For questions related to Euler's constant, which is defined to be the limiting difference between the natural logarithm and the harmonic series.

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7
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3answers
841 views

Integral representation of Euler's constant

Prove that : $$ \gamma=-\int_0^{1}\ln \ln \left ( \frac{1}{x} \right) \ \mathrm{d}x.$$ where $\gamma$ is Euler's constant ($\gamma \approx 0.57721$). This integral was mentioned in Wikipedia as in ...
9
votes
8answers
744 views

“How I wish I could calculate pi” analogs…

You might know the mnemonic for $\pi$ in the title or even this more elaborated one: Sir, I bear a rhyme excelling In mystic force, and magic spelling Celestial sprites elucidate All my own ...
3
votes
2answers
286 views

Elementary derivation of certian identites related to the Riemannian Zeta function and the Euler-Mascheroni Constant

Is the proof of these identities possible, only using elementary differential and integral calculus? If it is, can anyone direct me to the proofs? ( or give a hint for the solution ) ...
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vote
0answers
163 views

Help with interesting sum involving Euler's constant

Wikipedia gives an interesting infinite sum for Euler's constant $\gamma$ and I was wondering how one would evaluate this interesting sum. The sum is given as follows: Let $N_0 (x)$ and $N_1 (x)$ ...
5
votes
1answer
156 views

Define integral for $\gamma,\zeta(i) i\in\mathbb{N}$ and Stirling numbers of the first kind

Consider the integral $$\int\limits_0^{\infty}e^{-x}x^k\ln(x)^n\dfrac{dx}x$$ For $n=3$ we have ...
9
votes
2answers
2k views

Has Euler's Constant $\gamma$ been proven to be irrational?

I found a paper by Kaida Shi called "A Proof: Euler’s Constant γ is an Irrational Number" which claims to have proven the irrationality of $\gamma$. I know people have been trying to prove that ...
41
votes
10answers
2k views

What is the fastest/most efficient algorithm for estimating Euler's Constant $\gamma$?

What is the fastest algorithm for estimating Euler's Constant $\gamma \approx0.57721$? Using the definition: $$\lim_{n\to\infty} \sum_{x=1}^{n}\frac{1}{x}-\log n=\gamma$$ I finally get $2$ decimal ...
5
votes
2answers
477 views

Equality with Euler–Mascheroni constant

While trying to prove integral with exponential function and logarithm in an alternative way, I came to this solution: $$\sum_{k=0}^{+\infty}(-1)^{k+1}\frac{\log (k+1)+\gamma }{(k+1)}.$$ As both ...
9
votes
3answers
659 views

Limit of Zeta function

I'm looking for a reference for (or an elementary proof of) $$ \lim_{s \rightarrow 1} \left( \zeta(s) - \frac{1}{s-1} \right) = \gamma$$ Thanks for your help.