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

Jameson, G. J. O. (2019) A logarithmic estimate for harmonic sums and the digamma function, with an application to the Dirichlet divisor problem. Journal of Inequalities and Applications, 2019 (1). ISSN 1029-242X

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.