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Analytic Number Theory
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Prime number theorem for arithmetic progressions
Prime number theorem for arithmetic progressions
Generalizes the Prime Number Theorem to primes in arithmetic progressions.
📜
The statement of the theorem
The number of primes
≤
x
\le x
≤
x
in the progression
a
,
a
+
q
,
a
+
2
q
,
…
a, a+q, a+2q, \dots
a
,
a
+
q
,
a
+
2
q
,
…
is asymptotic to
1
ϕ
(
q
)
x
ln
x
\frac{1}{\phi(q)} \frac{x}{\ln x}
ϕ
(
q
)
1
l
n
x
x
.
Source: Wikipedia