Topics in Analytic Number Theory

An arithmetic function is a function f : N → C; there are many interesting and natural examples in analytic number theory. To begin with we consider what is perhaps the best known π(n) := xn: 1 P (x), the usual counting function of the primes. Various heuristic arguments suggest that one should expect x to be prime with probability 1/ log x, and coupled with a body of numerical evidence this prompted Gauss to conjecture that π(n) ∼ Li(n) := n 1 1 log x dx ∼ n log n. It is the purpose of the first section of the course to prove this result. As it stands π can be a little difficult to evaluate because the indicator function of the primes is not very smooth. To deal with this we introduce the von Mangoldt function Λ defined by Λ(n) = log p if n = p k 0 otherwise. It will turn out that Λ is smoother than the indicator function of the primes and while the von Mangoldt function is supported on a larger set (namely all powers of primes), in applications this difference will be negligible. For now we note that our interest lies in the sum of the von Mangoldt function, defined to be ψ(n) := xn Λ(x), and which is closely related to π as the next proposition shows. The key idea in the proof of the proposition is a technique called partial summation or, sometimes, Abel transformation. This is the observation that xn f (x)(g(x) − g(x − 1)) = f (n)g(n) − xn−1 g(x)(f (x + 1) − f (x)), and may be thought of as a discrete analogue of integration by parts. Proposition 1.1. ψ(n) ∼ n if and only if π(n) ∼ n/ log n.