The primzeta function is a mathematical function that plays a role in analytical number theory , a branch of mathematics . It is related to the Riemann zeta function . Like many other number theoretic functions, it gets its meaning from the connection to the prime numbers .
definition
For a complex number whose real part is greater than 1, the primzeta function is defined using a Dirichlet series that extends over all prime numbers.
s
{\ displaystyle s}
P
(
s
)
=
∑
p
p
r
i
m
1
p
s
=
1
2
s
+
1
3
s
+
1
5
s
+
1
7th
s
+
1
11
s
+
...
{\ displaystyle P (s) = \ sum _ {p \ \ mathrm {prim}} {\ frac {1} {p ^ {s}}} = {\ frac {1} {2 ^ {s}}} + {\ frac {1} {3 ^ {s}}} + {\ frac {1} {5 ^ {s}}} + {\ frac {1} {7 ^ {s}}} + {\ frac {1 } {11 ^ {s}}} + \ ldots}
Although this representation only converges on the right half-plane shifted by 1, there is a continuation to the complete right half-plane , which, however, is not meromorphic in all points .
H
=
{
s
∈
C.
∣
R.
e
s
>
0
}
{\ displaystyle \ mathbb {H} = \ {s \ in \ mathbb {C} \ mid \ mathrm {Re} \, s> 0 \}}
Connection to the Riemann zeta function
There is a connection between the primzeta function and the logarithmic Riemann zeta function. This applies to everyone and is expressed in a formula:
R.
e
s
>
0
{\ displaystyle \ mathrm {Re} \, s> 0}
log
ζ
(
s
)
=
∑
n
=
1
∞
P
(
n
s
)
n
=
P
(
s
)
+
P
(
2
s
)
2
+
P
(
3
s
)
3
+
P
(
4th
s
)
4th
+
...
{\ displaystyle \ log \ zeta (s) = \ sum _ {n = 1} ^ {\ infty} {\ frac {P (ns)} {n}} = P (s) + {\ frac {P (2s )} {2}} + {\ frac {P (3s)} {3}} + {\ frac {P (4s)} {4}} + \ ldots}
.
The Euler product of the zeta function serves as a simple way of proving this connection . With
ζ
(
s
)
=
∏
p
p
r
i
m
1
1
-
p
-
s
{\ displaystyle \ zeta (s) = \ prod _ {p \ \ mathrm {prim}} {\ frac {1} {1-p ^ {- s}}}}
is obtained by taking the logarithm on both sides:
log
ζ
(
s
)
=
log
∏
p
p
r
i
m
1
1
-
p
-
s
=
-
∑
p
p
r
i
m
log
(
1
-
1
p
s
)
=
∑
n
=
1
∞
P
(
n
s
)
n
{\ displaystyle \ log \ zeta (s) = \ log \ prod _ {p \ \ mathrm {prim}} {\ frac {1} {1-p ^ {- s}}} = - \ sum _ {p \ \ mathrm {prim}} \ log \ left (1 - {\ frac {1} {p ^ {s}}} \ right) = \ sum _ {n = 1} ^ {\ infty} {\ frac {P ( ns)} {n}}}
.
In the last step, the Taylor series of the natural logarithm was applied around the point .
x
=
1
{\ displaystyle x = 1}
Further representations
The frequently used representation is obtained via a Möbius inversion :
P
(
s
)
=
∑
n
=
1
∞
μ
(
n
)
n
log
ζ
(
n
s
)
=
log
ζ
(
s
)
-
1
2
log
ζ
(
2
s
)
-
1
3
log
ζ
(
3
s
)
-
1
5
log
ζ
(
5
s
)
+
1
6th
log
ζ
(
6th
s
)
+
...
{\ displaystyle P (s) = \ sum _ {n = 1} ^ {\ infty} {\ frac {\ mu (n)} {n}} \ log \ zeta (ns) = {\ log \ zeta (s ) - {\ frac {1} {2}} \ log \ zeta (2s) - {\ frac {1} {3}} \ log \ zeta (3s) - {\ frac {1} {5}} \ log \ zeta (5s) + {\ frac {1} {6}} \ log \ zeta (6s) + \ ldots}}
,
where the Möbius function is referred to here.
μ
{\ displaystyle \ mu}
properties
The primzeta function is a completely holomorphic function . It has a square-free positive integer singularities in the form of branch points at
{
s
∈
C.
