Euler's constant

 ( Mathematical Constants ) 

• The Weierstrass product formula for the gamma function and the Barnes G-function.
• The asymptotic expansion of the gamma function, $\backslash Gamma(1/x)\backslash sim\; x-\backslash gamma$.
• Evaluations of the digamma function at rational values.
• The Laurent series expansion for the Riemann zeta function*, where it is the first of the Stieltjes constants.
• Values of the derivative of the Riemann zeta function and Dirichlet beta function.
• In connection to the Laplace and Mellin transform.
  John Williams (1973). 9780045120215, Allen & Unwin. ISBN 9780045120215
• In the regularization/renormalization of the harmonic series as a finite value.
• Expressions involving the exponential and logarithmic integral.*
• A definition of the cosine integral.*
• In relation to Bessel function.
• Asymptotic expansions of modified Struve function.
• In relation to other special functions.

• An inequality for Euler's totient function.
• The growth rate of the divisor function.
  (2025). 9780199219865, Oxford University Press. ISBN 9780199219865
• A formulation of the Riemann hypothesis.
• The third of Mertens' theorems.*
• The calculation of the Meissel–Mertens constant.
• Lower bounds to specific prime gaps.
• An approximation of the average number of Divisor of all numbers from 1 to a given n.
• The Lenstra–Pomerance–Wagstaff conjecture on the frequency of Mersenne prime.
• An estimation of the efficiency of the euclidean algorithm.
• Sums involving the Möbius and von Mangolt function.
• Estimate of the divisor summatory function of the Dirichlet hyperbola method.
  Gérald Tenenbaum (2015). 9780821898543, American Mathematical Soc.. . ISBN 9780821898543

• In some formulations of Zipf's law.
• The answer to the coupon collector's problem.*
• The mean of the Gumbel distribution.
• An approximation of the Landau distribution.
• The information entropy of the Weibull and Lévy distributions, and, implicitly, of the chi-squared distribution for one or two degrees of freedom.
• An upper bound on Shannon entropy in quantum information theory.
• In dimensional regularization of in quantum field theory.
• In the BCS equation on the critical temperature in BCS theory of superconductivity.*
• Fisher–Orr model for genetics of adaptation in evolutionary biology.

is related to the [[digamma function]] , and hence the [[derivative]] of the [[gamma function]] , when both functions are evaluated at 1. Thus:
     

can also be expressed as an infinite sum whose terms involve the Riemann zeta function evaluated at positive integers:
     

&= \log\frac4{\pi} + \sum_{m=2}^{\infty} (-1)^m\frac{\zeta(m)}{2^{m-1}m}.\end{align} 
     

&= \lim_{n\to\infty}\left(\frac{2n-1}{2n} - \log n + \sum_{k=2}^n \left(\frac1{k} - \frac{\zeta(1-k)}{n^k}\right)\right) \\
&= \lim_{n\to\infty}\left(\frac{2^n}{e^{2^n}} \sum_{m=0}^\infty \frac{2^{mn}}{(m+1)!} \sum_{t=0}^m \frac1{t+1} - n \log 2+ O \left (\frac1{2^{n}\, e^{2^n}}\right)\right).\end{align}
     

can also be expressed as follows where  is the Glaisher–Kinkelin constant:
     

can also be expressed as follows, which can be proven by expressing the zeta function as a [[Laurent series]]:
     

   \gamma
   &= H_u - \frac{1}{2} \ln 2T_u - \sum_{k=1}^{v}\frac{R(k)}{T_{u}^{k}}-\Theta_{v}\,\frac{R(v+1)}{T_{u}^{v+1}}
     

$\backslash gamma\; =\; \backslash ln\; 2\backslash pi\; -\; \backslash sum\_\{k=2\}^\{\backslash infty\}\; \backslash frac\{\backslash zeta(k)\}\{T\_\{k\}\}$

