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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.

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Is this connection of (increasingly exclusive) integer partitions to the the Euler-Mascheroni constant useful?

$\mathbf{SETUP}$ From this previous question, I quote Cauchy's formula for the number of all possible cycle types \begin{align} N_{\lambda} = \frac{n!} {1^{\alpha_1} 2^{\alpha_2} ... n^{\...
julianiacoponi's user avatar
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About the infinite product $\prod_{n=1}^{\infty}\left(2n-1\right)^{\frac{(-1)^n}{2n-1}}$

I have found a rare infinite product that neither Wolfram Alpha and Mathematica can't evaluate. The product in question: $$\prod_{n=1}^{\infty}\left(2n-1\right)^{\frac{(-1)^n}{2n-1}}=\frac{e^{\pi\...
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Weierstrass' definition of Gamma function and Euler-Mascheroni constant

Recently came to know about Weierstrass' definition of Gamma function $\Gamma(z)$ and Euler-Mascheroni constant $\gamma$ $$\frac{1}{\Gamma(z)} = z e^{\gamma z} \prod_{n=1}^\infty \left( \left(1+ \frac{...
Jayadrata Banerjee's user avatar
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Gamma function as infinite product

Let´s consider the Gamma function expression due to Euler and Gauss: $$\Gamma(z) = \frac{1}{ze^{\gamma}\prod_{n=1}^{\infty} (1+\frac{z}{n})e^{\frac{-z}{n}}} $$ I am interested in showing that $\...
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Proof of $K_0(z)=-\left(\log\frac{z}{2}+\gamma\right)I_0(z)+\sum_{n=1}^\infty \frac{(z/2)^{2n}}{n!^2}H_n$

Let $I_{\nu}$ be the Bessel I function of order $\nu$ defined by $$I_{\nu}(z)=\sum_{n=0}^\infty \frac{(z/2)^{2n+\nu}}{n!\Gamma (n+\nu+1)}$$ and let $K_{0}$ be the Bessel K function of order $0$ ...
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Mathematica's algorithm for computing $\gamma$

Please note that this question is not a duplicate of What is the fastest/most efficient algorithm for estimating Euler's Constant $\gamma$? I was looking for algorithms for computing $\gamma$ (...
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Connection between the sawtooth Fourier series and the Euler-Mascheroni constant?

I've been studying the slope of the sawtooth waveform $$\sum_{n=1}^\infty \frac{sin(nx)}{x}$$ And found that, for $x \in (0, 2\pi)$, its graph seems to line up perfectly with the line $$y = \frac{x}{2}...
Alexandra's user avatar
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approximation to digamma function

I was learning about the harmonic series back in college to which the professor said "There is no known closed form for the harmonic sum", I felt that was strange given that the sum didn't ...
Motez Ouledissa's user avatar
6 votes
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Sum of over primes $p$ of $x^{-p}$

I was playing around with the following series $$S(x) = \frac{1}{x^2}+\frac{1}{x^3}+\frac{1}{x^5}+\frac{1}{x^7}+\frac{1}{x^{11}}+...=\sum_{p\in primes}\frac{1}{x^p}$$ for $x\in\mathbb{R}$ and $|x|>...
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Asymptotic expansion gives Euler-Mascheroni constant

Mathematica gives the following asymptotic expansion: $$ \int_0^\Lambda \frac{1-\cos(qx)}{q} \mathrm{d}q \overset{\Lambda\to\infty}{\sim}\gamma+\log\left(x\Lambda\right) +\mathcal{O}\left(\frac{1}{x\...
xzd209's user avatar
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A problem with Mertens Theorem

According to Mertens Theorem $$ \lim_{n \to \infty } (\frac{1}{ln(p_n)} \prod_{k=1}^n \frac{1}{1-\frac{1}{p_k}})=1.781072\dots$$ so we can say $$ 1.78 \ ln(p_n) \sim \prod_{k=1}^n \frac{1}{1-\frac{1}{...
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Where can I find the first proof of $\int_0^1H_xdx=\gamma$

We can prove $$\int_0^1H_xdx=\gamma$$ as such:$$\int_0^1H_xdx=\int_0^1\sum_{k=1}^\infty\left(\frac1k-\frac1{k+x}\right)dx=\sum_{k=1}^\infty\left(\frac1k-\ln\left(1+\frac1k\right)\right)=\gamma$$ where ...
Kamal Saleh's user avatar
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What series approaches to $\log(\log(\log(n)))$?

