A subset of the integers larger than 1 is primitive if no member divides another. Erdős proved in 1935 that the sum of 1/(a log a) for a running over a primitive set A is universally bounded over all choices for A. In 1988 he asked if this universal bound is attained for the set of prime numbers. In this paper we make some progress on several fronts and show a connection to certain prime number "races" such as the race between π(x) and li(x).

It has been known since Erdős that the sum of 1/(n log n) over numbers n with exactly k prime factors (with repetition) is bounded as k varies. We prove that as k tends to infinity, this sum tends to 1. Banks and Martin have conjectured that these sums decrease monotonically in k, and in earlier papers this has been shown to hold for k up to 3. However, we show that the conjecture is false in general, and in fact a global minimum occurs at k = 6.

It is a folklore conjecture that the Möbius function exhibits cancellation on shifted primes; that is, $\sum_{p{\,\leqslant} X}\mu(p+h) \ = \ o(\pi(X))$ as $X\to\infty$ for any fixed shift h > 0. This appears in print at least since Hildebrand in 1989. We prove the conjecture on average for shifts $h{\,\leqslant} H$, provided $\log H/\log\log X\to\infty$. We also obtain results for shifts of prime k-tuples, and for higher correlations of Möbius with von Mangoldt and divisor functions. Our argument combines sieve methods with a refinement of Matomäki, Radziwiłł and Tao’s work on an averaged form of Chowla’s conjecture.

There has been recent interest in a hybrid form of the celebrated conjectures of Hardy–Littlewood and of Chowla. We prove that for any $k,\ell \ge 1$ and distinct integers $h_2,\ldots ,h_k,a_1,\ldots ,a_\ell $ , we have: $$ \begin{align*}\sum_{n\leq X}\mu(n+h_1)\cdots \mu(n+h_k)\Lambda(n+a_1)\cdots\Lambda(n+a_{\ell})=o(X)\end{align*} $$ for all except $o(H)$ values of $h_1\leq H$ , so long as $H\geq (\log X)^{\ell +\varepsilon }$ . This improves on the range $H\ge (\log X)^{\psi (X)}$ , $\psi (X)\to \infty $ , obtained in previous work of the first author. Our results also generalise from the Möbius function $\mu $ to arbitrary (non-pretentious) multiplicative functions.

We investigate the reciprocal sum of primitive nondeficient numbers, or pnds. In 1934, Erdős showed that the reciprocal sum of pnds converges, which he used to prove that the abundant numbers have a natural density. We show the reciprocal sum of pnds is between 0.348 and 0.380.