A note concerning the coefficients of the laurent series of the Riemann zeta function

Stankus, Eugenijus

doi:10.1007/bf02108243

Cited by 5 publications

(7 citation statements)

References 3 publications

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“…The expansion (1.1) was independently established by several authors, and the yk have been studied quite extensively. For a short historical survey, see [3, p. 164 f.]; more recent references not included in [3] are [4,11,17,18,19]. The coefficients Ak are usually called Stieltjes constants, and the numbers yk are sometimes referred to as generalized Euler constants.…”

Section: Introductionmentioning

confidence: 99%

Generalized Euler constants for arithmetical progressions

Dilcher¹

1992

Math. Comp.

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Section: Introductionmentioning

confidence: 99%

Generalized Euler constants for arithmetical progressions

Dilcher¹

1992

Math. Comp.

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“…Currently, apart from γ 0 , no closed-form expressions are known for γ n . However, there are works devoted to their estimations and to the asymptotic series representations for them [8,46,58,60,81,72,96,97,94,89,98,99]. Besides, there are also works devoted to the behavior of their sign [11,71].…”

Section: I1 Introductionmentioning

confidence: 99%

A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations

Blagouchine

2015

Journal of Number Theory

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Recently, it was conjectured that the first generalized Stieltjes constant at rational argument may be always expressed by means of Euler's constant, the first Stieltjes constant, the Γ-function at rational argument(s) and some relatively simple, perhaps even elementary, function. This conjecture was based on the evaluation of γ 1 (1/2), γ 1 (1/3), γ 1 (2/3), γ 1 (1/4), γ 1 (3/4), γ 1 (1/6), γ 1 (5/6), which could be expressed in this way. This article completes this previous study and provides an elegant theorem which allows to evaluate the first generalized Stieltjes constant at any rational argument. Several related summation formulae involving the first gener-alized Stieltjes constant and the Digamma function are also presented. In passing, an interesting integral representation for the logarithm of the Γ-function at rational argument is also obtained. Finally, it is shown that similar theorems may be derived for higher Stieltjes constants as well; in particular, for the second Stieltjes constant the theorem is provided in an explicit form.

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“…(5)] by the method of finite differences. Several more complicated series representations for γ m with irrational coefficients may be found in [176,113,84,168,128,186,38,39]. 33 …”

Section: Iii1 Introductionmentioning

confidence: 99%

“…33 Apart from formula (41), in [84], Israilov also obtained several other series representations for γ m . Stankus [168] showed that first two Stieltjes constants may be represented by the series involving the divisor function. Works of several authors showing that Stieltjes constants are related to series containing nontrivial zeros of the ζ-function are summarized in [186].…”

Section: Iii2 Derivation Of the Series Expansionmentioning

confidence: 99%

Corrigendum to “Expansions of generalized Euler's constants into the series of polynomials in π−2 and into the formal enveloping series with rational coefficients only” [J. Number Theory 158 (2016) 365–396]

Blagouchine

2017

Journal of Number Theory

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In this work, two new series expansions for generalized Euler's constants (Stieltjes constants) γ m are obtained. The first expansion involves Stirling numbers of the first kind, contains polynomials in π −2 with rational coefficients and converges slightly better than Euler's series ∑ n −2 . The second expansion is a semi-convergent series with rational coefficients only. This expansion is particularly simple and involves Bernoulli numbers with a non-linear combination of generalized harmonic numbers. It also permits to derive an interesting estimation for generalized Euler's constants, which is more accurate than several well-known estimations. Finally, in Appendix A, the reader will also find two simple integral definitions for the Stirling numbers of the first kind, as well an upper bound for them.

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A note concerning the coefficients of the laurent series of the Riemann zeta function

Cited by 5 publications

References 3 publications

Generalized Euler constants for arithmetical progressions

Generalized Euler constants for arithmetical progressions

A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations

Corrigendum to “Expansions of generalized Euler's constants into the series of polynomials in π−2 and into the formal enveloping series with rational coefficients only” [J. Number Theory 158 (2016) 365–396]

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