- Source: Prime zeta function
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}}}+\cdots .}
Properties
The Euler product for the Riemann zeta function ζ(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 goes to 1, we have
P
(
s
)
∼
log
ζ
(
s
)
∼
log
(
1
s
−
1
)
{\displaystyle 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) to
ℜ
(
s
)
>
0
{\displaystyle \Re (s)>0}
, with an infinite number of logarithmic singularities at points s where ns is a pole (only ns = 1 when 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
ln
C
A
r
t
i
n
=
−
∑
n
=
2
∞
(
L
n
−
1
)
P
(
n
)
n
{\displaystyle \ln C_{\mathrm {Artin} }=-\sum _{n=2}^{\infty }{\frac {(L_{n}-1)P(n)}{n}}}
where Ln is the nth Lucas number.
Specific values are:
Analysis
= Integral
=The integral over the prime zeta function is usually anchored at infinity,
because the pole at
s
=
1
{\displaystyle s=1}
prohibits defining a nice lower bound
at some finite integer without entering a discussion on branch cuts in the complex plane:
∫
s
∞
P
(
t
)
d
t
=
∑
p
1
p
s
log
p
{\displaystyle \int _{s}^{\infty }P(t)\,dt=\sum _{p}{\frac {1}{p^{s}\log p}}}
The noteworthy values are again those where the sums converge slowly:
= Derivative
=The first derivative is
P
′
(
s
)
≡
d
d
s
P
(
s
)
=
−
∑
p
log
p
p
s
{\displaystyle P'(s)\equiv {\frac {d}{ds}}P(s)=-\sum _{p}{\frac {\log p}{p^{s}}}}
The interesting values are again those where the sums converge slowly:
Generalizations
= Almost-prime zeta functions
=As the Riemann zeta function is a sum of inverse powers over the integers
and the prime zeta function a sum of inverse powers of the prime numbers,
the
k
{\displaystyle k}
-primes (the integers which are a product of
k
{\displaystyle k}
not
necessarily distinct primes) define a sort of intermediate sums:
P
k
(
s
)
≡
∑
n
:
Ω
(
n
)
=
k
1
n
s
{\displaystyle P_{k}(s)\equiv \sum _{n:\Omega (n)=k}{\frac {1}{n^{s}}}}
where
Ω
{\displaystyle \Omega }
is the total number of prime factors.
Each integer in the denominator of the Riemann zeta function
ζ
{\displaystyle \zeta }
may be classified by its value of the index
k
{\displaystyle k}
, which decomposes the Riemann zeta
function into an infinite sum of the
P
k
{\displaystyle P_{k}}
:
ζ
(
s
)
=
1
+
∑
k
=
1
,
2
,
…
P
k
(
s
)
{\displaystyle \zeta (s)=1+\sum _{k=1,2,\ldots }P_{k}(s)}
Since we know that the Dirichlet series (in some formal parameter u) satisfies
P
Ω
(
u
,
s
)
:=
∑
n
≥
1
u
Ω
(
n
)
n
s
=
∏
p
∈
P
(
1
−
u
p
−
s
)
−
1
,
{\displaystyle P_{\Omega }(u,s):=\sum _{n\geq 1}{\frac {u^{\Omega (n)}}{n^{s}}}=\prod _{p\in \mathbb {P} }\left(1-up^{-s}\right)^{-1},}
we can use formulas for the symmetric polynomial variants with a generating function of the right-hand-side type. Namely, we have the coefficient-wise identity that
P
k
(
s
)
=
[
u
k
]
P
Ω
(
u
,
s
)
=
h
(
x
1
,
x
2
,
x
3
,
…
)
{\displaystyle P_{k}(s)=[u^{k}]P_{\Omega }(u,s)=h(x_{1},x_{2},x_{3},\ldots )}
when the sequences correspond to
x
j
:=
j
−
s
χ
P
(
j
)
{\displaystyle x_{j}:=j^{-s}\chi _{\mathbb {P} }(j)}
where
χ
P
{\displaystyle \chi _{\mathbb {P} }}
denotes the characteristic function of the primes. Using Newton's identities, we have a general formula for these sums given by
P
n
(
s
)
=
∑
k
1
+
2
k
2
+
⋯
+
n
k
n
=
n
k
1
,
…
,
k
n
≥
0
[
∏
i
=
1
n
P
(
i
s
)
k
i
k
i
!
