# 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^{\...
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### 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 $\gamma$ (...
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### 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 ...
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### 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/...
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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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### 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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### 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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