Very elementary interpretations of the Euler-Mascheroni constant from counting divisors in intervals

TL;DR

This paper provides elementary interpretations of the Euler-Mascheroni constant by analyzing divisor counts within specific intervals, revealing average behaviors related to divisor distributions and their connection to the constant.

Contribution

It introduces new elementary theorems linking divisor counts in intervals to the Euler-Mascheroni constant, offering accessible insights into its properties.

Findings

Average divisor counts relate to Euler's constant.

Explicit formulas connect divisor distributions to harmonic sums.

Results hold under broad conditions for divisor functions.

Abstract

Theorem 1 Let F:N-->R stand for any function which a) $F$ monotonically weakly increases; b) $F$ tends to infinity; and c) such that $q/F(q)$ tends to infinity. Let Z_F(q) equal the number of divisors of q less than sqrt{F(q)} minus the number of divisors of q between sqrt{F(q)} and F(q). Then, on the average, Z_F(q) equals Euler's constant Theorem 2 Fix a in (0,1). Write A for the average number of divisors of n that lie in (0,sqrt{a n}) minus the number of that lie in (sqrt{a n},a n)$. Then A= (sum_{i=1}^{\lceil {1-a}/a \rceil} \frac{1}{i}) - ln(1/a).