Background: Harmonic series approaches to $\lim_{n->\infty}\log(n)$. This gives the Euler–Mascheroni constant. In addition, the harmonic series summed only over the primes approaches to $\lim_{n-&...
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Generalizing this infinite series for Euler-Mascheroni constant.

Define $\gamma(t)$ as $$\gamma(t) = \sum_{n=1}^\infty \sum_{k=2^n}^\infty \frac{(-1)^k}{k+t}. $$ We have $\gamma(0)=\gamma = 0.577....$ I want to find a closed form when $t>0$ is an integer. I ...
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How to calculate convergence of a series related to the Euler's constant?

I understand that $$ \sum_{r=1}^n \frac{1}{r} \approx \int_{1}^n \frac{dx}{x} = \log n $$ I would like to verify whether the following holds true as well, for a series that is related to the above ...
Shatarupa18's user avatar
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Asymptotic analysis of $e^{H_n}$

To be precise, what is the asymptotic behavior of $e^{H_n}$, as $n$ tends to infinity, where $e$ is the Euler's number, a mathematical constant approximately equal to $2.71828$ and $H_n$ is the $n$-th ...
Ok-Virus2237's user avatar
4 votes
2 answers
362 views

Criteria for irrationality of Euler's constant

Define for $n\in\mathbb{N}$, $$I_n=\int_0^1\int_0^1 -\frac{(x(1-x)y(1-y))^n}{(1-xy)\log xy}dx dy$$ In this article it is proved that $$I_n=\binom{2n}{n}\gamma+L_n-A_n$$ where $L_n=d^{-1}_{2n}\log S_n$,...
Max's user avatar
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Integral of $\operatorname{Si}(x)/(x(x^2+1))$ from $0$ to infinity

Evaluate $$\int_0^\infty \frac{\operatorname{Si}(x)}{x(x^2+1)}\,dx.$$ I use the Feynman’s technique. Let $$I(a)=\int_0^\infty \frac{\operatorname{Si}(ax)}{x(x^2+1)}\,dx$$ then $I'(a)=\frac{\pi}{2a} ...
Integral's user avatar
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1 answer
139 views

Riemann Zeta function's behavior near the pole and Euler-Mascheroni constant

I want to show that $$\zeta(z) = \frac{1}{z-1} + \gamma + O(z-1)$$ for $z \rightarrow 1 $. Why is it enough to show this statement for real $z = s \in (1,2)$? We know that $\zeta$ is meromorphic and ...
hteica's user avatar
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2 votes
3 answers
202 views

Upper bound euler constant

Trying to show that the euler constant $\gamma \leq \frac{5}{6}$ . I'm stuck in this, I was able to bound it with 1 after manipulation with integrals but it doesn't good enough. then I tried this $\...
3xhaust's user avatar
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1 answer
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Euler's constant conundrum

It is known that $$\Gamma'(1)=-\gamma$$where $\Gamma(x)=(x-1)!$ at integer $x$ and $\gamma$ is the Euler-Mascheroni constant. We can redefine this as $$\Pi'(0)=-\gamma$$Where $\Pi(x)=x!$. By L'hopital'...
Kamal Saleh's user avatar
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2 votes
1 answer
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How to calculate $\lim\limits_{y\to+\infty}\left(\int_0^y\frac{\sin^4 x}{x}{\rm d}x-\frac38\ln y\right)$

