Riemann Zeta Function

Convergence of this function can be derived using integral test. The above-mentioned series is also referred to as p-series.

This function converges iff . And it converges to:

Otherwise it diverges.

Extension

The function can be extended to the set .

Ramanujan Sum

Which is why, it’s used as below:

This is known as the Ramanujan sum of the diverging series.

Riemann Hypothesis

The function has its zeros only at negative even integers and complex numbers with real part .

One of the most important unsolved problems in mathematics.