The Stieltjes Function--Definition and Properties

Bohman, Jan; Fröberg, Carl-Erik

doi:10.2307/2008591

Cited by 10 publications

(18 citation statements)

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“…We get, in less than 7 s of time and with a precision of at least 40 digits, the results in Table 6; to be sure about the correctness of such results we computed them twice using the formulae (31)-(32) and then we compared the outcomes. These values are in agreement with the data on p. 282 of Bohman-Fröberg [3] for n = 0, . .…”

Section: The General Casesupporting

confidence: 92%

Efficient computation of the Euler–Kronecker constants of prime cyclotomic fields

Languasco

2020

Res. number theory

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We introduce a new algorithm, which is faster and requires less computing resources than the ones previously known, to compute the Euler–Kronecker constants $${\mathfrak {G}}_q$$ G q for the prime cyclotomic fields $$ {\mathbb {Q}}(\zeta _q)$$ Q ( ζ q ) , where q is an odd prime and $$\zeta _q$$ ζ q is a primitive q-root of unity. With such a new algorithm we evaluated $${\mathfrak {G}}_q$$ G q and $${\mathfrak {G}}_q^+$$ G q + , where $${\mathfrak {G}}_q^+$$ G q + is the Euler–Kronecker constant of the maximal real subfield of $${\mathbb {Q}}(\zeta _q)$$ Q ( ζ q ) , for some very large primes q thus obtaining two new negative values of $${\mathfrak {G}}_q$$ G q : $${\mathfrak {G}}_{9109334831}= -0.248739\dotsc $$ G 9109334831 = - 0.248739 ⋯ and $${\mathfrak {G}}_{9854964401}= -0.096465\dotsc $$ G 9854964401 = - 0.096465 ⋯ We also evaluated $${\mathfrak {G}}_q$$ G q and $${\mathfrak {G}}^+_q$$ G q + for every odd prime $$q\le 10^6$$ q ≤ 10 6 , thus enlarging the size of the previously known range for $${\mathfrak {G}}_q$$ G q and $${\mathfrak {G}}^+_q$$ G q + . Our method also reveals that the difference $${\mathfrak {G}}_q - {\mathfrak {G}}^+_q$$ G q - G q + can be computed in a much simpler way than both its summands, see Sect. 3.4. Moreover, as a by-product, we also computed $$M_q=\max _{\chi \ne \chi _0} \vert L^\prime /L(1,\chi ) \vert $$ M q = max χ ≠ χ 0 | L ′ / L ( 1 , χ ) | for every odd prime $$q\le 10^6$$ q ≤ 10 6 , where $$L(s,\chi )$$ L ( s , χ ) are the Dirichlet L-functions, $$\chi $$ χ run over the non trivial Dirichlet characters mod q and $$\chi _0$$ χ 0 is the trivial Dirichlet character mod q. As another by-product of our computations, we will provide more data on the generalised Euler constants in arithmetic progressions.

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Section: The General Casesupporting

confidence: 92%

Efficient computation of the Euler–Kronecker constants of prime cyclotomic fields

Languasco

2020

Res. number theory

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“…There are infinitely many γ n of each sign 27 In Letter 71 Stieltjes uses the notation C0, C1, ..., but in Letter 75 he uses the same notation with a different meaning. We substitute A0, A1, ... for the Letter 71 coefficients following [36]. 28 The result (3.2.2) is misstated in Erdëlyi et al [87, (1.12.17)], making the right side equal (−1) n n!…”

Section: 2mentioning

confidence: 99%

Euler’s constant: Euler’s work and modern developments

Lagarias

2013

Bull. Amer. Math. Soc.

128

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Abstract. This paper has two parts. The first part surveys Euler's work on the constant γ = 0.57721 · · · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments involving Euler's constant, as well as another constant, the Euler-Gompertz constant. These developments include connections with arithmetic functions and the Riemann hypothesis, and with sieve methods, random permutations, and random matrix products. It also includes recent results on Diophantine approximation and transcendence related to Euler's constant.

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“…Throughout this paper when several "±" or "∓" are encountered in the same formula, it signifies that either the upper signs are used everywhere or the lower signs are used everywhere (but not the mix of them). 8 Note that since many formulas with the kernels decaying exponentially fast were already obtained in the past by Legendre, Poisson, Binet, Malmsten, Jensen, Hermite, Lindelöf and many others (see e.g. a formula for the digamma function on p. 541 [5], or [4] or [22]), it is possible that formulas similar or equivalent to those we derive in this section might appear in earlier sources of which we are not aware.…”

Section: Integral Representationsmentioning

confidence: 99%

“…Liang and Todd proposed computing γ n either via the Euler-Maclaurin summation formula for ζ(s), or, as an alternative, via the application of Euler's series transformation to the alternating zeta function. Bohman and Fröberg [8] later refined the limit formula technique. Formula (3) is a differentiated form of Hermite's integral representation for ζ(s, v), which can be interpreted as the Abel-Plana summation formula applied to the series for ζ(s, v).…”

Section: Introductionmentioning

confidence: 99%

Computing Stieltjes constants using complex integration

Johansson

Blagouchine

2018

Math. Comp.

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The generalized Stieltjes constants γn(v) are, up to a simple scaling factor, the Laurent series coefficients of the Hurwitz zeta function ζ(s, v) about its unique pole s = 1. In this work, we devise an efficient algorithm to compute these constants to arbitrary precision with rigorous error bounds, for the first time achieving this with low complexity with respect to the order n. Our computations are based on an integral representation with a hyperbolic kernel that decays exponentially fast. The algorithm consists of locating an approximate steepest descent contour and then evaluating the integral numerically in ball arithmetic using the Petras algorithm with a Taylor expansion for bounds near the saddle point. An implementation is provided in the Arb library. We can, for example, compute γn(1) to 1000 digits in a minute for any n up to n = 10 100 . We also provide other interesting integral representations for γn(v), ζ(s), ζ(s, v), some polygamma functions and the Lerch transcendent.Date: XXX and, in revised form, XXX.

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The Stieltjes Function--Definition and Properties

Cited by 10 publications

References 0 publications

Efficient computation of the Euler–Kronecker constants of prime cyclotomic fields

Efficient computation of the Euler–Kronecker constants of prime cyclotomic fields

Euler’s constant: Euler’s work and modern developments

Computing Stieltjes constants using complex integration

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