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Keywords:
divisor function; infinite series; inequality; monotonicity; $q$-digamma function; Euler's constant
divisor function; infinite series; inequality; monotonicity; $q$-digamma function; Euler's constant
Summary:
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 prove that $$ H(q) = T(q)- \frac {\log (1-q)}{\log (q)} $$ is strictly increasing on $ (0,1)$, while $$ F(q) = \frac {1-q}{q} H(q) $$ is strictly decreasing on $(0,1)$. 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 $\alpha =\frac 12$ and $\beta =1$.
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