Interesting Trig sum

Prove/Disprove that

\[\displaystyle \sum_{n=0}^{\infty} \left(\frac {\displaystyle\sum_{k=0}^{\left \lfloor \frac p2 \right\rfloor -1} \left(\cos \left(\frac {\pi}{p}(2k+1)(2n+1)\right)\right) }{(2n+1)^s}\right) =\left(\frac 12 -\frac {1}{2p^s }-\frac {\left\lfloor \frac p2\right\rfloor}{p^s}\right) (1-2^{-s})\zeta(s)\]

Where $p$ is a prime number, $s\in R$ and $s\gt 1$

Also $\lfloor. \rfloor$ denotes the floor function while $\zeta(s)$ denotes the Riemann Zeta function.

#Calculus

Easy Math Editor

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Comments

@Darkrai ~Rayquaza Any references???? I converted the sum into real parts of Polylogs but got stuck in finding the infinite summation.........

Basically, after a lot of simplifications, the problem breaks down to finding the real part of

$\displaystyle \left(\sum_{n=1}^{\infty}Li_s\left(a^{\left(2n-1\right)}\right)\right)-\frac{1}{2^s}\left(\sum_{n=1}^{\infty}Li_s\left(a^{\left(4n-2\right)}\right)\right)$

Here, $a$ is $\displaystyle e^{\frac{i\pi}{p}}$ and $\displaystyle Li_s\left(a\right)$ denotes the Polylogarithm of $a$ to the base $s$

Aaghaz Mahajan - 2 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$