Inequalities for Taylor series involving the divisor function

TL;DR

This paper investigates inequalities related to the divisor function's generating series, establishing monotonicity properties and refining bounds involving Euler's constant, thereby advancing understanding of divisor sum behaviors.

Contribution

It introduces new monotonicity results for functions based on the divisor series and refines existing inequalities with optimal constants.

Findings

H(q) is strictly increasing on (0,1)

F(q) is strictly decreasing on (0,1)

Refined bounds for T(q) with optimal constants  and 1

Abstract

Let $$ T(q)=\sum_{k=1}^\infty d(k) q^k, \quad |q|<1, $$ where $d(k)$ denotes the number of positive divisors of the natural number $k$. We present monotonicity properties of functions defined in terms of $T$. More specifically, we proved that $$ H(q) := T(q)- \frac{\log(1-q)}{\log(q)} $$ is strictly increasing in $ (0,1) $ while $$ F(q) := \frac{1-q}{q} \,H(q) $$ is strictly decreasing in $ (0,1) $. These results are then applied to obtain various inequalities, one of which states that the double-inequality $$ \alpha \,\frac{q}{1-q}+\frac{\log(1-q)}{\log(q)} < T(q)< \beta \,\frac{q}{1-q}+\frac{\log(1-q)}{\log(q)}, \quad 0<q<1, $$ holds with the best possible constant factors $\alpha=\gamma$ and $\beta=1$. Here, $\gamma$ denotes Euler's constant. This refines a result of Salem, who proved the inequalities with $\alpha=1/2$ and $\beta=1$.