Various representations in terms of integrals and infinite series are given in works of Jensen, Franel, Hermite, Hardy, Ramanujan, Ainsworth, Howell, Coppo, Connon, Coffey, Choi, Blagouchine and some other authors.[1][2][3][4][5][6] In particular, Jensen-Franel's integral formula, often erroneously attributed to Ainsworth and Howell, states that
where δn,k is the Kronecker symbol (Kronecker delta).[5][6] Among other formulae, we find
see.[1][5][7]
As concerns series representations, a famous series implying an integer part of a logarithm was given by Hardy in 1912[8]
Israilov[9] gave semi-convergent series in terms of Bernoulli numbers
Connon,[10] Blagouchine[6][11] and Coppo[1] gave several series with the binomial coefficients
where Gn are Gregory's coefficients, also known as reciprocal logarithmic numbers (G1=+1/2, G2=−1/12, G3=+1/24, G4=−19/720,... ). More general series of the same nature include these examples[11]
as well as semi-convergent series with rational terms only
where m=0,1,2,... In particular, series for the first Stieltjes constant has a surprisingly simple form
where Hn is the nth harmonic number.[6] More complicated series for Stieltjes constants are given in works of Lehmer, Liang, Todd, Lavrik, Israilov, Stankus, Keiper, Nan-You, Williams, Coffey.[2][3][6]
Bounds and asymptotic growth
The Stieltjes constants satisfy the bound
given by Berndt in 1972.[14] Better bounds in terms of elementary functions were obtained by Lavrik[15]
by Israilov[9]
with k=1,2,... and C(1)=1/2, C(2)=7/12,... , by Nan-You and Williams[16]
by Blagouchine[6]
where Bn are Bernoulli numbers, and by Matsuoka[17][18]
As concerns estimations resorting to non-elementary functions and solutions, Knessl, Coffey[19] and Fekih-Ahmed[20] obtained quite accurate results. For example, Knessl and Coffey give the following formula that approximates the Stieltjes constants relatively well for large n.[19] If v is the unique solution of
with , and if , then
where
Up to n = 100000, the Knessl-Coffey approximation correctly predicts the sign of γn with the single exception of n = 137.[19]
In 2022 K. Maślanka[21] gave an asymptotic expression for the Stieltjes constants, which is both simpler and more accurate than those previously known. In particular, it reproduces with a relatively small error the troublesome value for n = 137.
Namely, when
where are the saddle points:
is the Lambert function and is a constant:
Defining a complex "phase"
we get a particularly simple expression in which both the rapidly increasing amplitude and the oscillations are clearly seen:
For large n, the Stieltjes constants grow rapidly in absolute value, and change signs in a complex pattern.
Further information related to the numerical evaluation of Stieltjes constants may be found in works of Keiper,[23] Kreminski,[24] Plouffe,[25] Johansson[26][27] and Blagouchine.[27] First, Johansson provided values of the Stieltjes constants up to n = 100000, accurate to over 10000 digits each (the numerical values can be retrieved from the LMFDB [1]. Later, Johansson and Blagouchine devised a particularly efficient algorithm for computing generalized Stieltjes constants (see below) for large n and complex a, which can be also used for ordinary Stieltjes constants.[27] In particular, it allows one to compute γn to 1000 digits in a minute for any n up to n=10100.
Here a is a complex number with Re(a)>0. Since the Hurwitz zeta function is a generalization of the Riemann zeta function, we have γn(1)=γn The zeroth constant is simply the digamma-function γ0(a)=-Ψ(a),[28] while other constants are not known to be reducible to any elementary or classical function of analysis. Nevertheless, there are numerous representations for them. For example, there exists the following asymptotic representation
due to Berndt and Wilton. The analog of Jensen-Franel's formula for the generalized Stieltjes constant is the Hermite formula[5]
Similar representations are given by the following formulas:[27]
and
Generalized Stieltjes constants satisfy the following recurrence relation
The first generalized Stieltjes constant has a number of remarkable properties.
Malmsten's identity (reflection formula for the first generalized Stieltjes constants): the reflection formula for the first generalized Stieltjes constant has the following form
where m and n are positive integers such that m<n. This formula has been long-time attributed to Almkvist and Meurman who derived it in 1990s.[31] However, it was recently reported that this identity, albeit in a slightly different form, was first obtained by Carl Malmsten in 1846.[5][32]
Rational arguments theorem: the first generalized Stieltjes constant at rational argument may be evaluated in a quasi-closed form via the following formula
see Blagouchine.[5][28] An alternative proof was later proposed by Coffey[33] and several other authors.
Finite summations: there are numerous summation formulae for the first generalized Stieltjes constants. For example,
For more details and further summation formulae, see.[5][30]
Some particular values: some particular values of the first generalized Stieltjes constant at rational arguments may be reduced to the gamma-function, the first Stieltjes constant and elementary functions. For instance,
At points 1/4, 3/4 and 1/3, values of first generalized Stieltjes constants were independently obtained by Connon[34] and Blagouchine[30]
At points 2/3, 1/6 and 5/6
These values were calculated by Blagouchine.[30] To the same author are also due
Second generalized Stieltjes constant
The second generalized Stieltjes constant is much less studied than the first constant. Similarly to the first generalized Stieltjes constant, the second generalized Stieltjes constant at rational argument may be evaluated via the following formula
see Blagouchine.[5] An equivalent result was later obtained by Coffey by another method.[33]
References
^ abcCoppo, Marc-Antoine (1999). "Nouvelles expressions des constantes de Stieltjes". Expositiones Mathematicae. 17: 349–358.