How to calculate $$\lim_{y\rightarrow +\infty}\left( \int_0^y \frac {\sin^4(x)}{x} \mathrm{d}x - \frac38 \ln y\right).$$ In my view,there is $$\int_0^y \frac{\sin^4 x}{x} \mathrm{d} x =\int_0^y\frac{ \...
Mr.He's user avatar
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Two curious related sums on $\gamma$ and $\ln2$

Given these two curious formulas: How may you show that? $$\sum_{n=1}^{\infty}\frac{(2n-1)\zeta(2n+1)}{2^{2n+1}}\left[2n\left(1-\frac{1}{2^{2n}}\right)-\frac{(2n)^2+1}{(2n)^2-1}\right]=\gamma\tag1$$ $$...
Sibawayh's user avatar
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3 votes
1 answer
152 views

Does this mean that Euler's constant diverges?

I was playing around with the Euler-Mascheroni constant and the constant $\gamma$ is defined as the limit below. $$ \gamma = \lim_{n\to\infty} \sum_{k=1}^n \left(\frac{1}{k}\right) - \log{n} $$ It can ...
iwjueph94rgytbhr's user avatar
4 votes
1 answer
174 views

How can I evaluate this infinite sum?

I was tackling an integral I saw on the Maths 505 YouTube channel and I came across this sum. $$\sum_{n=1}^\infty(\frac{1}{2n}+1-(n+1)\ln(1+\frac{1}{n}))$$ I plugged it into Wolfram Alpha and it gave ...
Artur Stolf's user avatar
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0 answers
243 views

Evaluate the definite integral $\int_0^\infty \frac{W{(x)}}{xe^x}\,dx$

It is a well known identity that $$ \int_0^\infty \frac{\log{(x)}}{e^x}\,d{x} = -\gamma $$ This shows how the Euler–Mascheroni constant is directly connected to the exponential function and its ...
Peder's user avatar
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1 answer
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Creating an inequality for the numerator significance

An open mathematical problem is if the Euler-Mascheroni constant is irrational and if so transcendental. There has been some progress in (dis)proving this. It is known that: $\gamma\in\mathbb{Q}\...
Kamal Saleh's user avatar
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1 vote
2 answers
164 views

How to show that $\frac{d}{dx}\sum_{n=1}^\infty\left(\frac xn-\ln\left(1+\frac xn\right)\right)=H_x$

So I was trying to find a series expansion of $\Gamma(x+1)$ (which is the analytic continuation of $x!$ when I bumped into this sum $$\lambda(x)=\sum_{n=1}^\infty\left(\frac xn-\ln\left(1+\frac xn\...
Kamal Saleh's user avatar
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1 vote
1 answer
149 views

Expressions for $\sum_{k=1}^\infty\frac{(-1)^k}{k\cdot k!}$

It is known that $$\delta=-e\left(\gamma+\sum_{k=1}^\infty\frac{(-1)^k}{k\cdot k!}\right)$$Where $\delta$ is the Euler-Gompertz constant and $\gamma$ is the Euler-Mascheroni constant. I want to find ...
Kamal Saleh's user avatar
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3 votes
2 answers
123 views

Which functions besides $\ln{x}$ make $ \lim \limits_{N \to \infty} \sum_{n=1}^N f'(n) -f(N) $ converge?

Using a very hand-wavy argument, I convinced myself that if, instead of $f(x)=\ln{x}$, we let $f(x)=\sqrt{x}$, we should still get something finite and small. Wasn't really sure where to start to ...
LeiMagnus's user avatar
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1 answer
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Other closed forms of $\lim_{k\rightarrow\infty}\left(\sum_{i=0}^k\frac{1}{2i+1}-\sum_{i=1}^k\frac{1}{2i}\right)$

