संख्या सिद्धांत में, प्राइम ओमेगा कार्य करता है
ω
(
n
)
{\displaystyle \omega (n)}
Ω
(
n
)
{\displaystyle \Omega (n)}
n
.
{\displaystyle n.}
ω
(
n
)
{\displaystyle \omega (n)}
Ω
(
n
)
{\displaystyle \Omega (n)}
n
,
{\displaystyle n,}
n {\displaystyle n}
n
=
p
1
α
1
p
2
α
2
⋯
p
k
α
k
{\displaystyle n=p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}}\cdots p_{k}^{\alpha _{k}}}
p
i
{\displaystyle p_{i}}
1
≤
i
≤
k
{\displaystyle 1\leq i\leq k}
ω
(
n
)
=
k
{\displaystyle \omega (n)=k}
Ω
(
n
)
=
α
1
+
α
2
+
⋯
+
α
k
{\displaystyle \Omega (n)=\alpha _{1}+\alpha _{2}+\cdots +\alpha _{k}}
गुण और संबंध कार्यक्रम
ω
(
n
)
{\displaystyle \omega (n)}
Ω
(
n
)
{\displaystyle \Omega (n)}
ω
(
n
)
=
∑
p
∣
n
1
{\displaystyle \omega (n)=\sum _{p\mid n}1}
p {\displaystyle p} n {\displaystyle n}
ω
(
12
)
=
ω
(
2
2
3
)
=
2
{\displaystyle \omega (12)=\omega (2^{2}3)=2}
Ω
(
n
)
=
∑
p
α
∣
n
1
=
∑
p
α
∥
n
α
{\displaystyle \Omega (n)=\sum _{p^{\alpha }\mid n}1=\sum _{p^{\alpha }\parallel n}\alpha }
p {\displaystyle p} n {\displaystyle n}
α
≥
1
{\displaystyle \alpha \geq 1}
Ω
(
12
)
=
Ω
(
2
2
3
1
)
=
3
{\displaystyle \Omega (12)=\Omega (2^{2}3^{1})=3}
p
α
∥
n
{\displaystyle p^{\alpha }\parallel n}
α
{\displaystyle \alpha }
p {\displaystyle p}
n
{\displaystyle n}
Ω
(
n
)
≥
ω
(
n
)
{\displaystyle \Omega (n)\geq \omega (n)}
Ω
(
n
)
=
ω
(
n
)
{\displaystyle \Omega (n)=\omega (n)}
n {\displaystyle n} वर्गमुक्त है और मोबियस फ़ंक्शन से संबंधित है
μ
(
n
)
=
(
−
1
)
ω
(
n
)
=
(
−
1
)
Ω
(
n
)
{\displaystyle \mu (n)=(-1)^{\omega (n)}=(-1)^{\Omega (n)}}
अगर
Ω
(
n
)
=
1
{\displaystyle \Omega (n)=1}
n {\displaystyle n}
यह ज्ञात है कि भाजक फलन का औसत क्रम संतुष्ट करता है
2
ω
(
n
)
≤
d
(
n
)
≤
2
Ω
(
n
)
{\displaystyle 2^{\omega (n)}\leq d(n)\leq 2^{\Omega (n)}}
[1] अंकगणितीय कार्य ों की तरह इसके लिए कोई स्पष्ट सूत्र नहीं है
Ω
(
n
)
{\displaystyle \Omega (n)}
ω
(
n
)
{\displaystyle \omega (n)}
के औसत क्रम के लिए एक स्पर्शोन्मुख श्रृंखला
ω
(
n
)
{\displaystyle \omega (n)}
[2]
1
n
∑
k
=
1
n
ω
(
k
)
∼
log
log
n
+
B
1
+
∑
k
≥
1
(
∑
j
=
0
k
−
1
γ
j
j
!
−
1
)
(
k
−
1
)
!