^ abCoffey, Mark W. (2009). "Series representations for the Stieltjes constants". arXiv:0905.1111 [math-ph].
^ abCoffey, Mark W. (2010). "Addison-type series representation for the Stieltjes constants". J. Number Theory. 130 (9): 2049–2064. doi:10.1016/j.jnt.2010.01.003.
^Choi, Junesang (2013). "Certain integral representations of Stieltjes constants". Journal of Inequalities and Applications. 532: 1–10.
^ abcdefghBlagouchine, Iaroslav V. (2015). "A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations". Journal of Number Theory. 148: 537–592. arXiv:1401.3724. doi:10.1016/j.jnt.2014.08.009. And vol. 151, pp. 276-277, 2015. arXiv:1401.3724
^ abcdefgBlagouchine, Iaroslav V. (2016). "Expansions of generalized Euler's constants into the series of polynomials in π−2 and into the formal enveloping series with rational coefficients only". Journal of Number Theory. 158: 365–396. arXiv:1501.00740. doi:10.1016/j.jnt.2015.06.012. Corrigendum: vol. 173, pp. 631-632, 2017.
^"A couple of definite integrals related to Stieltjes constants". Stack Exchange.
^Hardy, G. H. (2012). "Note on Dr. Vacca's series for γ". Q. J. Pure Appl. Math. 43: 215–216.
^ abIsrailov, M. I. (1981). "On the Laurent decomposition of Riemann's zeta function [in Russian]". Trudy Mat. Inst. Akad. Nauk. SSSR. 158: 98–103.
^Donal F. Connon Some applications of the Stieltjes constants, arXiv:0901.2083
^ abBlagouchine, Iaroslav V. (2018). "Three notes on Ser's and Hasse's representations for the zeta-functions" (PDF). INTEGERS: The Electronic Journal of Combinatorial Number Theory. 18A (#A3): 1–45. arXiv:1606.02044.
^Actually Blagouchine gives more general formulas, which are valid for the generalized Stieltjes constants as well.
^Bruce C. Berndt. On the Hurwitz Zeta-function. Rocky Mountain Journal of Mathematics, vol. 2, no. 1, pp. 151-157, 1972.
^A. F. Lavrik. On the main term of the divisor's problem and the power series of the Riemann's zeta function in a neighbourhood of its pole (in Russian). Trudy Mat. Inst. Akad. Nauk. SSSR, vol. 142, pp. 165-173, 1976.
^Z. Nan-You and K. S. Williams. Some results on the generalized Stieltjes constants. Analysis, vol. 14, pp. 147-162, 1994.
^Y. Matsuoka. Generalized Euler constants associated with the Riemann zeta function. Number Theory and Combinatorics: Japan 1984, World Scientific, Singapore, pp. 279-295, 1985
^Y. Matsuoka. On the power series coefficients of the Riemann zeta function. Tokyo Journal of Mathematics, vol. 12, no. 1, pp. 49-58, 1989.
^ abcCharles Knessl and Mark W. Coffey. An effective asymptotic formula for the Stieltjes constants. Math. Comp., vol. 80, no. 273, pp. 379-386, 2011.
^Lazhar Fekih-Ahmed. A New Effective Asymptotic Formula for the Stieltjes Constants, arXiv:1407.5567
^Krzysztof Maślanka. Asymptotic Properties of Stieltjes Constants. Computational Methods in Science and Technology, vol. 28 (2022), p.123-131; https://arxiv.org/abs/2210.07244v1
^Choudhury, B. K. (1995). "The Riemann zeta-function and its derivatives". Proc. R. Soc. A. 450 (1940): 477–499. Bibcode:1995RSPSA.450..477C. doi:10.1098/rspa.1995.0096. S2CID 124034712.
^Keiper, J.B. (1992). "Power series expansions of Riemann ζ-function". Math. Comp. 58 (198): 765–773. Bibcode:1992MaCom..58..765K. doi:10.1090/S0025-5718-1992-1122072-5.
^Kreminski, Rick (2003). "Newton-Cotes integration for approximating Stieltjes generalized Euler constants". Math. Comp. 72 (243): 1379–1397. Bibcode:2003MaCom..72.1379K. doi:10.1090/S0025-5718-02-01483-7.
^Simon Plouffe. Stieltjes Constants, from 0 to 78, 256 digits each
^Johansson, Fredrik (2015). "Rigorous high-precision computation of the Hurwitz zeta function and its derivatives". Num. Alg. 69 (2): 253–570. arXiv:1309.2877. doi:10.1007/s11075-014-9893-1. S2CID 10344040.
^Connon, Donal F. (2009). "New proofs of the duplication and multiplication formulae for the gamma and the Barnes double gamma functions". arXiv:0903.4539 [math.CA].
^ abcdIaroslav V. Blagouchine Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results. The Ramanujan Journal, vol. 35, no. 1, pp. 21-110, 2014. Erratum-Addendum: vol. 42, pp. 777-781, 2017. PDF
^V. Adamchik. A class of logarithmic integrals. Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, pp. 1-8, 1997.