Euler's constant (sometimes also called the Euler–Mascheroni constant) is a
mathematical constant
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...
usually denoted by the lowercase Greek letter
gamma
Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter re ...
().
It is defined as the
limiting
In electronics, a limiter is a circuit that allows signals below a specified input power or level to pass unaffected while attenuating (lowering) the peaks of stronger signals that exceed this threshold. Limiting is a type of dynamic range comp ...
difference between the
harmonic series and the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
, denoted here by
:
Here,
represents the
floor function
In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least int ...
.
The numerical value of Euler's constant, to 50 decimal places, is:
:  
History
The constant first appeared in a 1734 paper by the
Swiss
Swiss may refer to:
* the adjectival form of Switzerland
* Swiss people
Places
* Swiss, Missouri
* Swiss, North Carolina
*Swiss, West Virginia
* Swiss, Wisconsin
Other uses
*Swiss-system tournament, in various games and sports
*Swiss Internation ...
mathematician
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
, titled ''De Progressionibus harmonicis observationes'' (Eneström Index 43). Euler used the notations and for the constant. In 1790,
Italian
Italian(s) may refer to:
* Anything of, from, or related to the people of Italy over the centuries
** Italians, an ethnic group or simply a citizen of the Italian Republic or Italian Kingdom
** Italian language, a Romance language
*** Regional Ita ...
mathematician
Lorenzo Mascheroni
Lorenzo Mascheroni (; May 13, 1750 – July 14, 1800) was an Italian mathematician.
Biography
He was born near Bergamo, Lombardy. At first mainly interested in the humanities (poetry and Greek language), he eventually became professor of mathem ...
used the notations and for the constant. The notation appears nowhere in the writings of either Euler or Mascheroni, and was chosen at a later time perhaps because of the constant's connection to the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
. For example, the
German
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ger ...
mathematician
Carl Anton Bretschneider
Carl Anton Bretschneider (27 May 1808 – 6 November 1878) was a mathematician from Gotha, Germany. Bretschneider worked in geometry, number theory, and history of geometry. He also worked on logarithmic integrals and mathematical tables. He was ...
used the notation in 1835 and
Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.
Appearances
Euler's constant appears, among other places, in the following (where '*' means that this entry contains an explicit equation):
* Expressions involving the
exponential integral
In mathematics, the exponential integral Ei is a special function on the complex plane.
It is defined as one particular definite integral of the ratio between an exponential function and its argument.
Definitions
For real non-zero values of&n ...
*
* The
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
* of the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
* The first term of the
Laurent series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...
expansion for the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
*, where it is the first of the
Stieltjes constants
In mathematics, the Stieltjes constants are the numbers \gamma_k that occur in the Laurent series expansion of the Riemann zeta function:
:\zeta(s)=\frac+\sum_^\infty \frac \gamma_n (s-1)^n.
The constant \gamma_0 = \gamma = 0.577\dots is known a ...
*
* Calculations of the
digamma function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
:\psi(x)=\frac\ln\big(\Gamma(x)\big)=\frac\sim\ln-\frac.
It is the first of the polygamma functions. It is strictly increasing and strictly ...
* A product formula for the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
* The asymptotic expansion of the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
for small arguments.
* An inequality for
Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
* The growth rate of the
divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including ...
* In
dimensional regularization
__NOTOC__
In theoretical physics, dimensional regularization is a method introduced by Giambiagi and Bollini as well as – independently and more comprehensively – by 't Hooft and Veltman for regularizing integrals in the evaluation of Fe ...
of
Feynman diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
s in
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
* The calculation of the
Meissel–Mertens constant
The Meissel–Mertens constant (named after Ernst Meissel and Franz Mertens), also referred to as Mertens constant, Kronecker's constant, Hadamard– de la Vallée-Poussin constant or the prime reciprocal constant, is a mathematical constant in n ...
* The third of
Mertens' theorems
In number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens.F. Mertens. J. reine angew. Math. 78 (1874), 46–6Ein Beitrag zur analytischen Zahlentheorie/ref> "Mertens' theorem" may a ...