(
log
n
)
k
,
{\displaystyle {\frac {1}{n}}\sum \limits _{k=1}^{n}\omega (k)\sim \log \log n+B_{1}+\sum _{k\geq 1}\left(\sum _{j=0}^{k-1}{\frac {\gamma _{j}}{j!}}-1\right){\frac {(k-1)!}{(\log n)^{k}}},}
कहाँ
B
1
≈
0.26149721
{\displaystyle B_{1}\approx 0.26149721}
γ
j
{\displaystyle \gamma _{j}}
स्टिल्टजेस स्थिरांक हैं।
कार्यक्रम
ω
(
n
)
{\displaystyle \omega (n)}
[3]
∑
d
∣
n
|
μ
(
d
)
|
=
2
ω
(
n
)
{\displaystyle \sum _{d\mid n}|\mu (d)|=2^{\omega (n)}}
∑
d
∣
n
|
μ
(
d
)
|
k
ω
(
d
)
=
(
k
+
1
)
ω
(
n
)
{\displaystyle \sum _{d\mid n}|\mu (d)|k^{\omega (d)}=(k+1)^{\omega (n)}}
∑
r
∣
n
2
ω
(
r
)
=
d
(
n
2
)
{\displaystyle \sum _{r\mid n}2^{\omega (r)}=d(n^{2})}
∑
r
∣
n
2
ω
(
r
)
d
(
n
r
)
=
d
2
(
n
)
{\displaystyle \sum _{r\mid n}2^{\omega (r)}d\left({\frac {n}{r}}\right)=d^{2}(n)}
∑
d
∣
n
(
−
1
)
ω
(
d
)
=
∏
p
α
|
|
n
(
1
−
α
)
{\displaystyle \sum _{d\mid n}(-1)^{\omega (d)}=\prod \limits _{p^{\alpha }||n}(1-\alpha )}
∑
(
k
,
m
)
=
1
1
≤
k
≤
m
gcd
(
k
2
−
1
,
m
1
)
gcd
(
k
2
−
1
,
m
2
)
=
φ
(
n
)
∑
d
2
∣
m
2
d
1
∣
m
1
φ
(
gcd
(
d
1
,
d
2
)
)
2
ω
(
lcm
(
d
1
,
d
2
)
)
,
m
1
,
m
2
odd
,
m
=
lcm
(
m
1
,
m
2
)
{\displaystyle \sum _{\stackrel {1\leq k\leq m}{(k,m)=1}}\gcd(k^{2}-1,m_{1})\gcd(k^{2}-1,m_{2})=\varphi (n)\sum _{\stackrel {d_{1}\mid m_{1}}{d_{2}\mid m_{2}}}\varphi (\gcd(d_{1},d_{2}))2^{\omega (\operatorname {lcm} (d_{1},d_{2}))},\ m_{1},m_{2}{\text{ odd}},m=\operatorname {lcm} (m_{1},m_{2})}
∑
gcd
(
k
,
m
)
=
1
1
≤
k
≤
n
1
=
n
φ
(
m
)
m
+
O
(
2
ω
(
m
)
)
{\displaystyle \sum _{\stackrel {1\leq k\leq n}{\operatorname {gcd} (k,m)=1}}\!\!\!\!1=n{\frac {\varphi (m)}{m}}+O\left(2^{\omega (m)}\right)}
अभाज्य संख्याओं के विशिष्ट कार्य को डिरिचलेट कनवल्शन द्वारा व्यक्त किया जा सकता है
मोबियस फ़ंक्शन:[4]
χ
P
(
n
)
=
(
μ
∗
ω
)
(
n
)
=
∑
d
|
n
ω
(
d
)
μ
(
n
/
d
)
.