The series of inverse triangular numbers also features in the study of the Basel problem posed by Pietro Mengoli. Mengoli proved that $\backslash textstyle\; \backslash sum\_\{k\; =\; 1\}^\backslash infty\; \backslash frac\{1\}\{2T\_k\}\; =\; 1$, a result Jacob Bernoulli later used to estimate the value of $\backslash zeta(2)$, placing it between $1$ and $\backslash textstyle\; \backslash sum\_\{k\; =\; 1\}^\backslash infty\; \backslash frac\{2\}\{2T\_k\}\; =\; \backslash sum\_\{k\; =\; 1\}^\backslash infty\; \backslash frac\{1\}\{T\_\{k\}\}\; =\; 2$. This identity appears in a formula used by Bernhard Riemann to compute roots of the zeta function, where $\backslash gamma$ is expressed in terms of the sum of roots $\backslash rho$ plus the difference between Boya's expansion and the series of exact Unit fraction $\backslash textstyle\; \backslash sum\_\{k\; =\; 1\}^\{\backslash infty\}\; \backslash frac\{1\}\{T\_\{k\}\}$:

$\backslash gamma\; -\; \backslash ln\; 2\; =\; \backslash ln\; 2\backslash pi\; +\; \backslash sum\_\{\backslash rho\}\; \backslash frac\{2\}\{\backslash rho\}\; -\; \backslash sum\_\{k\; =\; 1\}^\{\backslash infty\}\; \backslash frac\{1\}\{T\_k\}$

equals the value of a number of definite [[integral]]s:
     

&= -\int_0^1\log\left(\log\frac 1 x \right) dx \\
&= \int_0^\infty \left(\frac1{e^x-1}-\frac1{x\cdot e^x} \right)dx \\
&= \int_0^1\frac{1-e^{-x}}{x} \, dx -\int_1^\infty \frac{e^{-x}}{x}\, dx\\
&= \int_0^1\left(\frac1{\log x} + \frac1{1-x}\right)dx\\
&= \int_0^\infty \left(\frac1{1+x^k}-e^{-x}\right)\frac{dx}{x},\quad k>0\\
&= 2\int_0^\infty \frac{e^{-x^2}-e^{-x}}{x} \, dx ,\\
     

&= \int_0^1 H_x \, dx, \\
&= \frac{1}{2}+\int_{0}^{\infty}\log\left(1+\frac{\log\left(1+\frac{1}{t}\right)^{2}}{4\pi^{2}}\right)dt \\
     

\end{align}
     
     

The third formula in the integral list can be proved in the following way:

= \int_0^{\infty} \frac{e^{-x} + x - 1}{x[e^x -1]} dx
= \int_0^{\infty} \frac{1}{x[e^x - 1]} \sum_{m = 1}^{\infty} \frac{(-1)^{m+1}x^{m+1}}{(m+1)!} dx \\[2pt]
     

= \sum_{m = 1}^{\infty} \int_0^{\infty} \frac{(-1)^{m+1}x^m}{(m+1)![e^x -1]} dx
= \sum_{m = 1}^{\infty} \frac{(-1)^{m+1}}{(m+1)!} \int_0^{\infty} \frac{x^m}{e^x - 1} dx \\[2pt]
     

= \sum_{m = 1}^{\infty} \frac{(-1)^{m+1}}{m+1}\zeta(m+1)
= \sum_{m = 1}^{\infty} \frac{(-1)^{m+1}}{m+1} \sum_{n = 1}^{\infty}\frac{1}{n^{m+1}}
= \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \frac{(-1)^{m+1}}{m+1}\frac{1}{n^{m+1}} \\[2pt]
     

= \sum_{n = 1}^{\infty} \left[\frac{1}{n} - \log\left(1+\frac{1}{n}\right)\right]
= \gamma
     

The integral on the second line of the equation is the definition of the Riemann zeta function, which is .

