It is well known that The Riemann zeta function discribe prime distribution such that many results related to the Riemann zeta function show prime distribution using indirect method like prime number theorem and von mangoldt function and also prime counting function such that there is a clear mathematical formula for computing probability to get a prime number , All these basics are derived from Riemann zeta function , Now since there is a Randomness in primes than there is a such random variable which Riemann zeta function follow it , Now my question here why :Why Riemann zeta function is not a density function however it describe prime distribution? by means why we do not saw any formual figure out that integral( zeta(s))=1 from 0 to infty ?
Riemann's zeta function is not a density function because it does not give the probability of finding prime numbers. It describes their distribution in an asymptotic sense, through the J(x) function (more precisely, J(x) is given by an integral involving zeta(x), and Riemann proved a formula relating pi(x) --the number of primes less than x-- to J(x)). Riemann proved that the main term in the analytical development of the J(x) function is the logarithmic integral Li(x) (here the limit x tending to infinity is used, hence the following results will have only an approximate character), and he finally arrived to an approximate expression for pi(x) involving Li(x), an approximation which becomes better for large values of x.
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