Editing Floor and ceiling functions (section)

===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)<ref>Titchmarsh, p. 13</ref> that if <math>\varphi(x)</math> is any function with a continuous derivative in the closed interval [''a'', ''b''],

:<math>\sum_{a<n\le b}\varphi(n) =
\int_a^b\varphi(x) \, dx +
\int_a^b\left(\{x\}-\tfrac12\right)\varphi'(x) \, dx +
\left(\{a\}-\tfrac12\right)\varphi(a) -
\left(\{b\}-\tfrac12\right)\varphi(b).
</math>

Letting <math>\varphi(n) = n^{-s}</math> for [[real part]] of ''s'' greater than 1 and letting ''a'' and ''b'' be integers, and letting ''b'' approach infinity gives

:<math>\zeta(s) = s\int_1^\infty\frac{\frac12-\{x\}}{x^{s+1}}\,dx + \frac{1}{s-1} + \frac 1 2.</math>

This formula is valid for all ''s'' with real part greater than &minus;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.<ref>Titchmarsh, pp.14–15</ref>

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

:<math>\zeta(s)=s\int_{-\infty}^\infty e^{-\sigma\omega}(\lfloor e^\omega\rfloor - e^\omega)e^{-it\omega}\,d\omega.</math>

In 1947 [[Balthasar van der Pol|van der Pol]] used this representation to construct an analogue computer for finding roots of the zeta function.<ref>Crandall & Pomerance, p. 391</ref>