{\displaystyle \chi _{\mathbb {P} }(n)=(\mu \ast \omega )(n)=\sum _{d|n}\omega (d)\mu (n/d).}
के लिए विभाजन-संबंधित सटीक पहचान
ω
(
n
)
{\displaystyle \omega (n)}
[5]
ω
(
n
)
=
log
2
[
∑
k
=
1
n
∑
j
=
1
k
(
∑
d
∣
k
∑
i
=
1
d
p
(
d
−
j
i
)
)
s
n
,
k
⋅
|
μ
(
j
)
|
]
,
{\displaystyle \omega (n)=\log _{2}\left[\sum _{k=1}^{n}\sum _{j=1}^{k}\left(\sum _{d\mid k}\sum _{i=1}^{d}p(d-ji)\right)s_{n,k}\cdot |\mu (j)|\right],}
कहाँ
p
(
n
)
{\displaystyle p(n)}
विभाजन (संख्या सिद्धांत) है,
μ
(
n
)
{\displaystyle \mu (n)}
s
n
,
k
{\displaystyle s_{n,k}}
s
n
,
k
=
[
q
n
]
(
q
;
q
)
∞
q
k
1
−
q
k
=
s
o
(
n
,
k
)
−
s
e
(
n
,
k
)
,
{\displaystyle s_{n,k}=[q^{n}](q;q)_{\infty }{\frac {q^{k}}{1-q^{k}}}=s_{o}(n,k)-s_{e}(n,k),}
अनंत q-पॉचहैमर प्रतीक और प्रतिबंधित विभाजन कार्यों के संदर्भ में
s
o
/
e
(
n
,
k
)
{\displaystyle s_{o/e}(n,k)}
k {\displaystyle k} n {\displaystyle n} [6]
जटिल तल की निरंतरता की एक निरंतरता
ω
(
n
)
{\displaystyle \omega (n)}
[7]
sinc
{\displaystyle \operatorname {sinc} }
sinc
(
x
)
=
sin
(
π
x
)
π
x
{\displaystyle \operatorname {sinc} (x)={\frac {\sin(\pi x)}{\pi x}}}
ω
(
z
)
=
log
2
(
∑
x
=
1
⌈
R
e
(
z
)
⌉
sinc
(
∏
y
=
1
⌈
R
e
(
z
)
⌉
+
1
(
x
2
+
x
−
y
z
)
)
)
{\displaystyle \omega (z)=\log _{2}\left(\sum _{x=1}^{\lceil Re(z)\rceil }\operatorname {sinc} \left(\prod _{y=1}^{\lceil Re(z)\rceil +1}\left(x^{2}+x-yz\right)\right)\right)}
औसत क्रम और सारांश फलन दोनों के अंकगणितीय फलन का औसत क्रम
ω
(
n
)
{\displaystyle \omega (n)}
Ω
(
n
)
{\displaystyle \Omega (n)}
log
log
n
{\displaystyle \log \log n}
n {\displaystyle n}
ω
(
n
)
=
1
{\displaystyle \omega (n)=1}
n {\displaystyle n} आदिम है तो कार्य उतना ही बड़ा है
ω
(
n
)
∼
log
n
log
log
n
{\displaystyle \omega (n)\sim {\frac {\log n}{\log \log n}}}
n {\displaystyle n} 2 की शक्ति है
Ω
(
n
)
∼
log
n
log
2
{\displaystyle \Omega (n)\sim {\frac {\log n}{\log 2}}}
[8] सारांश कार्यों के लिए स्पर्शोन्मुखता
ω
(
n
)
{\displaystyle \omega (n)}
Ω
(
n
)
{\displaystyle \Omega (n)}
ω
(
n
)
2
{\displaystyle \omega (n)^{2}}
[9] [10]
∑
n
≤
x
ω
(
n
)
=
x
log
log
x
+
B
1
x
+
o
(
x
)
∑
n
≤
x
Ω
(
n
)
=
x
log
log
x
+
B
2
x
+
o
(
x
)
∑
n
≤
x
ω
(
n
)
2
=
x
(
log
log
x
)
2
+
O
(
x
log
log
x
)
∑
n
≤
x
ω
(
n
)
k
=
x
(
log
log
x
)
k
+
O
(
x
(
log
log
x
)
k
−
1
)
,
k
∈
Z
+
,
{\displaystyle {\begin{aligned}\sum _{n\leq x}\omega (n)&=x\log \log x+B_{1}x+o(x)\\\sum _{n\leq x}\Omega (n)&=x\log \log x+B_{2}x+o(x)\\\sum _{n\leq x}\omega (n)^{2}&=x(\log \log x)^{2}+O(x\log \log x)\\\sum _{n\leq x}\omega (n)^{k}&=x(\log \log x)^{k}+O(x(\log \log x)^{k-1}),k\in \mathbb {Z} ^{+},\end{aligned}}}
कहाँ
B
1
≈
0.2614972128
{\displaystyle B_{1}\approx 0.2614972128}
B
2
{\displaystyle B_{2}}
B
2
=
B
1
+
∑
p
prime
1
p
(
p
−
1
)
≈
1.0345061758.
