Ratios of consecutive values of the divisor function

Sean Eberhard

Ratios of consecutive values of the divisor function

Sean Eberhard

Abstract

We show that the sequence of ratios $d(n+1) / d(n)$ of consecutive values of the divisor function attains every positive rational infinitely many times. This confirms a prediction of Erdős.

Ratios of consecutive values of the divisor function

Abstract

We show that the sequence of ratios

of consecutive values of the divisor function attains every positive rational infinitely many times. This confirms a prediction of Erdős.

Paper Structure (1 theorem, 10 equations)

This paper contains 1 theorem, 10 equations.

Key Result

Theorem 1

Let $a_1, a_2, a_3, r_1, r_2, r_3$ be positive integers with $(r_i, a_i) = (r_i, a_i - a_j) = (r_i, r_j) = 1$ for all $i \ne j$. Let $L_i(x) = a_i x + 1$. Let $C$ be any positive integer. Then there are indices $i,j$ with $1 \le i < j \le 3$ such that there are infinitely many positive integers $x$

Theorems & Definitions (1)

Theorem 1: GGPY*Corollary 2.1, special case $b_1=b_2=b_3=1$