Constants in the zeta function's Laurent series expansion
In mathematics, the Stieltjes constants are the numbers formula_1 that occur in the Laurent series expansion of the Riemann zeta function:
formula_2
The constant formula_3 is known as the Euler–Mascheroni constant.
Representations.
The Stieltjes constants are given by the limit
 formula_4
Cauchy's differentiation formula leads to the integral representation
formula_5
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. In particular, Jensen-Franel's integral formula, often erroneously attributed to Ainsworth and Howell, states that
 formula_6
where δ"n,k" is the Kronecker symbol (Kronecker delta). Among other formulae, we find
 formula_7
 formula_8
see.
As concerns series representations, a famous series implying an integer part of a logarithm was given by Hardy in 1912
 formula_9
Israilov gave semi-convergent series in terms of Bernoulli numbers formula_10
 formula_11
Connon, Blagouchine and Coppo gave several series with the binomial coefficients
 formula_12
where "G""n" are Gregory's coefficients, also known as reciprocal logarithmic numbers ("G"1=+1/2, "G"2=−1/12, "G"3=+1/24, "G"4=−19/720... ). 
More general series of the same nature include these examples
 formula_13
and 
 formula_14
or
 formula_15
where "ψn"("a") are the Bernoulli polynomials of the second kind and "Nn,r"("a") are the polynomials given by the generating equation
 formula_16
respectively (note that "Nn,1"("a") 
 "ψn"("a")).
Oloa and Tauraso showed that series with harmonic numbers may lead to Stieltjes constants
 formula_17
Blagouchine obtained slowly-convergent series involving unsigned Stirling numbers of the first kind
formula_18
 formula_19
as well as semi-convergent series with rational terms only
 formula_20
where "m"=0,1,2... In particular, series for the first Stieltjes constant has a surprisingly simple form
 formula_21
where "H""n" is the "n"th harmonic number.
More complicated series for Stieltjes constants are given in works of Lehmer, Liang, Todd, Lavrik, Israilov, Stankus, Keiper, Nan-You, Williams, Coffey.
Bounds and asymptotic growth.
The Stieltjes constants satisfy the bound
formula_22
given by Berndt in 1972. Better bounds in terms of elementary functions were obtained by Lavrik
formula_23
by Israilov
formula_24
with "k"=1,2... and "C"(1)=1/2, "C"(2)=7/12... , by Nan-You and Williams
formula_25
by Blagouchine
formula_26
where "B""n" are Bernoulli numbers, and by Matsuoka
formula_27
As concerns estimations resorting to non-elementary functions and solutions, Knessl, Coffey and Fekih-Ahmed obtained quite accurate results. For example, Knessl and Coffey give the following formula that approximates the Stieltjes constants relatively well for large "n". If "v" is the unique solution of
formula_28
with formula_29, and if formula_30, then
formula_31
where
formula_32
formula_33
formula_34
formula_35
Up to n = 100000, the Knessl-Coffey approximation correctly predicts the sign of γ"n" with the single exception of n = 137.
In 2022 K. Maślanka 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 formula_36
formula_37
where formula_38 are the saddle points:
formula_39
formula_40 is the Lambert function and formula_41 is a constant:
formula_42
Defining a complex "phase" formula_43
formula_44
we get a particularly simple expression in which both the rapidly increasing
amplitude and the oscillations are clearly seen:
formula_45
Numerical values.
The first few values are 
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, Kreminski, Plouffe, Johansson and Blagouchine. 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 . 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. In particular, it allows one to compute "γ""n" to 1000 digits in a minute for any "n" up to "n"
10100.
Generalized Stieltjes constants.
General information.
More generally, one can define Stieltjes constants γ"n"(a) that occur in the Laurent series expansion of the Hurwitz zeta function:
formula_46
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 γ"n"(1)=γ"n" The zeroth constant is simply the digamma-function γ0(a)=-Ψ(a), 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
formula_47
due to Berndt and Wilton. The analog of Jensen-Franel's formula for the generalized Stieltjes constant is the Hermite formula
formula_48
Similar representations are given by the following formulas:
formula_49
and
formula_50
Generalized Stieltjes constants satisfy the following recurrence relation
formula_51
as well as the multiplication theorem
formula_52
where formula_53 denotes the binomial coefficient (see and, pp. 101–102).
First generalized Stieltjes constant.
The first generalized Stieltjes constant has a number of remarkable properties. 
formula_54
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. However, it was recently reported that this identity, albeit in a slightly different form, was first obtained by Carl Malmsten in 1846.
formula_55
see Blagouchine. An alternative proof was later proposed by Coffey and several other authors.
formula_56
For more details and further summation formulae, see.
formula_57
At points 1/4, 3/4 and 1/3, values of first generalized Stieltjes constants were independently obtained by Connon and Blagouchine
formula_58
At points 2/3, 1/6 and 5/6
formula_59
These values were calculated by Blagouchine. To the same author are also due 
formula_60
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
formula_61
see Blagouchine.
An equivalent result was later obtained by Coffey by another method.
References.
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