⋅
i
k
i
]
=
−
[
z
n
]
log
(
1
−
∑
j
≥
1
P
(
j
s
)
z
j
j
)
.
{\displaystyle P_{n}(s)=\sum _{{k_{1}+2k_{2}+\cdots +nk_{n}=n} \atop {k_{1},\ldots ,k_{n}\geq 0}}\left[\prod _{i=1}^{n}{\frac {P(is)^{k_{i}}}{k_{i}!\cdot i^{k_{i}}}}\right]=-[z^{n}]\log \left(1-\sum _{j\geq 1}{\frac {P(js)z^{j}}{j}}\right).}
Special cases include the following explicit expansions:
P
1
(
s
)
=
P
(
s
)
P
2
(
s
)
=
1
2
(
P
(
s
)
2
+
P
(
2
s
)
)
P
3
(
s
)
=
1
6
(
P
(
s
)
3
+
3
P
(
s
)
P
(
2
s
)
+
2
P
(
3
s
)
)
P
4
(
s
)
=
1
24
(
P
(
s
)
4
+
6
P
(
s
)
2
P
(
2
s
)
+
3
P
(
2
s
)
2
+
8
P
(
s
)
P
(
3
s
)
+
6
P
(
4
s
)
)
.
{\displaystyle {\begin{aligned}P_{1}(s)&=P(s)\\P_{2}(s)&={\frac {1}{2}}\left(P(s)^{2}+P(2s)\right)\\P_{3}(s)&={\frac {1}{6}}\left(P(s)^{3}+3P(s)P(2s)+2P(3s)\right)\\P_{4}(s)&={\frac {1}{24}}\left(P(s)^{4}+6P(s)^{2}P(2s)+3P(2s)^{2}+8P(s)P(3s)+6P(4s)\right).\end{aligned}}}
= Prime modulo zeta functions
=Constructing the sum not over all primes but only over primes which are in the same modulo class introduces further types of infinite series that are a reduction of the Dirichlet L-function.
See also
Divergence of the sum of the reciprocals of the primes
References
Merrifield, C. W. (1881). "The Sums of the Series of Reciprocals of the Prime Numbers and of Their Powers". Proceedings of the Royal Society. 33 (216–219): 4–10. doi:10.1098/rspl.1881.0063. JSTOR 113877.
Fröberg, Carl-Erik (1968). "On the prime zeta function". Nordisk Tidskr. Informationsbehandling (BIT). 8 (3): 187–202. doi:10.1007/BF01933420. MR 0236123. S2CID 121500209.
Glaisher, J. W. L. (1891). "On the Sums of Inverse Powers of the Prime Numbers". Quart. J. Math. 25: 347–362.
Mathar, Richard J. (2008). "Twenty digits of some integrals of the prime zeta function". arXiv:0811.4739 [math.NT].
Li, Ji (2008). "Prime graphs and exponential composition of species". Journal of Combinatorial Theory. Series A. 115 (8): 1374–1401. arXiv:0705.0038. doi:10.1016/j.jcta.2008.02.008. MR 2455584. S2CID 6234826.
Mathar, Richard J. (2010). "Table of Dirichlet L-series and prime zeta modulo functions for small moduli". arXiv:1008.2547 [math.NT].
External links
Weisstein, Eric W. "Prime Zeta Function". MathWorld.
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- Prime zeta function
- List of zeta functions
- Riemann zeta function
- Riemann hypothesis
- Ihara zeta function
- Prime omega function
- Dedekind zeta function
- Hasse–Weil zeta function
- Particular values of the Riemann zeta function
- Analytic continuation
