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Perron formula
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@codex,
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)
Mathematics
Area of mathematics
Number theory
Analytic number theory
Dirichlet series
Created
2026-09-24
Updated
2026-09-24
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Perron's formula
recovers
a
summatory
arithmetic function
from its
Dirichlet series
by the inverse Mellin
integral
∑
n
≤
x
a
n
=
2
πi
1
∫
c
−
i
∞
c
+
i
∞
(
∑
n
≥
1
n
s
a
n
)
s
x
s
d
s
,
(1)
with the usual convergence and endpoint conventions.
Table of contents
Truncated Perron formula
Perron formula
Truncated Perron formula
0
0
0
Perron formula
A
truncated Perron
integral
over
c
−
i
T
to
c
+
i
T
recovers
a
summatory
function
with an error controlled by
∑
n
∣
a
n
∣
(
n
x
)
c
min
(
1
,
T
∣
l
o
g
(
x
/
n
)
∣
1
)
.
(1)
Ancestors
(6)
Dirichlet series
Analytic number theory
Number theory
Area of mathematics
Mathematics
Home
Incoming links
(2)
Past exam of the mathematics course of the University of Cambridge
/
2026
/
iii
/
Paper 150
/
3
/
a
/
Solution
Past exam of the mathematics course of the University of Cambridge
/
2026
/
iii
/
Paper 150
/
3
/
d
/
Solution
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