{\displaystyle B_{2}=B_{1}+\sum _{p{\text{ prime}}}{\frac {1}{p(p-1)}}\approx 1.0345061758.}
प्राइम ओमेगा फ़ंक्शंस के दो प्रकारों से संबंधित अन्य योगों में शामिल हैं [11]
∑
n
≤
x
{
Ω
(
n
)
−
ω
(
n
)
}
=
O
(
x
)
,
{\displaystyle \sum _{n\leq x}\left\{\Omega (n)-\omega (n)\right\}=O(x),}
और
#
{
n
≤
x
:
Ω
(
n
)
−
ω
(
n
)
>
log
log
x
}
=
O
(
x
(
log
log
x
)
1
/
2
)
.
{\displaystyle \#\left\{n\leq x:\Omega (n)-\omega (n)>{\sqrt {\log \log x}}\right\}=O\left({\frac {x}{(\log \log x)^{1/2}}}\right).}
उदाहरण I: एक संशोधित सारांश फलन इस उदाहरण में हम सारांश कार्यों का एक प्रकार सुझाते हैं
S
ω
(
x
)
:=
∑
n
≤
x
ω
(
n
)
{\displaystyle S_{\omega }(x):=\sum _{n\leq x}\omega (n)}
x {\displaystyle x}
S
ω
(
x
)
{\displaystyle S_{\omega }(x)}
[12]
S
odd
(
x
)
:=
∑
n
≤
x
ω
(
n
)
[
n
odd
]
,
{\displaystyle S_{\operatorname {odd} }(x):=\sum _{n\leq x}\omega (n)[n{\text{ odd}}],}
कहाँ
[
⋅
]
{\displaystyle [\cdot ]}
इवरसन ब्रैकेट को दर्शाता है। फिर हमारे पास वह है
S
odd
(
x
)
=
x
2
log
log
x
+
(
2
B
1
−
1
)
x
4
+
{
x
4
}
−
[
x
≡
2
,
3
mod
4
]
+
O
(
x
log
x
)
.
{\displaystyle S_{\operatorname {odd} }(x)={\frac {x}{2}}\log \log x+{\frac {(2B_{1}-1)x}{4}}+\left\{{\frac {x}{4}}\right\}-\left[x\equiv 2,3{\bmod {4}}\right]+O\left({\frac {x}{\log x}}\right).}
इस परिणाम का प्रमाण पहले उसका अवलोकन करने से होता है
ω
(
2
n
)
=
{
ω
(
n
)
+
1
,
if
n
is odd;
ω
(
n
)
,
if
n
is even,
{\displaystyle \omega (2n)={\begin{cases}\omega (n)+1,&{\text{if }}n{\text{ is odd; }}\\\omega (n),&{\text{if }}n{\text{ is even,}}\end{cases}}}
और फिर सारांश कार्य के लिए हार्डी और राइट से प्राप्त स्पर्शोन्मुख परिणाम को लागू करना
ω
(
n
)
{\displaystyle \omega (n)}
S
ω
(
x
)
:=
∑
n
≤
x
ω
(
n
)
{\displaystyle S_{\omega }(x):=\sum _{n\leq x}\omega (n)}
S
ω
(
x
)
=
S
odd
(
x
)
+
∑
n
≤
⌊
x
2
⌋
ω
(
2
n
)
=
S
odd
(
x
)
+
∑
n
≤
⌊
x
4
⌋
(
ω
(
4
n
)
+
ω
(
4
n
+
2
)
)
=
S
odd
(
x
)
+
∑
n
≤
⌊
x
4
⌋
(
ω
(
2
n
)
+
ω
(
2
n
+
1
)
+
1
)
=
S
odd
(
x
)
+
S
ω
(
⌊
x
2
⌋
)
+
⌊
x
4
⌋
.
