In mathematics, the Stieltjes constants are the numbers that occur in the Laurent series expansion of the Riemann zeta function:
The constant is known as the Euler–Mascheroni constant.
Representations
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. 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). Among other formulae, we find
see.
As concerns series representations, a famous series implying an integer part of a logarithm was given by Hardy in 1912
Israilov gave semi-convergent series in terms of Bernoulli numbers
Connon, Blagouchine and Coppo 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
and
or
where are the Bernoulli polynomials of the second kind and are the polynomials given by the generating equation
respectively (note that ). Oloa and Tauraso showed that series with harmonic numbers may lead to Stieltjes constants
Blagouchine obtained slowly-convergent series involving unsigned Stirling numbers of the first kind
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. 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
given by Berndt in 1972. Better bounds in terms of elementary functions were obtained by Lavrik
by Israilov
with k=1,2,... and C(1)=1/2, C(2)=7/12,... , by Nan-You and Williams
by Blagouchine
where Bn are Bernoulli numbers, and by Matsuoka
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
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.
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
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:
Numerical values
The first few values are
{ | class="wikitable" | n || approximate value of γn || OEIS |- | 0 || +0.5772156649015328606065120900824024310421593359 || |- | 1 || −0.0728158454836767248605863758749013191377363383 || |- | 2 || −0.0096903631928723184845303860352125293590658061 || |- | 3 || +0.0020538344203033458661600465427533842857158044 || |- | 4 || +0.0023253700654673000574681701775260680009044694 || |- | 5 || +0.0007933238173010627017533348774444448307315394 || |- | 6 || −0.0002387693454301996098724218419080042777837151 || |- | 7 || −0.0005272895670577510460740975054788582819962534 || |- | 8 || −0.0003521233538030395096020521650012087417291805 || |- | 9 || −0.0000343947744180880481779146237982273906207895 || |- | 10 || +0.0002053328149090647946837222892370653029598537 || |- | 100 || −4.2534015717080269623144385197278358247028931053 × 1017 || |- | 1000 || −1.5709538442047449345494023425120825242380299554 × 10486 || |- | 10000 || −2.2104970567221060862971082857536501900234397174 × 106883 || |- | 100000 || +1.9919273063125410956582272431568589205211659777 × 1083432 || |}
For large n, the Stieltjes constants grow rapidly in absolute value, and change signs in a complex pattern.