Stieltjes constants

that occur in the Laurent series expansion of the Riemann zeta function: The constant

The Stieltjes constants are given by the limit (In the case n = 0, the first summand requires evaluation of 00, which is taken to be 1.)

Cauchy's differentiation formula leads to the integral representation 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.

More general series of the same nature include these examples[11] and or where ψn(a) are the Bernoulli polynomials of the second kind and Nn,r(a) are the polynomials given by the generating equation respectively (note that Nn,1(a) = ψn(a)).

[12] Oloa and Tauraso[13] showed that series with harmonic numbers may lead to Stieltjes constants Blagouchine[6] obtained slowly-convergent series involving unsigned Stirling numbers of the first kind

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.

[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

, then where Up to n = 100000, the Knessl-Coffey approximation correctly predicts the sign of γn with the single exception of n = 137.

In 2022 K. Maślanka[21] gave an asymptotic expression for the Stieltjes constants, which is both simpler and more accurate than those previously known.

we get a particularly simple expression in which both the rapidly increasing amplitude and the oscillations are clearly seen: The first few values are [22] 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.

More generally, one can define Stieltjes constants γn(a) that occur in the Laurent series expansion of the Hurwitz zeta function: Here a is a complex number with Re(a)>0.

For example, there exists the following asymptotic representation due to Berndt and Wilton.

[31] However, it was recently reported that this identity, albeit in a slightly different form, was first obtained by Carl Malmsten in 1846.