A logarithmic estimate for harmonic sums and the digamma function, with an application to the Dirichlet divisor problem

• Main Author: G. J. O. Jameson
• Format: Article
• Language: English
• Published: SpringerOpen 2019-05-01
• Series: Journal of Inequalities and Applications
• Subjects: Harmonic sum; Euler’s constant; Digamma function; Divisor problem
• Online Access: http://link.springer.com/article/10.1186/s13660-019-2104-9

Abstract Let Hn=∑r=1n1/r $H_{n} = \sum_{r=1}^{n} 1/r$ and Hn(x)=∑r=1n1/(r+x) $H_{n}(x) = \sum_{r=1}^{n} 1/(r+x)$. Let ψ(x) $\psi(x)$ denote the digamma function. It is shown that Hn(x)+ψ(x+1) $H_{n}(x) + \psi(x+1)$ is approximated by 12logf(n+x) $\frac{1}{2}\log f(n+x)$, where f(x)=x2+x+13 $f(x) = x^{2} + x + \frac{1}{3}$, with error term of order (n+x)−5 $(n+x)^{-5}$. The cases x=0 $x = 0$ and n=0 $n = 0$ equate to estimates for Hn−γ $H_{n} - \gamma $ and ψ(x+1) $\psi(x+1)$ itself. The result is applied to determine exact bounds for a remainder term occurring in the Dirichlet divisor problem.