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Zeta, gamma and the harmonic number

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Leonard Euler discovered one the classical formula of analytical number theory;

$\displaystyle{\sum_{r=1}^{n} \frac{1}{r}=\ln n+\gamma+O\Big(\frac{1}{n}\Big)}.$

where $\gamma$ is the Euler–Mascheroni constant. The sum on the RHS is called the harmonic number and is denoted by $H_n$. In this article, I shall represent this classical asymptotic formula of Euler in terms of the Riemann zeta function.

Entry 1. If $|x| \gg 1$ then

$\displaystyle{\sum_{r=1}^{n} \zeta(r + 1/x) = n + x + \gamma + O(1/|x| + 1/n)}.$

Entry 2. Taking $x = -n$ we obtain a new formula for the Euler–Mascheroni constant

$\displaystyle{\lim_{n \rightarrow \infty}\sum_{r=1}^{n} \zeta(r - 1/n) = \gamma}.$

Entry 3. If $\{y\}$ denotes the fractional part of $y$ then

$\displaystyle{\sum_{r=1}^{n} \{\zeta(r + 1/n)\} = 1 + \gamma + O\Big(\frac{1}{n}\Big)}.$

Entry 4. If $|x| \gg 1$ then

$\displaystyle{\sum_{r=1}^{n} \frac{\zeta(r + 1/x)}{r} = \ln n + x + \gamma + O(1/|x| + 1/n)}.$

Entry 5. If $\{y\}$ denotes the fractional part of $y$ then

$\displaystyle{\sum_{r=1}^{n} \bigg \{\frac{\zeta(r + 1/n)}{r}\bigg \}=\ln n+\gamma+O\Big(\frac{1}{n}\Big)}.$

The RHS of the above asymptotic formula is exactly equal to that of Euler’s formula and thus we have

Entry 6.

$\displaystyle{\sum_{r=1}^{n} \bigg \{\frac{\zeta(r + 1/n)}{r}\bigg \}=H_n+O\Big(\frac{1}{n}\Big)}.$