math.NT — Number Theory

Monotonicity of Erdős sums

Contributed by Jaideep Padhi

For $k \geq 1$ let $f_k = \sum_{\Omega(n)=k} 1/(n\log n)$, where $\Omega$ counts prime factors with multiplicity, and let $g_k$ be the same sum restricted to odd $n$. Lichtman showed that $f_k \to 1$, with a global minimum at $k=6$, and Gorodetsky, Lichtman and Wong that $f_k = 1 - 2^{-k}(a_1 k^2 + O(k \log(k+1)))$. We prove that $f_k < f_{k+1}$ for every $k \geq 6$, and that $g_k > g_{k+1}$ for every $k \geq 1$. The two statements come from one analytic distinction. Writing $\sum_k P_k(s) z^k = \zeta(s)^z H(s,z)$, at $s=1$ the point $z=2$ lies on the first singular circle of $H$, and strictly inside the first singular circle of the product with the prime $2$ removed. We show that this distinction decides the direction in general. If the complement of a set of primes $\mathcal{P}$ is thin and its least element is at least $3$, the corresponding sums eventually decrease, so the second theorem is one case of a general phenomenon. The inequalities used at the last step of the first theorem are certified by interval arithmetic.