Definite integrals in which appears include:

We also have Catalan's 1875 integral

$$\backslash gamma\; =\; \backslash int\_0^1\; \backslash left(\backslash frac1\{1+x\}\backslash sum\_\{n=1\}^\backslash infty\; x^\{2^n-1\}\backslash right)\backslash ,dx.$$

One can express using a special case of Hadjicostas's formula as a double integral 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 Binary number expansion of .

$$\backslash gamma\; =\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}\backslash left(\backslash frac\{1\}\{1\}+\backslash frac\{1\}\{2\}+\backslash frac\{1\}\{3\}\; +\; \backslash ldots\; +\; \backslash frac\{1\}\{n\}\; -\; \backslash log(n+\backslash alpha)\; \backslash right)\; \backslash equiv\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash gamma\_n(\backslash alpha)$$

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 approaches : $$\backslash gamma\; =\; \backslash sum\_\{k=1\}^\backslash infty\; \backslash left(\backslash frac\; 1\; k\; -\; \backslash log\backslash left(1+\backslash frac\; 1\; k\; \backslash right)\backslash right).$$

The series for is equivalent to a series Nielsen found in 1897:

$$\backslash gamma\; =\; 1\; -\; \backslash sum\_\{k=2\}^\backslash infty\; (-1)^k\backslash frac\{\backslash left\backslash lfloor\backslash log\_2\; k\backslash right\backslash rfloor\}\{k+1\}.$$

In 1910, Vacca found the closely related series

where is the binary logarithm and is the floor function.

This can be generalized to:

$$\backslash gamma=\; \backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{\backslash left\backslash lfloor\backslash log\_B\; k\backslash right\backslash rfloor\}\{k\}\; \backslash varepsilon(k)$$where:

In 1926 Vacca found a second series:

From the Malmsten–Ernst Kummer expansion for the logarithm of the gamma function we get:

$$\backslash gamma\; =\; \backslash log\backslash pi\; -\; 4\backslash log\backslash left(\backslash Gamma(\backslash tfrac34)\backslash right)\; +\; \backslash frac\; 4\; \backslash pi\; \backslash sum\_\{k=1\}^\backslash infty\; (-1)^\{k+1\}\backslash frac\{\backslash log(2k+1)\}\{2k+1\}.$$

Ramanujan, in his lost notebook gave a series that approaches :

$$\backslash gamma\; =\; \backslash log\; 2\; -\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash sum\_\{k=\backslash frac\{3^\{n-1\}+1\}\{2\}\}^\{\backslash frac\{3^\{n\}-1\}\{2\}\}\; \backslash frac\{2n\}\{(3k)^3-3k\}$$

An important expansion for Euler's constant is due to Gregorio Fontana and Mascheroni

$$\backslash gamma\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac$$

$$\backslash begin\{align\}$$

\gamma &= H_{k-1}  - \log k + \sum_{n=1}^{\infty}\frac{(n-1)!|G_n|}{k(k+1) \cdots (k+n-1)} && \\
    &= H_{k-1} - \log k + \frac1{2k} + \frac1{12k(k+1)} + \frac1{12k(k+1)(k+2)} + \frac{19}{120k(k+1)(k+2)(k+3)} + \cdots &&
     

convergent for

A similar series with the Cauchy numbers of the second kind is

$$\backslash gamma\; =\; 1\; -\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{C\_\{n\}\}\{n\; \backslash ,\; (n+1)!\}\; =1-\; \backslash frac\{1\}\{4\}\; -\backslash frac\{5\}\{72\}\; -\; \backslash frac\{1\}\{32\}\; -\; \backslash frac\{251\}\{14400\}\; -\; \backslash frac\{19\}\{1728\}\; -\; \backslash ldots$$

Blagouchine (2018) found a generalisation of the Fontana–Mascheroni series

$$\backslash gamma=\backslash sum\_\{n=1\}^\backslash infty\backslash frac\{(-1)^\{n+1\}\}\{2n\}\backslash Big\backslash \{\backslash psi\_\{n\}(a)+\; \backslash psi\_\{n\}\backslash Big(-\backslash frac\{a\}\{1+a\}\backslash Big)\backslash Big\backslash \},\; \backslash quad\; a>-1$$

where are the Bernoulli polynomials of the second kind, which are defined by the generating function

