linguistic:voice_change
Riemann zeta function (ζ)

The fractional part function also shows up in integral representations of the Riemann zeta function. It is straightforward to prove (using integration by parts) that if φ (x) {\displaystyle \varphi (x)} is any function with a continuous derivative in the closed interval,

Letting φ (n) = n − s {\displaystyle \varphi (n)=n^{-s}} for real part of s greater than 1 and letting a and b be integers, and letting b approach infinity gives

This formula is valid for all s with real part greater than − 1, (except s = 1, where there is a pole) and combined with the Fourier expansion for { x } can be used to extend the zeta function to the entire complex plane and to prove its functional equation.

For s = σ + it in the critical strip 0 < σ < 1,

In 1947 van der Pol used this representation to construct an analogue computer for finding roots of the zeta function.