*
* Solution of the second kind to
Bessel's equation
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
* In the regularization/
renormalization
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering v ...
of the
harmonic series as a finite value
* The
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
of the
Gumbel distribution
In probability theory and statistics, the Gumbel distribution (also known as the type-I generalized extreme value distribution) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions.
Thi ...
* The
information entropy
In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent to the variable's possible outcomes. Given a discrete random variable X, which takes values in the alphabet \ ...
of the
Weibull
Weibull is a Swedish locational surname. The Weibull family share the same roots as the Danish / Norwegian noble family of Falsenbr>They originated from and were named after the village of Weiböl in Widstedts parish, Jutland, but settled in Skà ...
and
Lévy distributions, and, implicitly, of the
chi-squared distribution
In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squa ...
for one or two degrees of freedom.
* The answer to the
coupon collector's problem
In probability theory, the coupon collector's problem describes "collect all coupons and win" contests. It asks the following question: If each box of a brand of cereals contains a coupon, and there are ''n'' different types of coupons, what is th ...
*
* In some formulations of
Zipf's law
* A definition of the
cosine integral
In mathematics, trigonometric integrals are a family of integrals involving trigonometric functions.
Sine integral
The different sine integral definitions are
\operatorname(x) = \int_0^x\frac\,dt
\operatorname(x) = -\int_x^\infty\frac ...
*
* Lower bounds to a
prime gap
A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g'n'' or ''g''(''p'n'') is the difference between the (''n'' + 1)-th and the
''n''-th prime numbers, i.e.
:g_n = p_ - p_n.\
W ...
* An upper bound on
Shannon entropy
Shannon may refer to:
People
* Shannon (given name)
* Shannon (surname)
* Shannon (American singer), stage name of singer Shannon Brenda Greene (born 1958)
* Shannon (South Korean singer), British-South Korean singer and actress Shannon Arrum Wi ...
in
quantum information theory
Quantum information is the information of the quantum state, state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information re ...
*
Fisher–Orr model for genetics of adaptation in evolutionary biology
[Connallon, T., Hodgins, K.A., 2021. Allen Orr and the genetics of adaptation. Evolution 75, 2624–2640. https://doi.org/10.1111/evo.14372]
Properties
The number has not been proved
algebraic or
transcendental. In fact, it is not even known whether is
irrational
Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
. Using a
continued fraction
In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...
analysis, Papanikolaou showed in 1997 that if is
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
, its denominator must be greater than 10
244663. The ubiquity of revealed by the large number of equations below makes the irrationality of a major open question in mathematics.
However, some progress was made. Kurt Mahler showed in 1968 that the number
is transcendental (here,
and
are
Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
s). In 2009 Alexander Aptekarev proved that at least one of Euler's constant and the
Euler–Gompertz constant In mathematics, the Gompertz constant or Euler–Gompertz constant, denoted by \delta, appears in integral evaluations and as a value of special functions. It is named after Benjamin Gompertz.
It can be defined by the continued fraction
: \delt ...
is irrational; Tanguy Rivoal proved in 2012 that at least one of them is transcendental. In 2010
M. Ram Murty
Maruti Ram Pedaprolu Murty, FRSC (born 16 October 1953 in Guntur, India)
is an Indo-Canadian mathematician at Queen's University, where he holds a Queen's Research Chair in mathematics.
Biography
M. Ram Murty is the brother of mathematician ...
and N. Saradha showed that at most one of the numbers of the form
:
with and is algebraic; this family includes the special case . In 2013 M. Ram Murty and A. Zaytseva found a different family containing , which is based on sums of reciprocals of integers not divisible by a fixed list of primes, with the same property.
Relation to gamma function
is related to the
digamma function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
:\psi(x)=\frac\ln\big(\Gamma(x)\big)=\frac\sim\ln-\frac.
It is the first of the polygamma functions. It is strictly increasing and strictly ...
, and hence the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
, when both functions are evaluated at 1. Thus:
:
This is equal to the limits:
:
Further limit results are:
:
A limit related to the
beta function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
: \Beta(z_1,z_2) = \int_0^1 t^(1 ...
