A generalization of the Erd\H{o}s-Sierpi\'nski conjecture

TL;DR

This paper studies the solutions of the equation (n+1) = k(n) for integers k>1, revealing their zero density and providing explicit bounds, with conditional results on their infinitude.

Contribution

It extends probabilistic number theory methods to analyze the distribution of solutions and proves they have zero density, also establishing conditional infinitude for k=2.

Findings

Solutions have zero natural density.

Explicit upper bound A_k(x)  x / \u221a{  }log log log x.

Conditional infinitude of solutions for k=2.

Abstract

In this paper, we investigate the combinatorial structure and asymptotic distribution of the solution set of the equation $\sigma(n+1) = k\sigma(n)$ for a given integer $k>1$. From a combinatorial perspective, the solutions to this equation are closely related to the concept of $k$-layered numbers, which are a generalization of Zumkeller numbers. In the analytic section, which constitutes the core of this research, we employ the framework of probabilistic number theory and an extension of the classical Kubilius model to study the oscillatory and local behavior of the sum-of-divisors function. Utilizing the truncation technique for arithmetic functions and applying the Chinese Remainder Theorem, the problem is reduced to a synthetic measure space equipped with independent random variables. Subsequently, by applying the optimized version of the Kolmogorov-Rogozin anti-concentration inequality (Petrov's theorem) to the difference of additive variables and finely tuning the error parameters, we prove that the natural density of this set is zero. The main quantitative outcome of this approach is the derivation of the explicit upper bound $A_k(x) \ll_k \frac{x}{\sqrt{\log \log \log x}}$ for the counting function of the solutions. Finally, alongside the zero asymptotic density, relying on the framework of polynomials and Schinzel's H Hypothesis, we establish the conditional infinitude of the solution set for the case $k=2$ and formulate the existential results.