Rigidity of Averages over the Two Largest Prime Factors

Abstract

Let \(P_1(n)\) and \(P_2(n)\) be the largest and second-largest distinct prime factors of \(n\), respectively. Alladi and Johnson asked whether there exists a bounded function \(f\) on the primes for which both limits \(\frac{1}{x}\sum_{2\le n\le x} f(P_1(n)) \longrightarrow κ_1\) and \(\frac{1}{x}\sum_{2\le n\le x} f(P_2(n)) \longrightarrow κ_2\) exist with \(κ_1\neqκ_2\), where we set \(f(P_2(n))=0\) when \(n\) is a prime power. We prove that this is impossible: convergence of the first average forces convergence of the second to the same limit. On the \(\log\log\)-scale, the two averages are expressed as convolutions with explicit Dickman kernels. The Fourier transform of the Dickman kernel associated with the \(P_1\)-average has no real zeros. Wiener's Tauberian theorem then yields vague convergence of the translated measures associated with the weighted prime sums. The convergence of the second average follows.

29 pages