stieltjes and Li constants and Brownian motion [5].…”

Section: We Havementioning

confidence: 95%

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Toward Verification of the Riemann Hypothesis: Application of the Li Criterion

2005

Math Phys Anal Geom

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“…γ k are called the Stieltjes constants [2,3,5,7,8,12,14,16], where γ 0 = γ, the Euler constant. These constants have many uses in analytic number theory and elsewhere.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Recent analytic results for the Stieltjes constants may be found in [3] and [4]. The latter includes an addition formula for the constants together with series representations.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

An effective asymptotic formula for the Stieltjes constants

Knessl

2010

Math. Comp.

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Abstract. The Stieltjes constants γ k appear in the coefficients in the regular part of the Laurent expansion of the Riemann zeta function ζ(s) about its only pole at s = 1. We present an asymptotic expression for γ k for k 1. This form encapsulates both the leading rate of growth and the oscillations with k. Furthermore, our result is effective for computation, consistently in close agreement (for both magnitude and sign) for even moderate values of k.Comparison to some earlier work is made.

Functional equations for the Stieltjes constants

2015

Ramanujan J

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The Stieltjes constants γ k (a) appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function ζ(s, a) about s = 1. We present the evaluation of γ 1 (a) and γ 2 (a) at rational arguments, this being of interest to theoretical and computational analytic number theory and elsewhere. We give multiplication formulas for γ 0 (a), γ 1 (a), and γ 2 (a), and point out that these formulas are cases of an addition formula previously presented. We present certain integral evaluations generalizing Gauss' formula for the digamma function at rational argument. In addition, we give the asymptotic form of γ k (a) as a → 0 as well as a novel technique for evaluating integrals with integrands with ln(− ln x) and rational factors.