English

Abstract

Let $$T(q) = \sum\limits_{k = 1}^\infty {d(k){q^k},\,\,\,\,\left| q \right| < 1,} $$ 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 $$H(q) = T(q) - {{\log (1 - q)} \over {\log (q)}}$$ is strictly increasing on (0, 1), while $$F(q) = {{1 - q} \over q}H(q)$$ is strictly decreasing on (0, 1). These results are then applied to obtain various inequalities, one of which states that the double inequality $$\alpha {q \over {1 - q}} + {{\log (1 - q)} \over {\log (q)}} < T(q) < \beta {q \over {1 - q}} + {{\log (1 - q)} \over {\log (q)}},\,\,\,\,\,\,0 < q < 1,$$ holds with the best possible constant factors α = γ and β = 1. Here, γ denotes Euler’s constant. This refines a result of Salem, who proved the inequalities with $$\alpha = {1 \over 2}$$ and β = 1.