The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm Krzysztof Ma slanka

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The High Precision Numerical Calculation of

Stieltjes Constants. Simple and Fast Algorithm

Krzysztof Ma´slanka

Polish Academy of Sciences

Institute for the History of Science

Nowy ´

Swiat 72, 00-330 Warsaw, Poland

e-mail krzysiek2357@gmail.com

Andrzej Kole˙zy´nski

University of Science and Technology

Faculty of Materials Science and Ceramics

Mickiewicza 30, 30-059 Cracow, Poland

e-mail kolezyn@agh.edu.pl

October 11, 2022

Abstract

We present a simple but eﬃcient method of calculating Stieltjes con-

stants at a very high level of precision, up to about 80000 signiﬁcant

digits. This method is based on the hypergeometric-like expansion for the

Riemann zeta function presented by one of the authors in 1997 [17]. The

crucial ingredient in this method is a sequence of high-precision numerical

values of the Riemann zeta function computed in equally spaced real argu-

ments, i.e. ζ(1 + ε), ζ(1 + 2ε), ζ(1 + 3ε), ... where εis some real parameter.

(Practical choice of εis described in the main text.) Such values of zeta

may be readily obtained using the PARI/GP program, which is especially

suitable for this.

Keywords: Riemann zeta function, Stieltjes constants, experimental

mathematics, PARI/GP computer algebra system

1 Introduction: the Riemann ζfunction

Fundamental formulas in number theory are seldom numerically eﬃcient. Al-

though deep and absolutely precise, they may even hide the most important

features of involved quantities. As a prominent example we consider the cel-

ebrated zeta function ζ(s) discovered by Euler in 1737 and published in 1744

[9] as a function of real variable and meticulously investigated by Riemann in

arXiv:2210.04609v1 [math.NT] 27 Sep 2022

the complex domain in his famous memoir submitted in 1859 to the Prussian

Academy [20]:

ζ(s) =

∞

n=1

nsRe(s)>1 (1)

This is a special case of more general class of functions called Dirichlet series. It

is divergent in the most interesting area of the complex plane, i.e., in the so called

critical strip 0 ≤Re s≤1 where all complex zeros of zeta lie. However, as was

shown by Riemann, the deﬁnition (1) does contain information about the zeta

function on the entire complex plane but the process of analytic continuation

must be used in order to reveal global behavior of this function. There is no

universal procedure how to achieve this in practice and usually various ingenious

tricks are required. For example, considering simply alternating version of (1)

leads to another Dirichlet series which is convergent for Re s > 0 (except s= 1),

i.e. also inside the critical strip:

ζ(s) = 1

1−21−s

∞

n=1

(−1)n

nsRe(s)>0, s 6= 1

However, in order to obtain globally convergent representation for ζone has to

use more sophisticated techniques. We shall describe such an approach below.

The Riemann zeta function contains the (heavily encoded) puzzle of the

distribution of prime numbers. According to the famous saying of Paul Erd¨os

(1913-1996) – the solution to this puzzle may appear only ”in millions of years,

but even then it will not be complete, because in this case we are facing Inﬁnity”.

We know, however, that this secret lies in the distribution of the zeros of the zeta

function, i.e. the roots of the ”simple” equation ζ(s) = 0, on the complex plane.

In 1859 Riemann hypothesized that all these roots (except for the so-called

trivial ones) lie precisely on the line Re s=1

Despite the passage of more than a century and a half and the persistent

eﬀorts of many top-class mathematical talents, the Riemann hypothesis remains

unsettled. We simply do not know whether it is true or false. (Some think

that it is undecidable.) Computer experiments based on billions of numerically

calculated complex roots seem to conﬁrm it. However, exact proof remains, so

far, beyond the reach of mathematicians. It seems no one has even had a good

idea of how to attack this problem so far. Some have suggested that some ”new

math” is needed for this, but this view is too vague to be of any practical help.

2 Stieltjes constants

The Stieltjes constants are closely related to the Riemann zeta function, and

since this function is extremely important in analytical number theory, these

constants are equally important.

Formulas for the Stieltjes constants may serve as another example of strict

and deep but numerically ineﬃcient formulas. These constants are essentially

Figure 1: Plot of the zeta function for real variable (blue curve). Euler dis-

covered the zeta function in 1737 and found its deep connection with prime

numbers [9] (see box on the right). But it was Riemann who in 1859 rigorously

proved certain fundamental equation for it and made its analytical continuation

to the entire complex plane, except for a single pole for s= 1 [20] (box on the

left). Values of zeta for s= 2n,n= 1,2, ... were found by Euler in closed form

(red dots). ζ(−2n) = 0 are so called trivial zeros (green dots).

Figure 2: The zeta function shows its essence and its true meaning only in the

complex domain, and we owe knowledge about it to Riemann. The upper graph

is the real part of the zeta function ζ(s), the lower graph is its imaginary part

in the complex domain. The blue plane is the plane of the complex variable s.

Figure 3: Both surfaces shown in Fig. 2 intersects the plane of the complex vari-

able salong certain irregular curves.After overlapping these surfaces, it turns

out that these curves intersect themselves at certain points – these are the com-

plex zeros of the zeta function (indicated by vertical blue lines). The Riemann

hypothesis says that all these zeros are placed exactly on the line Re s= 1/2.

coeﬃcients of the Laurent series expansion of the zeta function around its only

simple pole at s= 1:

ζ(s) = 1

s−1+

∞

n=0

(−1)n

n!γn(s−1)n(2)

Primary deﬁnition of these fundamental constants was found by Thomas Jan

Stieltjes and presented in a letter to his close friend and collaborator Charles

Hermite dated June 23, 1885 [12]:

γn= lim

m→∞ m

k=1

(ln k)n

k−(ln m)n+1

n+ 1 !(3)

When n= 0 the numerator in the ﬁrst summand in (3) is formally 00which

is taken to be 1. In this case, (3) reduces simply to the well-known Euler-

Mascheroni constant

γ0= lim

m→∞ m

k=1

k−ln m!

which, roughly speaking, measures the rate of divergence of the harmonic series.

Eﬀective numerical computing of the constants γnis quite a challenge be-

cause the formulas (3) are extremely slowly convergent. Even for n= 0, in

order to obtain just 10 accurate digits one has to sum up exactly 12366 terms

whereas in order to obtain 10000 digits (which is indeed required in some appli-

cations) one would have to sum up unrealistically large number of terms: nearly

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TheHighPrecisionNumericalCalculationofStieltjesConstants.SimpleandFastAlgorithmKrzysztofMaslankaPolishAcademyofSciencesInstitutefortheHistoryofScienceNowySwiat72,00-330Warsaw,Polande-mailkrzysiek2357@gmail.comAndrzejKole_zynskiUniversityofScienceandTechnologyFacultyofMaterialsScienceandCeramicsMi...

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The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm Krzysztof Ma slanka

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