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

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47
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11answers
3k views

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 ...
44
votes
2answers
1k views

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} ...
18
votes
1answer
395 views

Infinite Series $\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\left\lfloor\frac{\log(n)}{\log(2)}\right\rfloor$

How to prove that $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\left\lfloor\frac{\log(n)}{\log(2)}\right\rfloor=\gamma$$ Can we find a known value for ...
17
votes
1answer
510 views

A closed form for the series $\sum_{n=1}^{\infty} \frac{H_n^2-(\gamma + \ln n)^2}{n}$

I have found a closed form for the following new series involving non-linear harmonic numbers. Proposition. $$\sum_{n=1}^{\infty} \dfrac{H_n^2-(\gamma + \ln n)^2}{n} = ...
15
votes
2answers
507 views

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 ...
14
votes
1answer
894 views

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 ...
13
votes
1answer
264 views

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 ...
12
votes
2answers
160 views

Showing $\gamma < \sqrt{1/3}$ without a computer

In 1735 Euler gave the value of $\gamma$ as $0.577218.$ The constant is generally defined as the limit of the difference between the harmonic series and $\log n:~\gamma= ...
12
votes
1answer
393 views

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 ...
11
votes
3answers
1k views

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 ...
11
votes
3answers
264 views

Possible new definition of Gamma (Euler-Mascheroni Constant): $\lim_{x \to 0} (-\ln ( \sqrt[x]{x!} )) = \gamma$

I think I've discovered a new definition for the Euler-Mascheroni Constant (Gamma) I can't find it online anywhere, has anyone seen it before? $$\lim_{x \to 0} (-\ln ( \sqrt[x]{x!} )) = \gamma$$
10
votes
3answers
927 views

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.
10
votes
8answers
788 views

“How I wish I could calculate pi” analogs…

You might know the mnemonic for $\pi$ in the title or even this more elaborated one: Sir, I bear a rhyme excelling In mystic force, and magic spelling Celestial sprites elucidate All my own ...
9
votes
2answers
2k views

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 ...
9
votes
2answers
293 views

Why does $\gamma=\lim_{s\to1^+}\sum_{n=1}^{\infty}\left(\frac{1}{n^s}-\frac{1}{s^n}\right)=\lim_{s\to0}\frac{\zeta(1+s)+\zeta(1-s)}{2}$?

To be clear, I'm having trouble with proving both equalities, and would appreciate a hint. I'm also not sure why $1^+$ must be used as opposed to $1^-$. I'm not sure about the definition of $\zeta(x), ...
8
votes
2answers
131 views

Why is $-\gamma = \int_0^1 \frac{e^{-z}-1}{z}dz+\int_1^\infty \frac{e^{-z}}{z}dz$

It seems like the sum of the two RHS integrals is "well known"$^\dagger$ to be Euler's constant: $$\gamma \equiv \int_1^\infty \frac{1}{\lfloor z\rfloor} - \frac{1}{z}dz \quad\stackrel{?}{=}\quad ...
8
votes
1answer
78 views

Evaluate $\lim_{R\to\infty}\left(\int_0^R\left|\frac{\sin x}{x}\right|dx-\frac{2}{\pi}\log R\right)$

Is there a closed form of $$\lim_{R\to\infty}\left(\int_0^R\left|\frac{\sin x}{x}\right|dx-\frac{2}{\pi}\log R\right)$$ I am pretty interested whether we can find out a closed form of this limit. We ...
7
votes
1answer
108 views

Integrating:$ \displaystyle \int_0 ^ {\infty}e^{-x^2}\ln(x)dx $

$$ \displaystyle \int_0 ^ {\infty}e^{-x^2}\ln(x)dx = -\frac{1}{4}(\gamma +2\ln(2))\sqrt{\pi} $$ This is a well known integral. But I want to know how to solve it?? Also, please refrain using contour ...
6
votes
6answers
436 views

Finding $\sum_{k=1}^{\infty} \left[\frac{1}{2k}-\log \left(1+\frac{1}{2k}\right)\right]$

