4))}$

Asked Dec 09 '18 at 19:45

Active Dec 09 '18 at 21:17

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In the title, $\gamma$ is the Euler-Mascheroni constant and $\Gamma(3/4)$ represents the extension of the factorial function.

This isn't a homework question or something, someone left it on a board in one of the buildings in my university and I'm just really surprised by it. The only thing I've tried is writing out the first few terms and trying to manipulate them into some sort of pattern, but I don't see where to go from there.

$$\frac{\ln{3}}{3}-\frac{\ln{5}}{5}+\frac{\ln{7}}{7}-\frac{\ln{9}}{9}+...$$ $$\ln{(3^{1/3})}-\ln{(5^{1/5})}+\ln{(7^{1/7})}-\ln{(9^{1/9})}+...$$ $$\ln{\Bigg(\frac{3^{1/3}}{5^{1/5}}\Bigg)}+\ln{\Bigg(\frac{7^{1/7}}{9^{1/9}}\Bigg)}+...$$

From here, I know that I could combine log terms even more, multiplying the numerators/denominators, but I don't think that's the right path to follow for this.

edited Dec 09 '18 at 21:17

clathratus

18,002

asked Dec 09 '18 at 19:45

Calvin Godfrey

1,574

haha, sounds like something out of Good Will Hunting. – zoidberg Dec 09 '18 at 19:51
I have verified using software that the left side appears to converge somewhere close to the right side. – Calvin Godfrey Dec 09 '18 at 19:52
A straightforward consequence of differentiation of the Dirichlet Beta function, or of Kummer-Malmstein Fourier series of $\log\Gamma$. – Jack D'Aurizio Dec 09 '18 at 20:14
Thanks for that, Jack! I found the paper by Malmsten where he proved this, looking through it now. – Calvin Godfrey Dec 09 '18 at 20:25
See How are values of the Dirichlet Beta function derivative derived?. More specifically, the derivation of $\beta'(1)$. – projectilemotion Dec 09 '18 at 20:36
2

The original paper by Malmsten is here, with the statement in question on page 24 of the pdf, and the proof preceding it. – Calvin Godfrey Dec 09 '18 at 20:41
See related https://math.stackexchange.com/q/1284941/72031 – Paramanand Singh Jul 25 '20 at 03:48

Prove $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}\ln{(2n+1)}}{2n+1}=\pi/4(\gamma-\ln{\pi})+\pi\ln{(\Gamma(3/4))}$

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