For any rational this series contains rational terms only. For example, at , it becomes

$$\backslash gamma=\backslash frac\{3\}\{4\}\; -\; \backslash frac\{11\}\{96\}\; -\; \backslash frac\{1\}\{72\}\; -\; \backslash frac\{311\}\{46080\}\; -\; \backslash frac\{5\}\{1152\}\; -\; \backslash frac\{7291\}\{2322432\}\; -\; \backslash frac\{243\}\{100352\}\; -\; \backslash ldots$$ Other series with the same polynomials include these examples:

$$\backslash gamma=\; -\backslash log(a+1)\; -\; \backslash sum\_\{n=1\}^\backslash infty\backslash frac\{(-1)^n\; \backslash psi\_\{n\}(a)\}\{n\},\backslash qquad\; \backslash Re(a)>-1$$

and

$$\backslash gamma=\; -\backslash frac\{2\}\{1+2a\}\; \backslash left\backslash \{\backslash log\backslash Gamma(a+1)\; -\backslash frac\{1\}\{2\}\backslash log(2\backslash pi)\; +\; \backslash frac\{1\}\{2\}\; +\; \backslash sum\_\{n=1\}^\backslash infty\backslash frac\{(-1)^n\; \backslash psi\_\{n+1\}(a)\}\{n\}\backslash right\backslash \},\backslash qquad\; \backslash Re(a)>-1$$

where is the gamma function.

A series related to the Akiyama–Tanigawa algorithm is

$$\backslash gamma=\; \backslash log(2\backslash pi)\; -\; 2\; -\; 2\; \backslash sum\_\{n=1\}^\backslash infty\backslash frac\{(-1)^n\; G\_\{n\}(2)\}\{n\}=\; \backslash log(2\backslash pi)\; -\; 2\; +\; \backslash frac\{2\}\{3\}\; +\; \backslash frac\{1\}\{24\}+\; \backslash frac\{7\}\{540\}\; +\; \backslash frac\{17\}\{2880\}+\; \backslash frac\{41\}\{12600\}\; +\; \backslash ldots$$

where are the Gregory coefficients of the second order.

As a series of :

$$\backslash gamma\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\backslash left(\backslash log\; n\; -\; \backslash sum\_\{p\backslash le\; n\}\backslash frac\{\backslash log\; p\}\{p-1\}\backslash right).$$

equals the following asymptotic formulas (where  is the th [[harmonic number]]):
     

• $\backslash gamma\; \backslash sim\; H\_n\; -\; \backslash log\; n\; -\; \backslash frac1\{2n\}\; +\; \backslash frac1\{12n^2\}\; -\; \backslash frac1\{120n^4\}\; +\; \backslash cdots$ ( Euler)
• $\backslash gamma\; \backslash sim\; H\_n\; -\; \backslash log\backslash left(\{n\; +\; \backslash frac1\{2\}\; +\; \backslash frac1\{24n\}\; -\; \backslash frac1\{48n^2\}\; +\; \backslash cdots\}\backslash right)$ ( Negoi)
• $\backslash gamma\; \backslash sim\; H\_n\; -\; \backslash frac\{\backslash log\; n\; +\; \backslash log(n+1)\}\{2\}\; -\; \backslash frac1\{6n(n+1)\}\; +\; \backslash frac1\{30n^2(n+1)^2\}\; -\; \backslash cdots$ ( Cesàro)

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):

equals the following limit, where  is the th [[prime number]]:
     

$$e^\backslash gamma\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\backslash frac1\{\backslash log\; p\_n\}\; \backslash prod\_\{i=1\}^n\; \backslash frac\{p\_i\}\{p\_i-1\}.$$

This restates the third of Mertens' theorems.

We further have the following product involving the three constants , and :

$$\backslash frac\{\backslash pi^2\}\{6e^\backslash gamma\}=\backslash lim\_\{n\backslash to\backslash infty\}\; \backslash log\; p\_n\; \backslash prod\_\{i=1\}^n\; \backslash frac\{p\_i\}\{p\_i+1\}.$$

Other relating to include:

These products result from the Barnes -function.

