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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)}.

 

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