{\displaystyle {\begin{aligned}S_{\omega }(x)&=S_{\operatorname {odd} }(x)+\sum _{n\leq \left\lfloor {\frac {x}{2}}\right\rfloor }\omega (2n)\\&=S_{\operatorname {odd} }(x)+\sum _{n\leq \left\lfloor {\frac {x}{4}}\right\rfloor }\left(\omega (4n)+\omega (4n+2)\right)\\&=S_{\operatorname {odd} }(x)+\sum _{n\leq \left\lfloor {\frac {x}{4}}\right\rfloor }\left(\omega (2n)+\omega (2n+1)+1\right)\\&=S_{\operatorname {odd} }(x)+S_{\omega }\left(\left\lfloor {\frac {x}{2}}\right\rfloor \right)+\left\lfloor {\frac {x}{4}}\right\rfloor .\end{aligned}}}
हार्डी और राइट के अध्याय 22.11 में विस्तारित गणनाएँ सारांश फ़ंक्शन के लिए स्पर्शोन्मुख अनुमान प्रदान करती हैं
ω
(
n
)
{
ω
(
n
)
−
1
}
,
{\displaystyle \omega (n)\left\{\omega (n)-1\right\},}
इन दो घटकों के उत्पाद का आकलन करके ओमेगा इस प्रकार कार्य करता है
ω
(
n
)
{
ω
(
n
)
−
1
}
=
∑
p
,
q
prime
p
≠
q
p
q
∣
n
1
=
∑
p
,
q
prime
p
q
∣
n
1
−
∑
p
prime
p
2
∣
n
1.
{\displaystyle \omega (n)\left\{\omega (n)-1\right\}=\sum _{\stackrel {pq\mid n}{\stackrel {p\neq q}{p,q{\text{ prime}}}}}1=\sum _{\stackrel {pq\mid n}{p,q{\text{ prime}}}}1-\sum _{\stackrel {p^{2}\mid n}{p{\text{ prime}}}}1.}
हम फ़ंक्शन के तथाकथित तथ्यात्मक क्षण पर संबंधित सारांश कार्यों के लिए समान रूप से एसिम्प्टोटिक सूत्रों की गणना कर सकते हैं
ω
(
n
)
{\displaystyle \omega (n)}
एक ज्ञात डिरिचलेट श्रृंखला शामिल है
ω
(
n
)
{\displaystyle \omega (n)}
रीमैन ज़ेटा फ़ंक्शन द्वारा दिया गया है [13]
∑
n
≥
1
2
ω
(
n
)
n
s
=
ζ
2
(
s
)
ζ
(
2
s
)
,
ℜ
(
s
)
>
1.
{\displaystyle \sum _{n\geq 1}{\frac {2^{\omega (n)}}{n^{s}}}={\frac {\zeta ^{2}(s)}{\zeta (2s)}},\ \Re (s)>1.}
वो तो हम भी देख सकते हैं
∑
n
≥
1
z
ω
(
n
)
n
s
=
∏
p
(
1
+
z
p
s
−
1
)
,
|
z
|
<
2
,
ℜ
(
s
)
>
1
,
{\displaystyle \sum _{n\geq 1}{\frac {z^{\omega (n)}}{n^{s}}}=\prod _{p}\left(1+{\frac {z}{p^{s}-1}}\right),|z|<2,\Re (s)>1,}
∑
n
≥
1
z
Ω
(
n
)
n
s
=
∏
p
(
1
−
z
p
s
)
−
1
,
|
z
|
<
2
,
ℜ
(
s
)
>
1
,
{\displaystyle \sum _{n\geq 1}{\frac {z^{\Omega (n)}}{n^{s}}}=\prod _{p}\left(1-{\frac {z}{p^{s}}}\right)^{-1},|z|<2,\Re (s)>1,}
कार्यक्रम
Ω
(
n
)
