Not to be confused with
Euler's number, e ≈ 2.71828, the base of the natural logarithm.
This article uses technical mathematical notation for logarithms. All instances of log(x) without a subscript base should be interpreted as a
natural logarithm, commonly notated as ln(x) or loge(x).
Euler's constant (sometimes called the Euler–Mascheroni constant) is a
mathematical constant, usually denoted by the lowercase Greek letter
gamma (γ), defined as the
limiting difference between the
harmonic series and the
natural logarithm, denoted here by log:
The constant first appeared in a 1734 paper by the
Swiss mathematician
Leonhard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43). Euler used the notations C and O for the constant. In 1790, the
Italian mathematician
Lorenzo Mascheroni used the notations A and a for the constant. The notation γ 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.[2] For example, the
German mathematician
Carl Anton Bretschneider used the notation γ in 1835,[3] and
Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.[4]
Appearances
Euler's constant appears, among other places, in the following (where '*' means that this entry contains an explicit equation):
The number γ has not been proved
algebraic or
transcendental. In fact, it is not even known whether γ is
irrational. Using a
continued fraction analysis, Papanikolaou showed in 1997 that if γ is
rational, its denominator must be greater than 10244663.[7][8] The ubiquity of γ revealed by the large number of equations below makes the irrationality of γ a major open question in mathematics.[9]
However, some progress has been made. Kurt Mahler showed in 1968 that the number is transcendental (here, and are
Bessel functions).[10][2] In 2009 Alexander Aptekarev proved that at least one of Euler's constant γ and the
Euler–Gompertz constantδ is irrational;[11] Tanguy Rivoal proved in 2012 that at least one of them is transcendental.[12][2] In 2010
M. Ram Murty and N. Saradha showed that at most one of the numbers of the form
with q ≥ 2 and 1 ≤ a < q is algebraic; this family includes the special case γ(2,4) = γ/4.[2][13] In 2013 M. Ram Murty and A. Zaytseva found a different family containing γ, which is based on sums of reciprocals of integers not divisible by a fixed list of primes, with the same property.[2][14]
The constant can also be expressed in terms of the sum of the reciprocals of
non-trivial zeros of the zeta function:[16]
Other series related to the zeta function include:
The error term in the last equation is a rapidly decreasing function of n. 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:[17]
where ⌈ ⌉ are
ceiling 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 γ (rather than 0.5) as n tends to infinity.
Closely related to this is the
rational zeta series expression. By taking separately the first few terms of the series above, one obtains an estimate for the classical series limit:
where ζ(s, k) is the
Hurwitz zeta function. The sum in this equation involves the
harmonic numbers, Hn. Expanding some of the terms in the Hurwitz zeta function gives:
γ can also be expressed as follows, which can be proven by expressing the
zeta function as a
Laurent series:
Relation to triangular numbers
Numerous formulations have been derived that express in terms of sums and logarithms of
triangular numbers.[18][19][20][21] One of the earliest of these is a formula[22][23] for the thharmonic number attributed to
Srinivasa Ramanujan where is related to in a series that considers the powers of (an earlier, less-generalizable proof[24][25] by
Ernesto Cesà ro gives the first two terms of the series, with an error term):
The series of inverse triangular numbers also features in the study of the
Basel problem[27][28] posed by
Pietro Mengoli. Mengoli proved that , a result
Jacob Bernoulli later used to estimate the
value of , placing it between and . This identity appears in a formula used by
Bernhard Riemann to compute
roots of the zeta function,[29] where is expressed in terms of the sum of roots plus the difference between Boya's expansion and the series of exact
unit fractions:
Integrals
γ equals the value of a number of definite
integrals:
for any α > −n. However, the rate of convergence of this expansion depends significantly on α. In particular, γn(1/2) exhibits much more rapid convergence than the conventional expansion γn(0).[33][34] 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 approaches γ:
The series for γ is equivalent to a series
Nielsen found in 1897:[15][35]
The third formula is also called the Ramanujan expansion.
Alabdulmohsin derived closed-form expressions for the sums of errors of these approximations.[44] He showed that (Theorem A.1):
Exponential
The constant eγ is important in number theory. Some authors denote this quantity simply as γ′. eγ equals the following
limit, where pn is the nth
prime number:
The
continued fraction expansion of γ begins [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...],[51] which has no apparent pattern. The continued fraction is known to have at least 475,006 terms,[7] and it has infinitely many terms
if and only ifγ is irrational.
Generalizations
Euler's generalized constants are given by
for 0 < α < 1, with γ as the special case α = 1.[52] This can be further generalized to
for some arbitrary decreasing function f. For example,
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.