How do we find $$S=\sum_{k=1}^{\infty} \left[\frac{1}{2k} -\log\left(1+\dfrac{1}{2k}\right)\right]$$ I know that $\displaystyle\sum_{k=1}^{\infty} \left[\frac{1}{k} ...
6
votes
2answers
545 views

Equality with Euler–Mascheroni constant

While trying to prove integral with exponential function and logarithm in an alternative way, I came to this solution: $$\sum_{k=0}^{+\infty}(-1)^{k+1}\frac{\log (k+1)+\gamma }{(k+1)}.$$ As both ...
5
votes
3answers
550 views

Euler-Mascheroni constant expression, further simplification

The Euler-Mascheroni constant gamma is defined as: $$\gamma=\lim\limits_{n \rightarrow \infty}\left(\sum\limits_{m=1}^{n} \frac{1}{m} - \log(n)\right)$$ From this previous question Do these series ...
5
votes
4answers
247 views

How fundamental is Euler's identity, really?

Euler's identity, obviously, states that $e^{i \pi} = -1$, deriving from the fact that $e^{ix} = \cos(x) + i \sin(x)$. The trouble I'm having is that that second equation seems to be more of a ...
5
votes
2answers
125 views

Integral $\int_0^{\infty} \log(x) e^{-x^2} \mathrm{d}x = -\frac{1}{4}\sqrt{\pi} (\gamma + \log(4)).$

While trying to compute the expected value $E[\log(X)]$ for a normally distributed variable $X$ I found the following integral $$\int_{0}^{\infty}\log\left(x\right) {\rm e}^{-x^{2}}\,{\rm d}x =-\,{1 ...
5
votes
1answer
170 views

Define integral for $\gamma,\zeta(i) i\in\mathbb{N}$ and Stirling numbers of the first kind

Consider the integral $$\int\limits_0^{\infty}e^{-x}x^k\ln(x)^n\dfrac{dx}x$$ For $n=3$ we have ...
5
votes
1answer
119 views

Asymptotics of coefficients $[x^n] \frac{1}{\Gamma(1+x)}$ as $n$ is great

I am interested in the behaviour, as $n$ is great, of the coefficients $g_n$ in the Maclauren expansion of $\displaystyle \frac{1}{\Gamma(1+x)} $. We have $$ \frac{1}{\Gamma(1+x)}=\sum_{n=0}^\infty ...
5
votes
2answers
88 views

Characterizations of $e^x$

I've been thinking about the following problem for a while: (AFAIK) the 'exponential function', $e^x$ can be characterized as the unique solution to the following differential equation with initial ...
4
votes
4answers
209 views

Compute $\sum (1+\frac12+\dotsb+\frac1n-\ln (n+\frac12)-\gamma)$

Compute $$\sum_{n=1}^\infty \Big(1+\frac12+\dotsb+\frac1n-\ln (n+\frac12)-\gamma\Big)$$ where $\gamma$ is Euler's constant It seems to be difficult, I have no idea go get started Thank you very ...
4
votes
1answer
5k views

Simple proof Euler–Mascheroni $\gamma$ constant

I'm searching for a really simple and beautiful proof that the sequence $(u_n)_{n \in \mathbb{N}} = \sum\nolimits_{k=1}^n \frac{1}{k} - \log(n)$ converges. At first I want to know if my answer is OK. ...
4
votes
6answers
214 views

Graph of the function $x^y = y^x$, and $e$ (Euler's number).

Earlier, I was using the Desmos Graphing Calculator, and I wanted to remind myself of what the graph of the function $x^y = y^x$ looked like. If you have never seen what it looks like before, it is ...
4
votes
2answers
403 views

Elementary derivation of certian identites related to the Riemannian Zeta function and the Euler-Mascheroni Constant

Is the proof of these identities possible, only using elementary differential and integral calculus? If it is, can anyone direct me to the proofs? ( or give a hint for the solution ) ...
4
votes
1answer
89 views

What is known about the 'Double log Eulers constant', $\lim_{n \to \infty}{\sum_{k=2}^n\frac{1}{k\ln{k}}-\ln\ln{n}}$?

