On Erd\H{o}s sums of almost primes

In 1935, Erd\H{o}s proved that the sums $f_k=\sum_n 1/(n\log n)$, over integers $n$ with exactly $k$ prime factors, are bounded by an absolute constant, and in 1993 Zhang proved that $f_k$ is maximized by the prime sum $f_1=\sum_p 1/(p\log p)$. According to a 2013 conjecture of Banks and Martin, the sums $f_k$ 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 $f_k$ increase monotonically in $k$, sufficiently large. Our main result gives an asymptotic for $f_k$ which identifies the (negative) secondary term, namely $f_k = 1 - (a+o(1))k^2/2^k$ for an explicit constant $a= 0.0656\cdots$. 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 $f_k=1+O_{\varepsilon}(k^{\varepsilon-1/2})$. 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 to $e^{-\gamma}$, which may be of independent interest.