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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How to proof these two series of Euler-constant [closed]

Those are called vacca series,but I don’t know how to proof it
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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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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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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 ...
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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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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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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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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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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 $\...
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3 votes
1 answer
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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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$\frac {\zeta(2)}{e^2} + \frac {\zeta(3)}{e^3} + \frac {\zeta(4)}{e^4} + \frac {\zeta(5)}{e^5}.....?$

$$\frac {\zeta(2)}{e^2} + \frac {\zeta(3)}{e^3} + \frac {\zeta(4)}{e^4} + \frac {\zeta(5)}{e^5}.....?$$ I tried to solve it by using this product formula, $$\frac 1{\Gamma (x)}=xe^{\gamma x} \prod_{n=...
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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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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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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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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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Does $\sum_{n=0}^\infty \frac{1}{nH_n^2}$ (& 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_{...
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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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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–...
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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.
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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 ...
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0 votes
1 answer
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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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1 answer
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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?
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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 $\gamma$ ...
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2 votes
0 answers
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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
271 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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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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Dos 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 $\int_0^1 \sqrt{1-x^2}dx=\frac\pi4$, I wonder whether the area under ...
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5 votes
2 answers
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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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5 votes
1 answer
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Bounds on the difference between the polylogarithm with negative base and the gamma function

Trying to understand intuitively the Gamma function I started to think of it as a way to measure how much each factorial power "helps" $x^n$ in the infinite sum of $e^x$, thus trying to ...
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2 votes
3 answers
96 views

Does $c_{\ln(x)}=\lim_{n\to\infty}\left(\sum_{k=1}^n \ln(k)-\int_1^n \ln(x)\ dx\right)$ converge?

Does $c_{\ln(x)}=\displaystyle\lim_{n\to\infty}\left(\sum_{k=1}^n \ln(k)-\int_1^n \ln(x)\ dx\right)$ converge? First of all, I always thought Mascheroni was spelled (and pronounced) Masechroni, but ...
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Stieltjes constants found in the limit as $x \to \,1^+$

Mathematica 12 evaluates the following sequence of limits thus (tested up to $s=100$) $$\underset{x\to \,1^+}{\text{lim}}\;\left(\frac{\partial ^n }{\partial x^n}\left(\frac{1}{1-x}+\sum _{k=1}^{\...
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1 vote
1 answer
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Good review article for Euler's Constant $\gamma$

I am writing a paper on Euler's constant (or Euler-Mascheroni constant) $\gamma$ with 4 other people and we are looking for a good review article on the subject. So far we haven't found anything. A ...
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1 vote
1 answer
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Does $\lim_{k\to\infty}2^k \log_2(2^k-1)\equiv\gamma\mod 1$ where $k\in\mathbb{N}$ (Euler-Mascheroni Constant)

I was working with the function $f(k)=2^k \log_2(2^k-1)$ and had noticed that it's factional component seemed to converge when $k\in\mathbb{N}$. For example, $f(13)\equiv 0.557216896821\mod 1$, ...
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1 answer
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What's the typical role of the constant $e^{-\gamma}$?

I often encounter this constant in my research, but I wonder what typical roles does it play in other areas of mathematics? Wikipedia mentions probability theory but nothing exact. Also, I am ...
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1 vote
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Set of divergent integrals

So, we take $\frac{\text{sgn}(x-1)}{x}$ and apply $\mathcal{L}_t[t f(t)](x)$ four times. The transform keeps area. These integrals are minus Euler-Mascheroni constant: $$\int_0^\infty \frac{\text{sgn}(...
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4 votes
1 answer
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Why is $\ln 0\ne-\ln \infty$?

The title of this post is intentionally sensational, but what I am really going to do is to compare the divergent integrals $\int_0^1\frac1xdx$ and $\int_1^\infty\frac1xdx$. Let's consider the ...
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3 votes
0 answers
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Infinite sum related to hypergeometric series

I was working on some integrals and I came across the following series: $$\sum_{k=1}^{\infty}\frac{1}{k(k+n)!}$$ Wolfram Alpha evaluates it to be $$\frac{_2F_2\left(1,1;2,n+2;1\right)}{(n+1)!}$$ Which ...
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2 votes
1 answer
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On a certain integral related to Euler-Masheroni constant

When trying to derive Mertens' theorem using Perron's formula, I encountered the following integral: $$ \int_0^1{1-e^{-t}\over t}\mathrm dt-\int_1^\infty{e^{-t}\over t}\mathrm dt $$ which, according ...
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1 vote
4 answers
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Prove the limit equation: $\lim_{x\rightarrow\infty}e^{-x}\sum_{n=0}^{\infty}f_{n}\left(x\right)=-\gamma.$ FYI$,\quad\gamma$ is Euler's constant

Define a function sequence $\left \{ f_{n} \right \}_{n= 0}^{\infty}$ on $\left ( 0, \infty \right )$ as $$f_{0}\left ( x \right )= \ln x,\quad f_{n+ 1}\left ( x \right )= \int_{0}^{x}f_{n}\left ( t \...
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1 vote
1 answer
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Euler-Mascheroni and von Mangoldt function

I know that by their respective definitions: $$\gamma_0 = \sum_{n=1}^{\infty}{\frac{1}{n}-\ln(\frac{n+1}{n})}$$ $$\ln(n)=\sum_{d|n}{\Lambda(d)}$$ and I want to get there: $$\gamma_0 = \sum_{n=1}^{\...
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1 vote
1 answer
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Expressing Gamma Function as an euler constant

Q. Use the definition $$ \Gamma(x) = \lim_{n\to \infty} \frac{n!n^x}{x(x+1)\cdots(x+n)} $$ in this problem. a) Show that $ \Gamma (x+1) = x\Gamma(x) $ for $ x>0 $ b) First, express $\mathbb{ln}\...
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1 vote
1 answer
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Negative continued fraction of Euler Mascheroni constant

$\gamma$, the Euler-Mascheroni constant, has the following simple regular continued fraction: $$\gamma=[0; 1, 1, 2, 1, 2, 1,\dots]=0+ \cfrac{1}{ 1+\cfrac{1}{ 1+\cfrac{1}{ 2+\cfrac{1}{ ...
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0 votes
1 answer
62 views

Prove complex logarithm problem involving Euler Mascheroni constant

I want to prove/disprove the following equation: $$\text{Log}{\frac{1}{\sqrt{z^2+1}}} = -\frac12 \ln\left(\frac{1}{z^2}+1\right)-\ln\lvert z\rvert - \gamma$$ Which $\gamma$ is the Euler Mascheroni ...
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5 votes
2 answers
236 views

Closed form of the sum $\sum_{r\ge2}\frac{\zeta(r)}{r^2}$

Note:This is the same question, but it doesn't answer my question, the answer doesn't give a closed form. In fact, the answer is not accepted. Moreover, I don't think that a 9 month old inactive ...
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0 votes
1 answer
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Generalized Euler Constants

According to Wikipedia, the generalized Euler constants are $$\gamma_\alpha=\lim_{n\to \infty} \Big(\sum_{k=1}^n \frac{1}{k^\alpha} - \int_1^n \frac{1}{x^\alpha} \,dx \Big)$$ for $0<\alpha<1$. I ...
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