The Euler constant is defined as $$\gamma = \lim_{n \to \infty}{\sum_{k=1}^n\frac{1}{k}-\ln{n}}$$ Let $$q = \lim_{n \to \infty}{\sum_{k=2}^n\frac{1}{k\ln{k}}-\ln\ln{n}}$$ I managed to prove that ...
4
votes
2answers
105 views

Evaluate the double integral $\int _0^1\int _0^1\frac{x+i}{(1-ix y) \ln (x y)} \,dx\,dy$

We know that $$\int _0^1\int _0^1\frac{x-1}{(1+x y) \ln (x y)} \, dx\,dy=\gamma$$ $$\int _0^1\int _0^1\frac{x+1}{(1-x y) \ln (x y)}\,dx\,dy=\ln \frac4\pi$$ I wonder what would be $$\int ...
4
votes
1answer
51 views

How to prove this continued fraction connection between $\gamma$ and $e$?

There is apparently a curious connection between Euler-Mascheroni constant $\gamma$ and $e$ in the form of an infinite series and continued fraction: $$e \gamma=e \sum_{n=1}^{\infty} ...
4
votes
1answer
74 views

How to prove this series: $\displaystyle \sum_{n=1}^{\infty }\frac{\left ( -1 \right )^{n}\ln n}{n}=\gamma \ln 2-\frac{1}{2}\ln^22$ [duplicate]

How to prove this series $$\sum_{n=1}^{\infty }\frac{\left ( -1 \right )^{n}\ln n}{n}=\gamma \ln 2-\frac{1}{2}\ln^22$$ and \begin{align*} \sum_{n=1}^{\infty }\frac{\left ( -1 \right )^{n}\ln \left ( ...
3
votes
2answers
191 views

Need to show that $\lim_{x\to\infty}\left(\sum_{n\le x}^{}\frac{1}{n}-\ln x \right)$ exist and is less than $1$ [duplicate]

Need some help here. I need prove that the following limit exist and is less than $1$ $$\lim_{x\to\infty}\left(\sum_{n\le x}^{}\frac{1}{n}-\ln x\right)$$ I feel a little lost here, this is my first ...
3
votes
2answers
118 views

Unusual integral

I got a clock as a gift recently. It has a very novel face in that the hour positions are given by a complex formula. For the most part, I have been able to verify the calculations presented as ...
3
votes
1answer
81 views

Why Can my Phone Calculator do $e^{\pi\sqrt{-1}}$ but not $\sqrt{-1}$?

When I type in the identity $e^{\pi\sqrt{-1}}$ on my phone calculator (LG phone running Android), I get the correct result of $-1$ However, when I simply type $\sqrt{-1}$, it returns an error. Why ...
3
votes
1answer
135 views

Euler Mascheroni Constant is Zero

$$ H_n - \ln n = \int_0^1 \frac{1 - x^n}{1 - x} dx - \int_1^n \frac{dy}{y} $$ Let $x = \frac{y - 1}{n - 1}$, or $y = (n-1)x + 1$. Then, $dy = (n - 1) dx$. $$ H_n - \ln n = \int_0^1 \frac{1 - x^n}{1 ...
3
votes
1answer
91 views

Infinite product: $(1-0.5^2)(1-0.5^3)(1-0.5^4)…$

Find a closed form for the value of the infinite product $(1-0.5^2)(1-0.5^3)(1-0.5^4)...$ I know it converges. At first I thought it was the Euler–Mascheroni constant, but it's only accurate to about ...
3
votes
2answers
103 views

How to prove this limit about $\gamma=\lim_{N\to \infty }\left(\sum_{n=1}^N\frac{1}{n}-\ln N\right)$

I have no idea how to prove it. $$\lim_{m\rightarrow \infty }\left [ -\frac{1}{2m}+\ln\left ( \frac{e}{m} \right )+\sum_{n=2}^{m}\left ( \frac{1}{n}-\frac{\zeta \left ( 1-n \right )}{m^{n}} \right ) ...
3
votes
2answers
39 views

How do we bound the function $\frac{-1}{2}\sum_{m=n}^{\infty}\frac{1}{m^{2}}$?

