On Erdős sums of almost primes

Abstract

In 1935, Erdős proved that the sums f<inf>k</inf> = ^P<inf>n</inf> 1/(n log n), over integers n with exactly k prime factors, are bounded by an absolute constant, and in 1993 Zhang proved that f<inf>k</inf> is maximized by the prime sum f<inf>1</inf> = ^P<inf>p</inf> 1/(p log p). According to a 2013 conjecture of Banks and Martin, the sums f<inf>k</inf> are predicted to decrease monotonically in k. In this article, we show that the sums restricted to odd integers are indeed monotonically decreasing in k, sufficiently large. By contrast, contrary to the conjecture we prove that the sums f<inf>k</inf> increase monotonically in k, sufficiently large. Our main result gives an asymptotic for f<inf>k</inf> which identifies the (negative) secondary term, namely f<inf>k</inf> =1−(a+o(1))k²/2^k for an explicit constant a =0.0656···. This is proven by a refined method combining real and complex analysis, whereas the classical results of Sathe and Selberg on products of k primes imply the weaker estimate f<inf>k</inf> =1+Oε(k^ε−^1/2). We also give an alternate, probability-theoretic argument related to the Dickman distribution. Here the proof reduces to showing a sequence of integrals converges exponentially quickly e^−γ, which may be of independent interest.