منابع مشابه
Series representations for the Stieltjes constants
The Stieltjes constants γk(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 series representations of these constants of interest to theoretical and computational analytic number theory. A particular result gives an addition formula for the Stieltjes constants. As a byproduct, expressions for derivatives of a...
متن کاملSeries of zeta values , the Stieltjes constants , and a sum
We present a variety of series representations of the Stieltjes and related constants, the Stieltjes constants being the coefficients of the Laurent expansion of the Hurwitz zeta function ζ(s, a) about s = 1. Additionally we obtain series and integral representations of a sum Sγ(n) formed as an alternating binomial series from the Stieltjes constants. The slowly varying sum Sγ(n) + n is an impo...
متن کاملAn effective asymptotic formula for the Stieltjes constants
The Stieltjes constants γk 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 γk 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 ma...
متن کاملComputing Stieltjes constants using complex integration
The Stieltjes constants γn are the coefficients appearing in the Laurent series of the Riemann zeta function at s = 1. We give a simple and efficient method to compute a p-bit approximation of γn with rigorous error bounds. Starting from an integral representation due to Blagouchine, we shift the contour to eliminate cancellation. The integral is then evaluated numerically in ball arithmetic us...
متن کاملNewton-Cotes integration for approximating Stieltjes (generalized Euler) constants
In the Laurent expansion ζ(s, a) = 1 s− 1 + ∞ ∑ k=0 (−1)γk(a) k! (s− 1) , 0 < a ≤ 1, of the Riemann-Hurwitz zeta function, the coefficients γk(a) are known as Stieltjes, or generalized Euler, constants. [When a = 1, ζ(s, 1) = ζ(s) (the Riemann zeta function), and γk(1) = γk.] We present a new approach to high-precision approximation of γk(a). Plots of our results reveal much structure in the gr...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2010
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2010.01.003