MathDB
two equivalent statements about prime numbers

Source:

February 26, 2011
calculusintegrationlogarithmsinequalitiesfunctionnumber theoryprime numbers

Problem Statement

suppose that v(x)=∑p≤x,p∈Plog(p)v(x)=\sum_{p\le x,p\in \mathbb P}log(p) (here P\mathbb P denotes the set of all positive prime numbers). prove that the two statements below are equivalent:
a) v(x)∼xv(x) \sim x when x⟶∞x \longrightarrow \infty
b) π(x)∼xln(x)\pi (x) \sim \frac{x}{ln(x)} when x⟶∞x \longrightarrow \infty. (here π(x)\pi (x) is number of the prime numbers less than or equal to xx).
Back to ProblemsView on AoPS