^
abHaible, Bruno; Papanikolaou, Thomas (1998). "Fast multiprecision evaluation of series of rational numbers". In Buhler, Joe P. (ed.). Algorithmic Number Theory. Lecture Notes in Computer Science. Vol. 1423. Springer. pp. 338–350.
doi:
10.1007/bfb0054873.
ISBN9783540691136.
^Mahler, Kurt; Mordell, Louis Joel (4 June 1968). "Applications of a theorem by A. B. Shidlovski". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 305 (1481): 149–173.
Bibcode:
1968RSPSA.305..149M.
doi:
10.1098/rspa.1968.0111.
S2CID123486171.
^Aptekarev, A. I. (28 February 2009). "On linear forms containing the Euler constant".
arXiv:0902.1768 [
math.NT].
^
abcdKrämer, Stefan (2005). Die Eulersche Konstante γ und verwandte Zahlen (in German). University of Göttingen.
^Wolf, Marek (2019). "6+infinity new expressions for the Euler-Mascheroni constant".
arXiv:1904.09855 [
math.NT]. The above sum is real and convergent when zeros and complex conjugate are paired together and summed according to increasing absolute values of the imaginary parts of . See formula 11 on page 3. Note the typographical error in the numerator of Wolf's sum over zeros, which should be 2 rather than 1.
^
abBoya, L.J. (2008).
"Another relation between Ï€, e, γ and ζ(n)". Revista de la Real Academia de Ciencias Exactas, FÃsicas y Naturales. Serie A. Matemáticas. 102: 199–202.
doi:
10.1007/BF03191819. γ/2 in (10) reflects the residual (finite part) of ζ(1)/2, of course. See formulas 1 and 10.
^Villarino, Mark B. (2007). "Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number".
arXiv:0707.3950 [
math.CA]. It would also be interesting to develop an expansion for n! into powers of m, a new Stirling expansion, as it were. See formula 1.8 on page 3.
^Whittaker, E.; Watson, G. (2021) [1902]. A Course of Modern Analysis (5th ed.). p. 271, 275.
doi:
10.1017/9781009004091.
ISBN9781316518939. See Examples 12.21 and 12.50 for exercises on the derivation of the integral form of the series .
^DeTemple, Duane W. (May 1993). "A Quicker Convergence to Euler's Constant". The American Mathematical Monthly. 100 (5): 468–470.
doi:
10.2307/2324300.
ISSN0002-9890.
JSTOR2324300.
^Hardy, G.H. (1912). "Note on Dr. Vacca's series for γ". Q. J. Pure Appl. Math. 43: 215–216.
^Vacca, G. (1926). "Nuova serie per la costante di Eulero, C = 0,577...". Rendiconti, Accademia Nazionale dei Lincei, Roma, Classe di Scienze Fisiche". Matematiche e Naturali (in Italian). 6 (3): 19–20.
^Kluyver, J.C. (1927). "On certain series of Mr. Hardy". Q. J. Pure Appl. Math. 50: 185–192.
^
abcBlagouchine, Iaroslav V. (2016). "Expansions of generalized Euler's constants into the series of polynomials in π−2 and into the formal enveloping series with rational coefficients only". J. Number Theory. 158: 365–396.
arXiv:1501.00740.
doi:
10.1016/j.jnt.2015.06.012.
^
abAlabdulmohsin, Ibrahim M. (2018). Summability Calculus. A Comprehensive Theory of Fractional Finite Sums.
Springer. pp. 147–8.
ISBN9783319746487.
^Sondow, Jonathan (2003). "An infinite product for eγ via hypergeometric formulas for Euler's constant, γ".
arXiv:math.CA/0306008.
^Choi, Junesang; Srivastava, H.M. (1 September 2010). "Integral Representations for the Euler–Mascheroni Constant γ". Integral Transforms and Special Functions. 21 (9): 675–690.
doi:
10.1080/10652461003593294.
ISSN1065-2469.
S2CID123698377.
Finch, Steven R. (2003). Mathematical Constants. Encyclopedia of Mathematics and its Applications. Vol. 94. Cambridge: Cambridge University Press.
ISBN0-521-81805-2.
Gerst, I. (1969). "Some series for Euler's constant". Amer. Math. Monthly. 76 (3): 237–275.
doi:
10.2307/2316370.
JSTOR2316370.
Lerch, M. (1897). "Expressions nouvelles de la constante d'Euler". Sitzungsberichte der Königlich Böhmischen Gesellschaft der Wissenschaften. 42: 5.
Mascheroni, Lorenzo (1790). Adnotationes ad calculum integralem Euleri, in quibus nonnulla problemata ab Eulero proposita resolvuntur. Galeati, Ticini.