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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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. ...
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An interesting integral about γ

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^...
5 votes
4 answers
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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 ...
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1 vote
0 answers
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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 ...
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1 vote
0 answers
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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-...
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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 ...
1 vote
1 answer
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How did Euler calculate his $\gamma$ constant to 16 digits? [closed]

I couldn't find any information online on how Euler calculated this to 16 digits of accuracy: $$ \gamma = - \ln{n} + \sum_{k=1}^\infty \frac1k $$ Obviously not like that, can somebody help please?
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Prove limit is the Euler-Mascheroni constant $\gamma$

How do I prove this result? $$\lim_{\varepsilon\to0^+}\left[\operatorname{li}(1-\varepsilon)-(1-\varepsilon)\ln\ln\frac{1}{1-\varepsilon}\right]=\gamma$$with$$\operatorname{li}(x):=\int_0^x\frac{dt}{\...
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Identity for the Euler-Mascheroni constant

Euler created the following expression for the Euler-Mascheroni constant: $$\gamma=\sum^\infty_{k=1} \left(\frac{1}{k}-\ln\left(1+\frac{1}{k} \right)\right)$$ This converges more rapidly than the ...
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1 answer
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How do I prove this Euler-Mascheroni limit?

There is a number called the Euler-Mascheroni constant that is defined as the limiting difference between the harmonic series and the natural logarithm. In other words: $$ \gamma = \lim_{N \to \infty}\...
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4 votes
1 answer
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Deriving $\gamma \approx H(n)-\ln(n+1)+\frac{1}{2(n+1)}+\frac{1}{12(n+1)^2}$

The Euler-Mascheroni constant can be represented geometrically by the infinite sum of the areas in blue in the following picture, which is the area between the curve $y=1/x$ and the harmonic numbers. ...
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Show without a computer that : $\frac{\gamma}{2-\gamma}\pi-\ln(3+\gamma)<0$ where $\gamma$ is the Euler-Mascheroni constant

Challenging problem : Show that : $$\frac{\gamma}{2-\gamma}\pi-\ln(3+\gamma)<0$$ I come up with this inequality with What is the limit $\lim_{x\to 0}\left(\frac{x!x!!!x!!!!!...}{x!!x!!!!x!!!!!!...}\...
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3 votes
1 answer
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Euler-Mascheroni Constant and the Digamma Function

I have been trying to prove without success that the digamma function evaluated at $1$ is equal to the Euler-Mascheroni constant $\gamma$. I would like to know if there is any way of doing it by just ...
1 vote
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Is there any intuition of why the both, regularized logarithm of zero is $-\gamma$ and the regularized logarithm of Bernoulli umbra is $-\gamma$?

If we take the MacLaurin series for $\ln(x+1)$ and evaluate it at $x=-1$, we will get the Harmonic series with the opposite sign: $-\sum_{k=1}^\infty \frac1x$. Since the regularized sum of the ...
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5 votes
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Elementary proof for $\varphi(n)\ge\frac n{2\log(\log(n))}$

Trying to derive an elementary bound for Euler totient function $\varphi(n)$ to be $\mathcal{O}(\frac n{\log(\log(n))})$, I thought to prove a weak version of the well-known inequality $${\...
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A series on Stieltjes constants, $\sum_{n=0}^\infty \frac{\gamma_n}{n!}$, where is the mistake?

Stieltjes constants are defined as: $$\gamma_n=\lim_{m\to\infty}\left[ \left( \sum_{k=1}^m\frac{(\ln k)^n}{k}\right)-\frac{(\ln m)^{n+1}}{n+1} \right]$$ I want to compute the following series: $$\...
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21 votes
5 answers
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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)}{...
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How to find an equivalent of $\int_{1}^{+\infty}\exp{(-x^n)}dx$?

It's an exercise, I need to find an equivalent of : $$\int_{1}^{+\infty}\exp{(-x^n)}dx$$ I tried this : Let $x=u^{1/n}.$ Then $dx=\frac{1}{n}u^{1/n-1}\,du,$ so : \begin{align} \int_1^{+\infty} e^{-x^n}...
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1 answer
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Generalization of $\gamma$ as limit involving primes

The following limit is utilized in Merten’s theorem. $$ \gamma = \lim\limits_{n\to\infty}\left(-\log\log n - \sum_{p\text{ prime}}^n\log\left(1-\frac{1}{p}\right)\right) $$ I’m interested in the ...
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1 vote
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Evaluation of $\sum_{n=1}^{\infty}\frac{1}{n(n!)}$

WolframAlpha gives the evaluation as $\text{Ei}(1)-\gamma$. It does not offer a step-by-step solution. Where does this come from?
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Where does the Euler Mascheroni constant come from in the formula $\sum_{n≤x} 1/n= \log(x)+\gamma +O(1/x)$

From this question, Asymptotic formula for $\sum_{n\le x}\frac 1n$ The author claims without proof that this identity holds. $$\sum_{n≤x} 1/n= \log(x)+\gamma +O(1/x) .$$ It is apparently well known ...
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1 answer
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How to show that Euler constant lies in $(0,1)$ [duplicate]

