# 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 solve this post? [closed]

How long does it take to download a two-hour HD movie from the iTunes store? According to Apple’s technical support site, support.apple.com/en-us/HT201587, downloading such a movie using a 15 Mbit/s ...
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### 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 ...
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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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### Euler-Mascheroni constant in Bessel function integral

I am currently juggling some integrals. In a physics textbook, Chaikin-Lubensky , 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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### 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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