It is known that $$\lim_{k\rightarrow\infty}\left(\sum_{i=0}^k\frac{1}{2i+1}-\sum_{i=1}^k\frac{1}{2i}\right)=\beta$$converges. I wonder if there are any other closed forms for this limit. At first I ...
Kamal Saleh's user avatar
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6 votes
1 answer
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Another strange limit which seems to converges to $\gamma$

Hi in trying to show that the Euler Mascheroni constant is irrational or not I find empiricaly : $$\lim_{n\to\infty,x\to 0}f_n(x)=\lim_{n\to\infty,x\to 0}\frac{x!!!...!^{x!!...!^{x!...!^{...^{x!}}}}-x!...
Miss and Mister cassoulet char's user avatar
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1 answer
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Given that $\lim_{z\rightarrow0}az-\frac{\ln z}{z!}=\gamma$, find the value of $a$ that gives the quickest convergence

You could prove that $$\lim_{z\rightarrow0}az-\frac{\ln (z!)}{z}=\gamma$$For any $a\in\mathbb{R}$. Where $z!$ is the factorial extended to all real numbers (except negative integers). Which value ...
Kamal Saleh's user avatar
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0 answers
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Sum versus product on Primes

Disclaimer: I had a hard time choosing the title. Hello MSE! So in this post I was wondering why $e^\gamma$ has importance in number theory while $\gamma$ and $e$ don't. But I have another question ...
Kamal Saleh's user avatar
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1 vote
2 answers
98 views

Is it always true that $\sum\limits_{j=1}^n\frac{1}{j} > \int_1^{n+1} \frac{dt}t$?

This inequality was presented as obvious in this answer. I was interested in understanding the details for this comparison so I started reading up the Euler-MacLaurin formula but this actually ...
Larry Freeman's user avatar
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0 answers
59 views

Is this substitution valid?

So I made this identity for Euler's constant $\gamma$:$$\int_{-\infty}^\infty e^{-(e^t+t)}tdt=-\gamma$$But this isn't right for some reason. So what's wrong with my proof?: $$\int_0^\infty e^{-t}\ln ...
Kamal Saleh's user avatar
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2 votes
1 answer
201 views

Euler's constant's role in the Prime numbers

I know that: $$\lim_{k\rightarrow\infty}\frac{1}{\log p_n}\prod_{i=1}^k\frac{p_i}{p_i-1}=e^\gamma$$Where $p_n$ is the $n$th prime and $\gamma$ is Euler's constant. But why does this make the number $\...
Kamal Saleh's user avatar
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4 votes
1 answer
110 views

Prove that the limit converges to $\gamma$

Let $$H(x)=\int_0^1\frac{t^x-1}{t-1}dt$$be the harmonic series and let $$s(x)=\int^\infty_0e^{-t}\ln(t+x)dt$$How do I prove that their difference converges? It seems to me that they approach $\gamma$ (...
Kamal Saleh's user avatar
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6 votes
3 answers
519 views

Proof of $\int_0^1\ln\left(\frac{1}{\ln \frac{1}{x}}\right)dx=\gamma$

After solving this problem I found out that: $$\int_0^1\ln^sxdx=(-1)^s\Gamma(s+1)\tag{1}$$Where $\Gamma(s)$ is the gamma function, defined as $(s-1)!$ when $s\in\mathbb{Z}$. $(1)$ can be proven by ...
Kamal Saleh's user avatar
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1 vote
1 answer
36 views

How to prove that $\lim_{k\rightarrow\infty} H_k-\log_b(x)$ converges/diverges?