In case the question didn't display in the title correctly: How do we bound the function $\frac{-1}{2}\sum_{m=n}^{\infty}\frac{1}{m^{2}}$? I think a way I can do this is to show that ...
3
votes
1answer
74 views

Does Euler-Mascheroni constant belong to the ring of periods?

I wonder whether $\gamma$ belongs to the ring of periods? UPDATE Well now I know it should not. But $e^{-\gamma}$ should.
2
votes
3answers
181 views

Is it possible to intuitively explain, how the three irrational numbers $e$, $i$ and $\pi$ are related?

I read a bit about this equation: $e^{i\pi}=-1$ For someone knowing high school maths this perplexes me. How are these three irrational numbers so seemingly smoothly related to one another? Can this ...
2
votes
2answers
257 views

Why is $1 + \frac{1}{2} + \frac{1}{3} + … + \frac{1}{n} \approx \ln(n) + \gamma$?

On StackExchange, I read that the harmonic series up to $\frac{1}{n}$ is approximately $\ln(n) + \gamma$, where $\gamma$ is the Euler-Mascheroni constant, which is close to $0.5772$. When I researched ...
2
votes
2answers
245 views

Euler-Mascheroni constant: understanding why $\lim_{m\rightarrow \infty} \sum_{n=1}^{m} (\ln (1 + \frac{1}{n})-\frac{1}{n+1})= 1 - \gamma$

I am trying to understand why the Euler-Mascheroni constant $\displaystyle \gamma = \lim_{n \rightarrow \infty} \left ( \sum_{k=1}^n \frac{1}{k} - \ln n \right )$ is equal to $1 - \displaystyle ...
2
votes
1answer
69 views

Does $ \lim_{n \to \infty} \frac{\operatorname{exp}(H_n)}{n+1} $ exist? [closed]

Question: Does $ \lim_{n \to \infty} \frac{\operatorname{exp}(H_n)}{n+1} $ exist? If so, what is its value? I know that the answer to the second part is $e^\gamma$, where $\gamma$ is the ...
2
votes
1answer
126 views

Pole of Riemann zeta and Riemann zeta zeros, prove this relation.

Prove this relation: $$\displaystyle \lim_{s\to 1} \, \left(\zeta (s)-\frac{\zeta '(s-1+\rho _n)}{\zeta \left(s-1+\rho _n\right)}\right)=\gamma -\frac{\zeta ''(\rho _n)}{2 \zeta '(\rho ...
2
votes
2answers
60 views

Zeta Function $\zeta(1\pm1/n)$ and Euler's constant.

How do I show that $$\lim_{n\to\infty}{\zeta(1+1/n)+\zeta(1-1/n)}=2\gamma$$ and $$\lim_{n\to\infty}{\zeta(1+1/n)-\zeta(1-1/n)}=\infty,$$ where $\gamma$ is the Euler's constant?
2
votes
1answer
31 views

Approximating end behavior of a function by plugging in infinity

In Algebra 2, I learned to be able to tell if the end behavior of a function has an asymptote, approaches infinity, approaches zero, etc, by plugging in numbers closer and closer to infinity, or by ...
2
votes
2answers
105 views

Proving $\lim_{n\to\infty}\sum_{k=n}^{\infty}\Big(\frac{1}{n\left\lfloor\frac kn\right\rfloor}-\frac1k\Big)=\gamma$

I remember arriving at the following equality: $$\lim_{n\to\infty}\sum_{k=n}^{\infty}\left(\frac{1}{n\left\lfloor\frac kn\right\rfloor}-\frac1k\right)=\gamma$$ where $\gamma$ denotes the ...