{\displaystyle \Omega (n)}
योगात्मक कार्य है, जहाँ
ω
(
n
)
{\displaystyle \omega (n)}
ω
(
n
)
{\displaystyle \omega (n)}
Ω
(
n
)
{\displaystyle \Omega (n)}
लेम्मा. लगता है कि f {\displaystyle f}
f
(
p
α
)
:=
f
0
(
p
,
α
)
{\displaystyle f(p^{\alpha }):=f_{0}(p,\alpha )}
f
(
p
1
α
1
⋯
p
k
α
k
)
=
f
0
(
p
1
,
α
1
)
+
⋯
+
f
0
(
p
k
,
α
k
)
{\displaystyle f(p_{1}^{\alpha _{1}}\cdots p_{k}^{\alpha _{k}})=f_{0}(p_{1},\alpha _{1})+\cdots +f_{0}(p_{k},\alpha _{k})}
p
i
{\displaystyle p_{i}}
α
i
≥
1
{\displaystyle \alpha _{i}\geq 1}
f {\displaystyle f}
∑
n
≥
1
f
(
n
)
n
s
=
ζ
(
s
)
×
∑
p
p
r
i
m
e
(
1
−
p
−
s
)
⋅
∑
n
≥
1
f
0
(
p
,
n
)
p
−
n
s
,
ℜ
(
s
)
>
min
(
1
,
σ
f
)
.
{\displaystyle \sum _{n\geq 1}{\frac {f(n)}{n^{s}}}=\zeta (s)\times \sum _{p\mathrm {\ prime} }(1-p^{-s})\cdot \sum _{n\geq 1}f_{0}(p,n)p^{-ns},\Re (s)>\min(1,\sigma _{f}).}
सबूत। हम देख सकते हैं कि
∑
n
≥
1
u
f
(
n
)
n
s
=
∏
p
p
r
i
m
e
(
1
+
∑
n
≥
1
u
f
0
(
p
,
n
)
p
−
n
s
)
.
{\displaystyle \sum _{n\geq 1}{\frac {u^{f(n)}}{n^{s}}}=\prod _{p\mathrm {\ prime} }\left(1+\sum _{n\geq 1}u^{f_{0}(p,n)}p^{-ns}\right).}
इसका अर्थ यह है कि
∑
n
≥
1
f
(
n
)
n
s
=
d
d
u
[
∏
p
p
r
i
m
e
(
1
+
∑
n
≥
1
u
f
0
(
p
,
n
)
p
−
n
s
)
]
|
u
=
1
=
∏
p
(
1
+
∑
n
≥
1
p
−
n
s
)
×
∑
p
∑
n
≥
1
f
0
(
p
,
n
)
p
−
n
s
1
+
∑
n
≥
1
p
−
n
s
=
ζ
(
s
)
×
∑
p
p
r
i
m
e
(
1
−
p
−
s
)
⋅
∑
n
≥
1
f
0
(
p
,
n
)
p
−
n
s
,
{\displaystyle {\begin{aligned}\sum _{n\geq 1}{\frac {f(n)}{n^{s}}}&={\frac {d}{du}}\left[\prod _{p\mathrm {\ prime} }\left(1+\sum _{n\geq 1}u^{f_{0}(p,n)}p^{-ns}\right)\right]{\Biggr |}_{u=1}=\prod _{p}\left(1+\sum _{n\geq 1}p^{-ns}\right)\times \sum _{p}{\frac {\sum _{n\geq 1}f_{0}(p,n)p^{-ns}}{1+\sum _{n\geq 1}p^{-ns}}}\\&=\zeta (s)\times \sum _{p\mathrm {\ prime} }(1-p^{-s})\cdot \sum _{n\geq 1}f_{0}(p,n)p^{-ns},\end{aligned}}}
जहां भी संबंधित श्रृंखला और उत्पाद अभिसरण होते हैं। पिछले समीकरण में, हमने रीमैन ज़ेटा फ़ंक्शन के यूलर उत्पाद प्रतिनिधित्व का उपयोग किया है।
⊡
{\displaystyle \boxdot }
ℜ
(
s
)
>
1
{\displaystyle \Re (s)>1}
D
ω
(
s
)
:=
∑
n
≥
1