(expressed in terms of
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
s) is
:
Relation to the zeta function
can also be expressed as an
infinite sum
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
whose terms involve the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
evaluated at positive integers:
:
Other series related to the zeta function include:
:
The error term in the last equation is a rapidly decreasing function of . As a result, the formula is well-suited for efficient computation of the constant to high precision.
Other interesting limits equaling Euler's constant are the antisymmetric limit:
:
and the following formula, established in 1898 by
de la Vallée-Poussin:
:
where
are
ceiling
A ceiling is an overhead interior surface that covers the upper limits of a room. It is not generally considered a structural element, but a finished surface concealing the underside of the roof structure or the floor of a story above. Ceilings ...
brackets.
This formula indicates that when taking any positive integer n and dividing it by each positive integer k less than n, the average fraction by which the quotient n/k falls short of the next integer tends to
(rather than 0.5) as n tends to infinity.
Closely related to this is the
rational zeta series In mathematics, a rational zeta series is the representation of an arbitrary real number in terms of a series consisting of rational numbers and the Riemann zeta function or the Hurwitz zeta function. Specifically, given a real number ''x'', t ...
expression. By taking separately the first few terms of the series above, one obtains an estimate for the classical series limit:
:
where is the
Hurwitz zeta function
In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables with and by
:\zeta(s,a) = \sum_^\infty \frac.
This series is absolutely convergent for the given values of and and can ...
. The sum in this equation involves the
harmonic number
In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers:
H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac.
Starting from , the sequence of harmonic numbers begins:
1, \frac, \frac, \frac, \frac, \dot ...
s, . Expanding some of the terms in the Hurwitz zeta function gives:
:
where
can also be expressed as follows where is the
Glaisher–Kinkelin constant In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted , is a mathematical constant, related to the -function and the Barnes -function. The constant appears in a number of sums and integrals, especially those ...
:
:
can also be expressed as follows, which can be proven by expressing the
zeta function
In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function
: \zeta(s) = \sum_^\infty \frac 1 .
Zeta functions include:
* Airy zeta function, related to the zeros of the Airy function
* ...
as a
Laurent series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...
:
:
Integrals
equals the value of a number of definite
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s:
:
where is the
fractional harmonic number.
The third formula in the integral list can be proved in the following way:
:
The integral on the second line of the equation stands for the
Debye function value of +∞, which is .
Definite integrals in which appears include:
:
One can express using a special case of
Hadjicostas's formula In mathematics, Hadjicostas's formula is a formula relating a certain double integral to values of the gamma function and the Riemann zeta function. It is named after Petros Hadjicostas.
Statement
Let ''s'' be a complex number with ''s'' ≠-1 an ...
as a
double integral
In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number ...
with equivalent series:
:
An interesting comparison by Sondow is the double integral and alternating series
:
It shows that may be thought of as an "alternating Euler constant".
The two constants are also related by the pair of series
:
where and are the number of 1s and 0s, respectively, in the
base 2 expansion of .
We also have
Catalan
Catalan may refer to:
Catalonia
From, or related to Catalonia:
* Catalan language, a Romance language
* Catalans, an ethnic group formed by the people from, or with origins in, Northern or southern Catalonia
Places
* 13178 Catalan, asteroid #1 ...
's 1875 integral
:
Series expansions
In general,
:
for any
. However, the rate of convergence of this expansion depends significantly on
. In particular,
exhibits much more rapid convergence than the conventional expansion
. This is because
:
while
:
Even so, there exist other series expansions which converge more rapidly than this; some of these are discussed below.
Euler showed that the following
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
approaches :
:
The series for is equivalent to a series
Nielsen found in 1897:
:
In 1910,
Vacca found the closely related series
:
where is the
logarithm to base 2 and is the
floor function
In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least int ...
.
In 1926 he found a second series:
:
From the
Malmsten–
Kummer expansion for the logarithm of the gamma function we get:
:
An important expansion for Euler's constant is due to
Fontana and
Mascheroni
:
where are
Gregory coefficients Gregory coefficients , also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind,Ch. Jordan. ''The Calculus of Finite Differences'' Chelsea Publishing Company, USA, 1947.L. Comtet. ''Adva ...
