On an logarithmic equation by primes

TL;DR

This paper proves that every large positive integer can be expressed as a sum of three terms involving primes and their logarithms, and provides an asymptotic count for such representations.

Contribution

It establishes a new representation of integers using primes and logs, and derives an asymptotic formula for the number of such representations.

Findings

Every sufficiently large integer can be represented as a sum involving primes and logs.

An asymptotic formula for the count of representations is derived.

The result extends understanding of prime-based additive number theory.

Abstract

Let $[\, \cdot\,]$ be the floor function. In this paper we show that every sufficiently large positive integer $N$ can be represented in the form \begin{equation*} N=[p_1\log p_1]+[p_2\log p_2]+[p_3\log p_3], \end{equation*} where $p_1,\, p_2,\, p_3$ are prime numbers. We also establish an asymptotic formula for the number of such representations, when $p_1,\, p_2,\, p_3$ do not exceed given sufficiently large positive number.