A logarithmic estimate for harmonic sums and the digamma function, with an application to the Dirichlet divisor problem
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...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2019-05-01
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| Series: | Journal of Inequalities and Applications |
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| Online Access: | http://link.springer.com/article/10.1186/s13660-019-2104-9 |