Dirichlet series

In mathematics, a Dirichlet series, one of a number of concepts named in honor of Johann Peter Gustav Lejeune Dirichlet, is a series of the form

Related Topics:
Mathematics - Johann Peter Gustav Lejeune Dirichlet - Series

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:f(s)=sum_{n=1}^{infty} rac{a_n}{n^s}.

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The most famous of Dirichlet series is

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:zeta(s)=sum_{n=1}^{infty} rac{1}{n^s},

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which is the Riemann zeta function.

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Other Dirichlet series are:

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:rac{1}{zeta(s)}=sum_{n=1}^{infty} rac{mu(n)}{n^s}

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where μ(n) is the Möbius function,

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:rac{zeta(s-1)}{zeta(s)}=sum_{n=1}^{infty}

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rac{arphi(n)}{n^s}

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where φ(n) is the totient function, and

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:zeta(s) zeta(s-a)=sum_{n=1}^{infty} rac{sigma_{a}(n)}{n^s}

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:rac{zeta(s)zeta(s-a)zeta(s-b)zeta(s-a-b)}{zeta(2s-a-b)}

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