ω
(
n
)
n
s
=
ζ
(
s
)
P
(
s
)
=
ζ
(
s
)
×
∑
n
≥
1
μ
(
n
)
n
log
ζ
(
n
s
)
D
Ω
(
s
)
:=
∑
n
≥
1
Ω
(
n
)
n
s
=
ζ
(
s
)
×
∑
n
≥
1
P
(
n
s
)
=
ζ
(
s
)
×
∑
n
≥
1
ϕ
(
n
)
n
log
ζ
(
n
s
)
D
Ω
λ
(
s
)
:=
∑
n
≥
1
λ
(
n
)
Ω
(
n
)
n
s
=
ζ
(
s
)
log
ζ
(
s
)
,
{\displaystyle {\begin{aligned}D_{\omega }(s)&:=\sum _{n\geq 1}{\frac {\omega (n)}{n^{s}}}=\zeta (s)P(s)\\&\ =\zeta (s)\times \sum _{n\geq 1}{\frac {\mu (n)}{n}}\log \zeta (ns)\\D_{\Omega }(s)&:=\sum _{n\geq 1}{\frac {\Omega (n)}{n^{s}}}=\zeta (s)\times \sum _{n\geq 1}P(ns)\\&\ =\zeta (s)\times \sum _{n\geq 1}{\frac {\phi (n)}{n}}\log \zeta (ns)\\D_{\Omega \lambda }(s)&:=\sum _{n\geq 1}{\frac {\lambda (n)\Omega (n)}{n^{s}}}=\zeta (s)\log \zeta (s),\end{aligned}}}
P
(
s
)
{\displaystyle P(s)}
प्राइम जीटा फ़ंक्शन है और
λ
(
n
)
=
(
−
1
)
Ω
(
n
)
{\displaystyle \lambda (n)=(-1)^{\Omega (n)}}
लिउविले लैम्ब्डा फ़ंक्शन ।
प्राइम ओमेगा फ़ंक्शंस के अंतर का वितरण अंतरों के विशिष्ट पूर्णांक मानों का वितरण
Ω
(
n
)
−
ω
(
n
)
{\displaystyle \Omega (n)-\omega (n)}
k
≥
0
{\displaystyle k\geq 0}
N
k
(
x
)
:=
#
(
{
n
∈
Z
+
:
Ω
(
n
)
−
ω
(
n
)
=
k
}
∩
[
1
,
x
]
)
.
{\displaystyle N_{k}(x):=\#(\{n\in \mathbb {Z} ^{+}:\Omega (n)-\omega (n)=k\}\cap [1,x]).}
इन कार्डिनलिटीज़ में सीमित घनत्वों का एक संगत क्रम होता है
d
k
{\displaystyle d_{k}}
x
≥
2
{\displaystyle x\geq 2}
N
k
(
x
)
=
d
k
⋅
x
+
O
(
(
3
4
)
k
x
(
log
x
)
4
3
)
.
{\displaystyle N_{k}(x)=d_{k}\cdot x+O\left(\left({\frac {3}{4}}\right)^{k}{\sqrt {x}}(\log x)^{\frac {4}{3}}\right).}
∑
k
≥
0
d
k
⋅
z
k
=
∏
p
(
1
−
1
p
)
(
1
+
1
p
−
z
)
.
{\displaystyle \sum _{k\geq 0}d_{k}\cdot z^{k}=\prod _{p}\left(1-{\frac {1}{p}}\right)\left(1+{\frac {1}{p-z}}\right).}
c
^
:=
1
4
×
∏
p
>
2
(
1
−
1
(
p
−
1
)
2
)
−
1
{\displaystyle {\hat {c}}:={\frac {1}{4}}\times \prod _{p>2}\left(1-{\frac {1}{(p-1)^{2}}}\right)^{-1}}
घनत्व
d
k
{\displaystyle d_{k}}
d
k
=
c
^
⋅
2
−
k
+
O
(
5
−
k
)
.
{\displaystyle d_{k}={\hat {c}}\cdot 2^{-k}+O(5^{-k}).}
[14] यह भी देखें टिप्पणियाँ
↑ This inequality is given in Section 22.13 of Hardy and Wright.
