Zeta Function

Zeta function is defined as $$\zeta (s) = \sum_{n=1}^{\infty}\frac{1}{n^s}$$
Leonhard Euler proved that $$\sum_{n=1}^{\infty}\frac{1}{n^s} = \prod_{p}\left (1 - \frac{1}{p^s} \right )^{-1} \hspace{8 mm} s > 1$$
where $p$ takes only prime numbers.

This equality gives an alternative proof of infiniteness of primes. When $s \to \infty$, left-hand side goes to infinity. If there are finite number of primes, then right-hand side finite product will be equal to an infinite sum, which is a contradiction.

The proof of the above identity is not very difficult. Consider separate infinite series for each prime $p$.
$$\left (1 - \frac{1}{p^s} \right )^{-1}=\sum_{e=0}^{1}\frac{1}{p^{es}}$$
Product of infinite series of all primes will contain terms of form $$\frac{1}{(p_1^{e_1}p_2^{e_2} \cdots p_m^{e_m})^s}$$
Each number have unique prime factorization, so rearranging the terms of product of infinite series of all primes we get the infinite sum on the left-hand side of the identity.

Exact values of zeta functions and some related sums for positive integral value of $s$ is considered in this post.

Let me start with evaluating $\zeta (2)$. Consider the function $f(x)=x^2$, $x \in (-\pi, \pi)$ and $f(x+2\pi)=f(x)$

Taking the Fourier series expansion of $f(x)$,
$$x^2=\frac{\pi ^2}{3}+\sum_{n=1}^{\infty}\frac{4\cos n\pi}{n^2}\cos nx$$
Substituting $x=\pi$, we get $\zeta (2)=\frac{\pi ^2}{6}$. We can obtain other interesting sums for different values of $x$.

Euler proved that $\zeta (2n)$ is rational multiple of $\pi ^{2n}$, where $n$ is positive integer. It is not known if $\zeta (2n+1)$ is irrational for all $n$.

We can obtain some interesting sums for odd positive integers. Integrating the Fourier series expansion of $f(x)$ defined above from $0$ to $x$,

$$\frac{x^3}{3}=\frac{\pi ^2 x}{3}+\sum_{n=1}^{\infty}\frac{4 \cos n\pi}{n^3} \sin nx$$

Substituting $x=\frac{\pi}{2}$ we get, $$\sum_{n=0}^{\infty}\left [ \frac{1}{(4n+1)^3}- \frac{1}{(4n+3)^3}\right ] = \frac{\pi ^3}{32}$$

Proceeding similarly we can derive such sums for larger odd positive integers.

$$\sum_{n=0}^{\infty}\left [ \frac{1}{(4n+1)^5}- \frac{1}{(4n+3)^5}\right ] = \frac{5\pi ^5}{1536}$$

$$\sum_{n=0}^{\infty}\left [ \frac{1}{(4n+1)^7}- \frac{1}{(4n+3)^7}\right ] = \frac{61\pi ^7}{184320}$$

It seems that we can generalize this sum for all odd positive integers. Apparently, for even positive integral value of $n$, this sum is not rational multiple of $\pi ^n$, exactly opposite holds for zeta function at positive integral value of $n$.

$$\sum_{n=0}^{\infty}\left [ \frac{1}{(4n+1)^{2k+1}}- \frac{1}{(4n+3)^{2k+1}}\right ] = \frac{E(k)\pi ^{2k+1}}{(2k)!2^{2k+2}}$$

where

$$E(0)=1$$

$$E(n)=(-1)^{n+1}\sum_{i=0}^{n-1}(-1)^iE(i)\binom{2n}{2i}$$

$E(n)$ is related to A000364

This formula is based on observation. I don't have any proof yet. But I am quite certain about its correctness. I verified for $n=3,5,7,11,13,15$ using Mathematica.

I found another interesting result, again without proof.

$$\operatorname{\mathbb{R}e} \left ( \sum_{n=0}^{\infty} \left [ \frac{1}{(an+1)^3} - \frac{1}{(an+a-1)^3} \right ] \right )= \left(\frac{\pi}{a}\right)^3\text{cos}\left(\frac{\pi}{a}\right)\text{cosec}^3\left(\frac{\pi}{a}\right)$$

for any positive real number $a>1$. It might be possible to extend it for other odd positive integers.

$\textbf{EDIT}:$ It is possible to get closed form formula of above sum for other odd positive integers. $$\operatorname{\mathbb{R}e} \left ( \sum_{n=0}^{\infty} \left [ \frac{1}{(an+1)^{2k+1}} - \frac{1}{(an+a-1)^{2k+1}} \right ] \right )= \frac{2}{(2k)!}\left(\frac{\pi}{a}\right)^{2k+1}\text{cosec}^{2k+1}\left(\frac{\pi}{a}\right)\sum_{i=1}^{k}b(k,i)\text{cos}\frac{(2i-1)\pi}{a}$$
where $b(k,i)=T(2k-1,2k-i-1)$

$T$ is known as Euler's Number Triangle and defined as
$$T(0,0)=1$$ $$T(i,j)=(i-j)T(i-1,j-1)+(j+1)T(i-1,j) \hspace{4 mm} i \ge 1, 0 \le j < i$$
First few terms are given below
\[\begin{array}{ccccccccccc}
  &    &    &    &    &    &  1 &    &    &    &    &    &   \cr
  &    &    &    &    &  1 &    &  1 &    &    &    &    &   \cr
  &    &    &    &  1 &    &  4 &    &  1 &    &    &    &   \cr
  &    &    &  1 &    &  11 &    &  11 &    &  1 &    &    &   \cr
  &    &  1 &    &  26 &    &  66 &    &  26 &    &  1 &    &   \cr
  &  1 &    &  57 &    & 302 &    & 302 &    &  57 &    & 1    & \cr
1 &    &  120 &    & 1191 &    & 2416 &    &  1191 &    & 120 &    & 1 \cr
\end{array}\]

I shall conclude this post by mentioning the most famous conjecture associated with zeta function, the Riemann hypothesis. It states that "Zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$". Riemann derived an explicit formula of prime counting function $\pi(x)$, which is related to the zeros of zeta function. Interested reader can read about this in detail here.

Thanks for reading :)