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Questions tagged [euler-mascheroni-constant]

For questions related to Euler's constant $\gamma$, which is defined to be the limiting difference between the natural logarithm and the harmonic series.

Euler's constant, also called the Euler-Mascheroni constant and typically denoted $\gamma$, is defined to be the limiting difference between the natural logarithm and the harmonic numbers:

$$\gamma=\lim_{n \to \infty}H_n-\log n$$ where

$$H_n=1+\frac{1}{2}+\cdots+\frac{1}{n}$$

Euler's constant arises in analysis and number theory, in part due to its connections with the gamma and zeta functions.

Note this is not the same as Euler's number $e$, defined by $e:=\sum_{n=0}^{\infty}\frac{1}{n!}$. Questions about this number should use the tag .

Source: the Wikipedia article on the Euler-Mascheroni constant.

459 questions
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Evaluating $\int_{0}^{1}\cdots\int_{0}^{1}\bigl\{\frac{1}{x_{1}\cdots x_{n}}\bigr\}^{2}\:\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$

Here is my source of inspiration for this question. I suggest to evaluate the following new one. $$ I_{n}:= \int_0^1 \! \cdots \! \int_0^1 \left\{\frac{1}{x_1x_2 \cdots x_n}\right\}^{2} \:\mathrm{d}x_1\,\mathrm{d}\,x_2 \cdots \mathrm{d}x_n…
Olivier Oloa
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What is the fastest/most efficient algorithm for estimating Euler's Constant $\gamma$?

What is the fastest algorithm for estimating Euler's Constant $\gamma \approx0.57721$? Using the definition: $$\lim_{n\to\infty} \sum_{x=1}^{n}\frac{1}{x}-\log n=\gamma$$ I finally get $2$ decimal places of accuracy when $n\geq180$. The third…
Argon
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Showing that $\lim\limits_{n\to\infty}\sum^n_{k=1}\frac{1}{k}-\ln(n)=0.5772\ldots$

How to show that $$\lim_{n\to\infty}\left[\sum^n_{k=1}\frac{1}{k}-\ln(n)\right]=0.5772\ldots$$ No clue at all. Need help! Appreciated!
user67253
30
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7 answers

Simple proof Euler–Mascheroni $\gamma$ constant

I'm searching for a really simple and beautiful proof that the sequence below converges: $$(u_n)_{n \in \mathbb{N}} = \displaystyle\sum_{k=1}^n \frac{1}{k} - \log(n)$$ At first I want to know if my answer is OK. My try: $\lim\limits_{n\to\infty}…
Thomas Produit
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A closed form of the series $\sum_{n=1}^{\infty} \frac{H_n^2-(\gamma + \ln n)^2}{n}$

I have found a closed form of the following new series involving non-linear harmonic numbers. Proposition. $$\sum_{n=1}^{\infty} \dfrac{H_n^2-(\gamma + \ln n)^2}{n} = \dfrac{5}{3}\zeta(3)-\dfrac{2}{3}\gamma^3-2\gamma \gamma_{1}-\gamma_{2} $$…
Olivier Oloa
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Euler-Mascheroni constant in Bessel function integral

I am currently juggling some integrals. In a physics textbook, Chaikin-Lubensky [1], Chapter 6, (6.1.26), I came upon an integral that goes \begin{equation} \int_0^{1} \textrm{d} y\, \frac{1 - J_0(y)}{y} - \int_{1}^{\infty} \textrm{d} y\,…
Grob
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Intuitively, why is the Euler-Mascheroni constant near $\sqrt{1/3}$?

Questions that ask for "intuitive" reasons are admittedly subjective, but I suspect some people will find this interesting. Some time ago, I was struck by the coincidence that the Euler-Mascheroni constant $\gamma$ is close to the square root of…
idmercer
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Limit of Zeta function

I'm looking for a reference for (or an elementary proof of) $$ \lim_{s \rightarrow 1} \left( \zeta(s) - \frac{1}{s-1} \right) = \gamma$$ Thanks for your help.
Simon S
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Integral representation of Euler's constant

Prove that : $$ \gamma=-\int_0^{1}\ln \ln \left ( \frac{1}{x} \right) \ \mathrm{d}x.$$ where $\gamma$ is Euler's constant ($\gamma \approx 0.57721$). This integral was mentioned in Wikipedia as in Mathworld , but the solutions I've got uses…
user55386
19
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1 answer

A fractional part integral giving $\frac{F_{n-1}}{F_n}-\frac{(-1)^n}{F_n^2}\ln\left(\!\frac{F_{n+2}-F_n\gamma}{F_{n+1}-F_n\gamma}\right)$

I've been asked to elaborate on the following evaluation: $$ \begin{align}\\ \displaystyle {\large\int_0^{1}} \!\cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots + \cfrac 1 { 1 + \psi (\left\{1/x\right\}+1)}}}} \:\mathrm{d}x & =…
Olivier Oloa
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Derivative of Riemann zeta, is this inequality true?

Is the following inequality true? $$\gamma -\frac{\zeta ''(-2\;n)}{2 \zeta '(-2\;n)} > \log (n)-\gamma$$ This for $n$ a positive integer, $n=1,2,3,4,5,...$, and more precisely when $n$ approaches infinity. $\gamma$ is the Euler-Mascheroni constant,…
Mats Granvik
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Is this Euler-Mascheroni constant calculation from double integrals a true identity?

A prime number is a number that is only divisible by itself and one, that is the number of divisors of a prime number is equal to $2$. One way to illustrate this is to plot a matrix such that if the column index (1,2,3,...) divides the row index…
Mats Granvik
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Closed form of Euler-type sum over zeta functions $\sum _{k=2}^{\infty } \frac{\zeta (k)}{k^2}$?

Revisiting the question on the integral over the harmonic number I stumbled over the nice formula $$\sum_{k\ge2} (-1)^{k+1}\frac{\zeta(k)}{k} = \gamma\tag{1}$$ where $\zeta(z)$ is the Riemann zeta function and $\gamma$ is Euler's gamma. Searching…
Dr. Wolfgang Hintze
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Has Euler's Constant $\gamma$ been proven to be irrational?

I found a paper by Kaida Shi called "A Proof: Euler’s Constant γ is an Irrational Number" which claims to have proven the irrationality of $\gamma$. I know people have been trying to prove that $\gamma$ is irrational or not for hundreds of years. I…
Argon
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Prove that $\int _{-\infty }^{+\infty }{\frac {\mathrm {d} z}{(\phi ^{n}z)^{2}+(F_{2n+1}-\phi F_{2n})(e^{\gamma }z^{2}+\zeta (3)z-\pi )^2}}=1$

Recently while dealing with few interesting integrals, I was quite fascinated by this one: $$\int _{-\infty }^{+\infty }{\dfrac {1}{(\phi ^{n}z)^{2}+(F_{2n+1}-\phi F_{2n})(e^{\gamma }z^{2}+\zeta (3)z-\pi )^2}}\,\mathrm {d} z=1$$ where $\phi$…
BooleanCoder
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