It is known that: $$\lim_{k\rightarrow\infty} H_k-\log(k)=\gamma$$Where $\gamma$ is Euler's constant. What if we change the base from $e$ to any other number $b$? How do we prove that this converges/...
Kamal Saleh's user avatar
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-1 votes
4 answers
198 views

Approximations of Euler's constant using $\pi$

Disclaimer: This is for recreational purposes. Hello MSE! So while my research paper about Euler's constant $\gamma$, I created this amazing approximation for it: $$\frac{\pi^2}{12000}-\ln((10^{-3})!)*...
Kamal Saleh's user avatar
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6 votes
1 answer
207 views

Integrals of the form $\int_0^1\left(\frac{1}{x+1}-\ln\left|\frac{x+1}{2}\right|-\ln\ln\left|\frac{x+1}{x-1}\right|\right)dx$

How could we evaluate the following integrals? (Integrands pictures above) Blue: \begin{align} \int_0^1 \left(\frac{1}{x+1}-\ln\left|\frac{x+1}{2}\right|-\ln\ln\left|\frac{x+1}{x-1}\right|\right)dx &...
tyobrien's user avatar
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2 votes
1 answer
69 views

Prove that $\lim_{x\rightarrow\infty}\sum_{k=0}^x\frac{1}{2k+1}-\sum_{k=1}^x\frac{1}{2k}$ converges

I thought up of the limit in the title by playing around with the limit definition of Euler's constant. I think I was able to prove that it is convergent. Is the following proof correct? Let $H_e(x)=\...
Kamal Saleh's user avatar
  • 6,520
2 votes
1 answer
58 views

How to prove that $\lim_{k\rightarrow\infty}\ln k - \frac{1}{2}H(k)$ diverges?

It is known that $$\lim_{k\rightarrow\infty}\ln k - H(k)=\gamma$$(Where $H(k)$ is the $k$th harmonic number and $\gamma$ is Euler's constant). I decided to put a random $\frac{1}{2}$ in the front of $...
Kamal Saleh's user avatar
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1 vote
1 answer
101 views

Where does Euler write the definition $\gamma=\sum^\infty_{k=1}\left(\frac1k-\ln\left(1+\frac1k\right)\right)$ of the Euler-Mascheroni Constant?

I am writing a research paper on the Euler-Mascheroni Constant when I mentioned this sum: $$\gamma=\sum^\infty_{k=1}\left(\frac{1}{k}-\ln\left(1+\frac{1}{k}\right)\right)$$ which was derived by Euler. ...
Kamal Saleh's user avatar
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-2 votes
1 answer
62 views

An interesting integral about γ [closed]

It is well known that $$\gamma = \lim_{n \to +\infty} (H_n − \ln(n)) = 0.5772(...)$$ where $H_n$ is the sum of reciprocals of all integers from $1$ to $n$. Prove that $$\int_1^\infty \frac{\{ x\}}{x^...
Pulkit Sabharwal's user avatar
6 votes
4 answers
545 views

Curiosities of the function $Q(x)=\sum_{n=1}^\infty \frac{P_n(x)}{n(2n+1)}$ where $P_n(x)$ is a sequence of all polynomials with unit coefficients

Backround: I have been studying the peculiar function $$Q(x)=\sum_{n=1}^\infty \frac{P_n(x)}{n(2n+1)}$$ where $P_n(x)$ is the set of all polynomials with unit coefficients, defined by the binary ...
Graviton's user avatar
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1 vote
0 answers
24 views

Construction, similar to Chow's EL-numbers? Is it valid? What are the properties?

The idea of EL-numbers, proposed by Chow, impressed me very much, so I decided to build something similar and look what this will turn out. Instead of $\exp(x)$ and $\ln(x)$ functions as the building ...
Anixx's user avatar
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1 vote
0 answers
54 views

A constant written in terms of non-elementary functions

I found out that: $$\frac{\int \Pi(x)H(x)dx-\Pi(x)}{\int \Pi(x)dx}=\gamma$$ Where $H(x)$ is the $x$th harmonic number, $\Pi(x)$ is the analytic continuation of $z!$, and $\gamma$ is the Euler-...
Kamal Saleh's user avatar
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1 vote
1 answer
112 views

Why rate of convergence is studied?

There are lots of papers on the Internet about sequences with different rates of convergence towards Euler's constant and every year more are published for better rates of convergence. Why is having ...
AnnaLena's user avatar

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