Let $$ C_n=\sum_{i=1}^n \frac{1}{i}-\log n,\quad n\geq 1 $$ It is well-known that the limit of $(C_n)_{n\geq 1}$ exists, where its limit is denoted $\gamma$, Euler's constant. I'll shortly write what ...
3 votes
1 answer
159 views

A sum that's possibly equal to the Euler-Mascheroni Constant $\sum_{n=1}^\infty \frac{\ln n!}{n^3}$

The following interesting sum seems to approach the Euler-Mascheroni constant $\gamma$. $$\sum_{n=1}^\infty \frac{\ln n!}{n^3} \overset{?}{=} \gamma$$ I've looked at the different ways to express the ...
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2 answers
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Is $\sum_{j=1}^\infty\sum_{n=1}^\infty\left(\frac{e^{-j/n}}{n^2}-\frac{e^{-n/j}}{j^2}\right)=\gamma ?$

A friend proposed the following problem: $$\sum_{j=1}^\infty\sum_{n=1}^\infty\left(\frac{e^{-j/n}}{n^2}-\frac{e^{-n/j}}{j^2}\right)=\gamma,$$ where $\gamma$ is the Euler-Mascheroni constant. The ...
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2 votes
2 answers
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Valid proof that Euler's Constant $\gamma$ is between $0$ and $1$?

I was wondering if there is any reasonable way/theory to do calculations with divergent limits of a sequence. I was trying to prove that Euler's constant $$\gamma = \displaystyle{\lim_{n \to \infty}} \...
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Modulus of the "pair" $(0,\infty)$ seems to be a finite value?

It is known that the split-complex numbers are isomorphic to the pairs of real numbers in the following way: $a + bj \leftrightarrow (a - b, a + b)$ with operations defined on the pairs element-wise. ...
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2 votes
1 answer
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How can I continue my proof that the difference of cosine integrals is the Euler Mascheroni constant?

$\newcommand{\d}{\,\mathrm{d}}\newcommand{\cn}{\mathrm{Cin}}\newcommand{\ci}{\mathrm{Ci}}$I am very new to the Ci, Si functions - I saw these an hour ago! In particular, I saw an identity involving ...
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0 votes
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Book suggestions to learn more about Euler-Mascheroni constant

I want to learn more about the Euler-Mascheroni constant, $\gamma$ and do research on a problem related to it. The problem does not necessarily has to be the question of rationality/irrationality of $\...
3 votes
1 answer
117 views

Triple product $\displaystyle\prod_{k=1}^{\infty} \prod_{n=1}^{\infty} \prod_{m=1}^{\infty}(k+n+m)^{\frac{(-1)^{k+m+n}}{k+m+n}}$

Prove that $$\prod_{k=1}^{\infty} \prod_{n=1}^{\infty} \prod_{m=1}^{\infty}(k+n+m)^{\frac{(-1)^{k+m+n}}{k+m+n}}=\frac{\mathbb{A}^{\frac{3}{2}}}{\pi^{\frac{3}{4}} e^{\frac{1}{8}-(\frac{7}{12}+\gamma) \...
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4 votes
3 answers
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$\frac {\zeta(2)}{e^2} + \frac {\zeta(3)}{e^3} + \frac {\zeta(4)}{e^4} + \frac {\zeta(5)}{e^5}.....?$

$$\frac {\zeta(2)}{e^2} + \frac {\zeta(3)}{e^3} + \frac {\zeta(4)}{e^4} + \frac {\zeta(5)}{e^5}.....?$$ I tried to solve it by using this product formula, $$\frac 1{\Gamma (x)}=xe^{\gamma x} \prod_{n=...
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2 votes
2 answers
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Proof that the limit of a series of exponential distributed random variables follows a Gumbel distribution

How can I prove that given $X_k \sim \text{Exp}(1/k),$ $$\lim_{n\to\infty}\left(-\log(n) + \sum_{k=1}^n X_k \right)\sim \text{Gumbel}(\mu =0,\beta = 1)$$ ? I see that $\log(n)$ is the integral of $1/x$...
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1 answer
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Little-o of a summation

I'm having trouble proving that $$\sum_{k=1}^{n}\frac{1}{2k-1} = \log(n)/2 + \log(2) + γ/2 + o(1)$$ as $n → ∞$. I honestly don't even know where to start, and I don't know what the $o(1)$ is supposed ...
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Evaluate a $\gamma$ integral [duplicate]

In a previous question I asked about convergence of the integral, now I would like to see how to evaluate it. I believe the result is $\gamma$. $$\int_0^1 \frac{1}{1-x} + \frac{1}{\log(x)} dx$$ I have ...
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0 votes
1 answer
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Seeking reference on a fact involving Euler's constant and the reciprocal of a uniform

I have seen the following statement on a Stack Exchange answer: Let $X = 1 - (1/U - \left\lfloor {1/U} \right\rfloor )$, where $U$ is a uniform random variable in $[0, 1]$. Then— $$\mathbb{E}[X] = \...
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1 answer
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On the irrationality of Euler Mascheroni constant [duplicate]

I saw one of the expansions of Euler Mascheroni constant in terms of Meissel Mertens constant as a consequence of Mertens theorem. $$ B = \gamma + \sum_p \left\{ \log\left( 1 - \frac 1p\right) + \frac ...
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2 votes
1 answer
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Does $\sum_{n=1}^\infty \frac{1}{nH_n^2}$ (& related) have any closed form representaions?