This series is the special case
of the expansions
:
convergent for
A similar series with the Cauchy numbers of the second kind is
:
Blagouchine (2018) found an interesting generalisation of the Fontana–Mascheroni series
:
where are the ''
Bernoulli polynomials of the second kind
The Bernoulli polynomials of the second kind , also known as the Fontana-Bessel polynomials, are the polynomials defined by the following generating function:
:
\frac= \sum_^\infty z^n \psi_n(x) ,\qquad , z, -1
and
:\gamma=\sum_^\infty\frac\B ...
'', which are defined by the generating function
:
For any rational this series contains rational terms only. For example, at , it becomes
:
Other series with the same polynomials include these examples:
:
and
:
where is the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
.
A series related to the Akiyama–Tanigawa algorithm is
:
where are the
Gregory coefficients Gregory coefficients , also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind,Ch. Jordan. ''The Calculus of Finite Differences'' Chelsea Publishing Company, USA, 1947.L. Comtet. ''Adva ...
of the second order.
Series of
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s:
:
Asymptotic expansions
equals the following asymptotic formulas (where is the th
harmonic number
In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers:
H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac.
Starting from , the sequence of harmonic numbers begins:
1, \frac, \frac, \frac, \frac, \dot ...
):
:
(''Euler'')
:
(''Negoi'')
:
(''
Cesà ro'')
The third formula is also called the
Ramanujan expansion.
Alabdulmohsin derived closed-form expressions for the sums of errors of these approximations. He showed that (Theorem A.1):
Exponential
The constant is important in number theory. Some authors denote this quantity simply as . equals the following
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
, where is the th
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
:
:
This restates the third of
Mertens' theorems
In number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens.F. Mertens. J. reine angew. Math. 78 (1874), 46–6Ein Beitrag zur analytischen Zahlentheorie/ref> "Mertens' theorem" may a ...
. The numerical value of is:
:.
Other
infinite product In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product
:
\prod_^ a_n = a_1 a_2 a_3 \cdots
is defined to be the limit of a sequence, limit of the Multiplication#Capital pi notation, partial products ''a' ...
s relating to include:
:
These products result from the
Barnes -function.
In addition,
:
where the th factor is the th root of
:
This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow using
hypergeometric functions.
It also holds that
:
Continued fraction
The
continued fraction
In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...
expansion of begins , which has no ''apparent'' pattern. The continued fraction is known to have at least 475,006 terms, and it has infinitely many terms
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
is irrational.
Generalizations
''Euler's generalized constants'' are given by
:
for , with as the special case . This can be further generalized to
:
for some arbitrary decreasing function . For example,
:
gives rise to the
Stieltjes constants
In mathematics, the Stieltjes constants are the numbers \gamma_k that occur in the Laurent series expansion of the Riemann zeta function:
:\zeta(s)=\frac+\sum_^\infty \frac \gamma_n (s-1)^n.
The constant \gamma_0 = \gamma = 0.577\dots is known a ...
, and
:
gives
:
where again the limit
:
appears.
A two-dimensional limit generalization is the
Masser–Gramain constant.
''Euler–Lehmer constants'' are given by summation of inverses of numbers in a common
modulo class:
:
The basic properties are
:
and if then
:
Published digits
Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Mascheroni attempted to calculate the constant to 32 decimal places, but made errors in the 20th–22nd and 31st-32nd decimal places; starting from the 20th digit, he calculated ...1811209008239 when the correct value is ...0651209008240.
References
*
*
*
Footnotes
Further reading
* Derives as sums over Riemann zeta functions.
*
*
*
*
*
*
*
*
*
*
*
* with an Appendix b
Sergey Zlobin
External links
*
*
Jonathan Sondow. E.A. Karatsuba (2005)
*Further formulae which make use of the constant
{{DEFAULTSORT:Euler's constant
Mathematical constants
Unsolved problems in number theory
Leonhard Euler