In mathematics, the prime zeta function is an analogue of the Riemann zeta function, studied by Glaisher (1891). It is defined as the following infinite series, which converges for ℜ ( s ) > 1 {\displaystyle \Re (s)>1} :
P ( s ) = ∑ p ∈ p r i m e s 1 p s = 1 2 s + 1 3 s + 1 5 s + 1 7 s + 1 11 s + … . {\displaystyle P(s)=\sum _{p\,\in \mathrm {\,primes} }{\frac {1}{p^{s}}}={\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+{\frac {1}{5^{s}}}+{\frac {1}{7^{s}}}+{\frac {1}{11^{s}}}+\dots \ .}
Properties The Euler product for the Riemann zeta function ζ ( s ) {\displaystyle \zeta (s)} implies that
log ζ ( s ) = ∑ n > 0 P ( n s ) n , {\displaystyle \log \zeta (s)=\sum _{n>0}{\frac {P(ns)}{n}},}
which by Möbius inversion gives
P ( s ) = ∑ n > 0 μ ( n ) log ζ ( n s ) n {\displaystyle P(s)=\sum _{n>0}\mu (n){\frac {\log \zeta (ns)}{n}}}
When s {\displaystyle s} goes to 1, we have P ( s ) ∼ log ζ ( s ) ∼ log ( 1 s − 1 ) {\displaystyle \textstyle P(s)\sim \log \zeta (s)\sim \log \left({\frac {1}{s-1}}\right)} . This is used in the definition of Dirichlet density. This gives the continuation of P ( s ) {\displaystyle P(s)} to ℜ ( s ) > 0 {\displaystyle \Re (s)>0} , with an infinite number of logarithmic singularities at points s {\displaystyle s} where n s {\displaystyle ns} is a pole (only n s = 1 {\displaystyle ns=1} when n {\displaystyle n} is a squarefree number greater than or equal to 1), or zero of the Riemann zeta function ζ(.). The line ℜ ( s ) = 0 {\displaystyle \Re (s)=0} is a natural boundary as the singularities cluster near all points of this line. If one defines a sequence
a n = ∏ p k ∣ n 1 k = ∏ p k ∣∣ n 1 k ! {\displaystyle a_{n}=\prod _{p^{k}\mid n}{\frac {1}{k}}=\prod _{p^{k}\mid \mid n}{\frac {1}{k!}}}
then
P ( s ) = log ∑ n = 1 ∞ a n n s . {\displaystyle P(s)=\log \sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}.}
(Exponentiation shows that this is equivalent to Lemma 2.7 by Li.) The prime zeta function is related to Artin's constant by
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