pi(x) - prime counting function

Hello,

Let pi(x) denote the prime counting function,
i.e. pi(x) = number of primes =< x.

Then I'm wondering about the following identity

sum_{n = 2}^infty pi(n) int_{n}^{n + 1} f(x) dx

= int_{n = 2}^infty pi(x) f(x) dx

where f(x) is 1/(x(x^s - 1)).

So, why are we allowed to pull P(n) on LHS
into the integral on the RHS?

(What is the integral of pi(x)?!)

Thank you!

Oliver

.

Relevant Pages

  • Equivalent Forms of the Riemann Hypothesis?
    ... Let R_0 denote sup, taken over all z such that zeta=0. ... where pi is the prime counting function, ... is the sum of mutaken over all positive integers n<=x, ... the sum of mu/n taken over all positive integers n<=x. ...
    (sci.math)
  • Re: pi(x) - prime counting function
    ... Let pidenote the prime counting function, ... into the integral on the RHS? ... William Hughes ...
    (sci.math)