In addition,

$$e^\{\backslash gamma\}\; =\; \backslash sqrt\{\backslash frac2\{1\}\}\; \backslash cdot\; \backslash sqrt3\{\backslash frac\{2^2\}\{1\backslash cdot\; 3\}\}\; \backslash cdot\; \backslash sqrt4\{\backslash frac\{2^3\backslash cdot\; 4\}\{1\backslash cdot\; 3^3\}\}\; \backslash cdot\; \backslash sqrt5\{\backslash frac\{2^4\backslash cdot\; 4^4\}\{1\backslash cdot\; 3^6\backslash cdot\; 5\}\}\; \backslash cdots$$

where the th factor is the th root of

$$\backslash prod\_\{k=0\}^n\; (k+1)^\{(-1)^\{k+1\}\{n\; \backslash choose\; k\}\}.$$

This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow using hypergeometric functions.

It also holds that

$$\backslash frac\{e^\backslash frac\{\backslash pi\}\{2\}+e^\{-\backslash frac\{\backslash pi\}\{2\}\}\}\{\backslash pi\; e^\backslash gamma\}=\backslash prod\_\{n=1\}^\backslash infty\backslash left(e^\{-\backslash frac\{1\}\{n\}\}\backslash left(1+\backslash frac\{1\}\{n\}+\backslash frac\{1\}\{2n^2\}\backslash right)\backslash right).$$

$$\backslash gamma\_\backslash alpha\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\backslash left(\backslash sum\_\{k=1\}^n\; \backslash frac1\{k^\backslash alpha\}\; -\; \backslash int\_1^n\; \backslash frac1\{x^\backslash alpha\}\backslash ,dx\backslash right)$$

for , with as the special case . Extending for gives:

$$\backslash gamma\_\{\backslash alpha\}\; =\; \backslash zeta(\backslash alpha)\; -\; \backslash frac1\{\backslash alpha-1\}$$

with again the limit:

$$\backslash gamma\; =\; \backslash lim\_\{a\backslash to\; 1\}\backslash left(\backslash zeta(a)\; -\; \backslash frac1\{a-1\}\backslash right)$$

This can be further generalized to

$$c\_f\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\backslash left(\backslash sum\_\{k=1\}^n\; f(k)\; -\; \backslash int\_1^n\; f(x)\backslash ,dx\backslash right)$$

for some arbitrary decreasing function . Setting

$$f\_n(x)\; =\; \backslash frac\{(\backslash log\; x)^n\}\{x\}$$

gives rise to the Stieltjes constants $\backslash gamma\_n$, that occur in the Laurent series expansion of the Riemann zeta function:

$\backslash zeta(1+s)=\backslash frac\{1\}\{s\}+\backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{(-1)^n\}\{n!\}\; \backslash gamma\_n\; s^n.$

with $\backslash gamma\_0\; =\; \backslash gamma\; =\; 0.577\backslash dots$

The basic properties are

and if the greatest common divisor then

$$q\backslash gamma(a,q)\; =\; \backslash frac\{q\}\{d\}\backslash gamma\backslash left(\backslash frac\{a\}\{d\},\backslash frac\{q\}\{d\}\backslash right)-\backslash log\; d.$$

$\backslash delta\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\; \backslash left(\; -\backslash log\; n\; +\; \backslash sum\_\{k=2\}^n\; \backslash frac\{1\}\{\backslash pi\; r\_k^2\}\; \backslash right)$

where $r\_k$ is the smallest radius of a disk in the complex plane containing at least $k$ Gaussian integer.

The following bounds have been established: .

• Harmonic series
• Riemann zeta function
• Stieltjes constants
• Meissel-Mertens constant
• Julian Havil (2025). 9780691099835, Princeton University Press. ISBN 9780691099835

• Derives as sums over Riemann zeta functions.
• Steven R. Finch (2025). 9780521818056, Cambridge University Press. ISBN 9780521818056
• Julian Havil (2003): GAMMA: Exploring Euler's Constant, Princeton University Press, ISBN 978-0-69114133-6.
• Donald Knuth (1997). 9780201896831, Addison-Wesley. ISBN 9780201896831
• with an Appendix by Sergey Zlobin

• Jonathan Sondow.
• Fast Algorithms and the FEE Method, E.A. Karatsuba (2005)
• Further formulae which make use of the constant: Gourdon and Sebah (2004).