We know that: $$\sum \frac{1}{n^{1+\epsilon}} \quad converges \ \forall \epsilon>0 \quad \& \quad \sum \frac{1}{n} \sim \ ln(n) \tag{1}\label{asymp1}$$ Using this we define: $$\gamma_1:= \lim_{...
2 votes
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Explaining Algebraic Independence of $\delta$ and $\gamma$

I have been attempting to read this paper by Alexander Aptekarev. In it, he proves that $\delta$ and $\gamma$ cannot both be rational simultaneously. He also notes that this result follows from a ...
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4 votes
1 answer
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How to derive this series for $\gamma$ that is only involving odd integer values of $\zeta(s)$?

With $\gamma$ being the Euler Mascheroni constant, this series is well known: $$1- \sum_{n=2}^{\infty} \frac{\zeta(n)-1}{n} = \gamma \tag{1}$$ The following series involving $\zeta(2n+1)$ also seems ...
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3 votes
2 answers
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Intermediate step in deriving integral representation of Euler–Mascheroni constant: $\int_0^1\frac{1-e^{-t}-e^{-1/t}}{t}dt$

I'm following a complex analysis course and am making an exercise in which I have to derive an integral representation for the Euler–Mascheroni constant. I have the following definition of the Euler–...
0 votes
1 answer
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How to derive Euler-Maclaurin sum formula from Taylor Series?

Page 152 at https://link.springer.com/content/pdf/10.1007%2F978-0-387-73468-2.pdf Hi readers, I have tried substituting y'(0) , y''(0) , y'''(0) and y''''(0) into equation A that is the first equation....
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1 answer
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Closed form of sum of n^n series? [closed]

Hi readers, may i know how this formula is derived? I thought of using https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula which is Euler-Maclaurian to find the formula but its not.
2 votes
0 answers
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What is an approximate closed form for sum of $n^n$ series?

I read about this https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula which is Euler-Maclaurain and I did my calculation to find the constants. However, the answer differ greatly from ...
0 votes
1 answer
121 views

What are some interesting problems in the intersection of Diophantine Approx and Algebraic Geometry?

I am a first year graduate student and I am eager to work on irrationality/transcendental proofs of specific numbers like Euler's constant gamma. Because backgrounds for Elliptic Curves include very ...
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0 votes
1 answer
97 views

Sum of 1.5-powers of natural numbers using Euler-Maclaurin Formula?

Hi, I have read this Sum of 1.5-powers of natural numbers and answers posted by Mark Viola. However, I am not too sure on how he got to this. Can anyone show me as I am quite new to series?
1 vote
1 answer
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Why is $\lim\limits_{x\to\infty}\frac{\sum_{i=1}^x(\sum_{j=1}^i\frac1j-\ln i-\gamma)}{\sum_{i=1}^x\frac1i}=\frac12$?

$$ \mbox{Why is}\quad\lim_{x\to\infty} \frac{\sum_{i = 1}^{x}\left[\sum_{j = 1}^{i}1/j -\ln\left(i\right)-\gamma\right]}{\sum_{i = 1}^{x}1/i} = \frac{1}{2}\ ?.$$ I learnt Euler's Constant $\gamma$ ...
2 votes
0 answers
193 views

Ergodic Theory and Euler-Mascheroni Constant

Originally posted on mathoverflow but didn't get an answer. I am highly interested in doing research on proving irrationality of some specific numbers like Euler-Mascheroni Constant or ζ(5). A ...
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14 votes
1 answer
282 views

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-\...
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0 answers
120 views

Do these constants appear in other areas of mathematics?

I decided to consider divergent (to infinity) integrals as some new kind of number. Towards this end, I began by establishing certain rules defining equivalence of the integrals. It seems the usual ...
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3 votes
0 answers
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Does the Euler-Mascheroni constant $\gamma$ correspond to infinite hyperbolic angle?

So, I am trying to find the regularized value of the divergent integral $I=\int_1^\infty \sqrt{x^2-1}dx$. Since the area of sector of a circle $\int_0^1 \sqrt{1-x^2}dx=\frac\pi4$, I wonder whether the ...
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5 votes
2 answers
251 views

Euler constant as limit of zeta function

I want to prove that $ \lim \limits_{s \to 0} (\zeta(1+s)+\zeta(1-s))= 2\gamma$ , I divide it to two limits, $\lim \limits_{s \to 1^{+}} (\zeta(s)-\frac{1}{s-1}) = \gamma$ which I proved using the ...
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