∣
R.
e
s
>
1
}
{\ displaystyle \ {s \ in \ mathbb {C} \ mid \ mathrm {Re} \, s \,> 1 \}}
K
{\ displaystyle K}
all places
s
=
1
/
K
{\ displaystyle s = 1 / K}
all digits , where any ( non-trivial ) zero of the Riemann zeta function denotes.
s
=
ρ
/
K
{\ displaystyle s = \ rho / K}
ρ
{\ displaystyle \ rho}
This is considering the illustration
P
(
s
)
=
∑
n
=
1
∞
μ
(
n
)
n
log
ζ
(
n
s
)
{\ displaystyle P (s) = \ sum _ {n = 1} ^ {\ infty} {\ frac {\ mu (n)} {n}} \ log \ zeta (ns)}
clear, since the logarithm is not defined in all places or and ( in the sum).
ζ
(
K
⋅
ρ
K
)
=
ζ
(
ρ
)
=
0
{\ displaystyle \ textstyle \ zeta (K \ cdot {\ frac {\ rho} {K}}) = \ zeta (\ rho) = 0}
ζ
(
K
⋅
1
K
)
=
ζ
(
1
)
=
∞
{\ displaystyle \ textstyle \ zeta (K \ cdot {\ frac {1} {K}}) = \ zeta (1) = \ infty}
μ
(
K
)
≠
0
{\ displaystyle \ mu (K) \ not = 0}
n
=
K
{\ displaystyle n = K}
Since the Riemann zeta function has an infinite number of non-trivial zeros in the so-called critical strip , there is a compression of singularities on the straight line , which can be seen as the natural limit of the domain of definition of the primzeta function.
S.
=
{
s
∈
C.
|
1
>
R.
e
s
>
0
}
{\ displaystyle S = \ {s \ in \ mathbb {C} | 1> \ mathrm {Re} \, s> 0 \}}
R.
e
s
=
0
{\ displaystyle \ mathrm {Re} \, s = 0}
Furthermore applies to all :
t
{\ displaystyle t}
lim
σ
→
∞
P
(
σ
+
i
t
)
=
0
{\ displaystyle \ lim _ {\ sigma \ to \ infty} P (\ sigma + \ mathrm {i} t) = 0}
.
Derivation
The primzeta function is completely holomorphic . A derivative expression is:
{
s
∈
C.
|
R.
e
s
>
1
}
{\ displaystyle \ {s \ in \ mathbb {C} | \ mathrm {Re} \, s> 1 \}}
P
′
(
s
)
=
-
∑
p
p
r
i
m
log
p
p
s
{\ displaystyle P '(s) = - \ sum _ {p \ \ mathrm {prim}} {\ frac {\ log p} {p ^ {s}}}}
.
The following applies to the -th derivative:
k
{\ displaystyle k}
P
(
k
)
(
s
)
=
(
-
1
)
k
∑
p
p
r
i
m
(
log
p
)
k
p
s
{\ displaystyle P ^ {(k)} (s) = (- 1) ^ {k} \ sum _ {p \ \ mathrm {prim}} {\ frac {(\ log p) ^ {k}} {p ^ {s}}}}
.
Indefinite integral
An antiderivative is given by:
∫
P
(
s
)
d
s
=
-
∑
p
p
r
i
m
1
p
s
log
p
+
C.
{\ displaystyle \ int P (s) \ \ mathrm {d} s = - \ sum _ {p \ \ mathrm {prim}} {\ frac {1} {p ^ {s} \ log p}} + C}
.
Special values
As Euler was able to prove, the series of reciprocal values of all prime numbers is divergent. The following applies:
P
(
1
)
=
1
2
+
1
3
+
1
5
+
1
7th
+
1
11
+
...
=
∞
{\ displaystyle P (1) = {\ frac {1} {2}} + {\ frac {1} {3}} + {\ frac {1} {5}} + {\ frac {1} {7} } + {\ frac {1} {11}} + \ ldots = \ infty}
.
To this day nothing is known about other integer values of the primzeta function. Decimal expansions are:
P
(
2
)
=
0.452
24
74200
41065
49850
...
{\ displaystyle P (2) = 0 {,} 45224 \ 74200 \ 41065 \ 49850 \ ldots}
(Follow A085548 in OEIS )
P
(
3
)
=
0.174
76
26392
99443
53642
...
{\ displaystyle P (3) = 0 {,} 17476 \ 26392 \ 99443 \ 53642 \ ldots}
(Follow A085541 in OEIS )
P
(
4th
)
=
0.076
99
31397
64246
84494
...
{\ displaystyle P (4) = 0 {,} 07699 \ 31397 \ 64246 \ 84494 \ ldots}
(Follow A085964 in OEIS )
Web links
Individual evidence
↑ Komaravolu Chandrasekharan: Introduction to Analytical Number Theory . Springer Verlag, 1965/66, Chapter XI, page 2
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