↑ S. R. Finch, Two asymptotic series, Mathematical Constants II, Cambridge Univ. Press, pp. 21-32, [1]
↑ Each of these started from the second identity in the list are cited individually on the pages Dirichlet convolutions of arithmetic functions , Menon's identity , and other formulas for Euler's totient function . The first identity is a combination of two known divisor sums cited in Section 27.6 of the NIST Handbook of Mathematical Functions .
↑ This is suggested as an exercise in Apostol's book. Namely, we write
f
=
μ
∗
ω
{\displaystyle f=\mu \ast \omega }
f
(
n
)
=
∑
d
|
n
μ
(
n
/
d
)
∑
r
|
d
(
π
(
r
)
−
π
(
r
−
1
)
)
{\displaystyle f(n)=\sum _{d|n}\mu (n/d)\sum _{r|d}\left(\pi (r)-\pi (r-1)\right)}
f {\displaystyle f}
D
f
(
s
)
:=
∑
n
≥
1
f
(
n
)
n
s
=
P
(
s
)
,
{\displaystyle D_{f}(s):=\sum _{n\geq 1}{\frac {f(n)}{n^{s}}}=P(s),}
P
(
s
)
{\displaystyle P(s)}
prime zeta function . Then it becomes obvious to see that
f
(
n
)
=
π
(
n
)
−
π
(
n
−
1
)
=
χ
P
(
n
)
{\displaystyle f(n)=\pi (n)-\pi (n-1)=\chi _{\mathbb {P} }(n)}
↑ This identity is proved in the article by Schmidt cited on this page below.
↑ This triangular sequence also shows up prominently in the Lambert series factorization theorems proved by Merca and Schmidt (2017–2018)
↑ Z. Hoelscher & E. Palsson, Counting restricted partitions of integers into fractions: symmetry and modes of the generating function and a connection to
ω
(
t
)
{\displaystyle \omega (t)}
The PUMP Journal of Undergraduate Research , 3 (2020), 277-307. [2]
↑ For references to each of these average order estimates see equations (3) and (18) of the MathWorld reference and Section 22.10-22.11 of Hardy and Wright.
↑ See Sections 22.10 and 22.11 for reference and explicit derivations of these asymptotic estimates.
↑ Actually, the proof of the last result given in Hardy and Wright actually suggests a more general procedure for extracting asymptotic estimates of the moments
∑
n
≤
x
ω
(
n
)
k
{\displaystyle \sum _{n\leq x}\omega (n)^{k}}
k
≥
2
{\displaystyle k\geq 2}
factorial moments of the form
∑
n
≤
x
[
ω
(
n
)
]
!
[
ω
(
n
)
−
m
]
!
{\displaystyle \sum _{n\leq x}{\frac {\left[\omega (n)\right]!}{\left[\omega (n)-m\right]!}}}
m
≥
2
{\displaystyle m\geq 2}
↑ Hardy and Wright Chapter 22.11.
↑ N.b., this sum is suggested by work contained in an unpublished manuscript by the contributor to this page related to the growth of the Mertens function . Hence it is not just a vacuous and/or trivial estimate obtained for the purpose of exposition here.
↑ This identity is found in Section 27.4 of the NIST Handbook of Mathematical Functions .
↑ Rényi, A.; Turán, P. (1958). "On a theorem of Erdös-Kac" (PDF) . Acta Arithmetica . 4 (1): 71–84. doi :10.4064/aa-4-1-71-84 .
संदर्भ G. H. Hardy and E. M. Wright (2006). An Introduction to the Theory of Numbers (6th ed.). Oxford University Press. H. L. Montgomery and R. C. Vaughan (2007). Multiplicative number theory I. Classical theory (1st ed.). Cambridge University Press. Schmidt, Maxie (2017). "Factorization Theorems for Hadamard Products and Higher-Order Derivatives of Lambert Series Generating Functions". arXiv :1712.00608 math.NT ]. Weisstein, Eric. "Distinct Prime Factors" . MathWorld . Retrieved 22 April 2018 .
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