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.

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“rnels and to clarify the role of the counterexamples in his work. Professor Tenenbaum also suggested that a shorter proof of this step should exist; a possible such proof, based on smoothing, is included in the appendix. The author used AI tools to assist with grammatical corrections, language polishing, certain LaTeX and coding-related issues, and literature searches, including locating potentially relevant references. At the final stage, AI tools were also used to identify typog”

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2026-08-23T21:13:31.617668Z

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