Sums over p and improper integrals
- Suppose you are trying to approximate sum over large primes p of a(p)
where a(p) is some monotonically decreasing to 0 function of p.
Instead of writing sum(a(p),p>=P) ~ intgeral(a(x),x=P to infinity), you
should rather approximate it by sum(a(p),p>=P) ~ intgeral(a(x)/log
(x),x=P to infinity), basically because of a frequency count (gap
size). Say p and q~p+log(p) are consecutive large primes and a(x) is
(roughly) constant on the interval [p,q]. Then integral(a(x),p<=x<=q)
~ a(p)+a(p+1)+ ... +a(q) ~ a(p)*log(p) and is too big to approximate
the summand a(p). Instead, you should use a(x)/log(x) for the
integrand. Another way to interpret this is via the dx differential
which should be weighted: a(p) = a(p)*1 ~ a(x) * dx/log(x).