Perron's formula

Perron's formula recovers a summatory arithmetic function from its Dirichlet series by the inverse Mellin integral

\sum_{n \leq x} a_{n} = \frac{1}{2 πi} \int_{c - i \infty}^{c + i \infty} (\sum_{n \geq 1} \frac{a _{n}}{n ^{s}}) \frac{x ^{s}}{s} d s,

with the usual convergence and endpoint conventions.

Table of contents
- Logarithmically smoothed Perron formula Perron's formula
  - Unsmoothing a logarithmically weighted sum Logarithmically smoothed Perron formula
- Truncated Perron formula Perron's formula
  - Short-interval Perron bound for the second Chebyshev function Truncated Perron formula
  - Truncated Perron kernel estimate Truncated Perron formula

Logarithmically smoothed Perron formula ( $\sum_{n \leq x} a_{n} lo g (x / n)$ )

If the Dirichlet series

F (s) = \sum_{n} a_{n} n^{- s}

is absolutely convergent at

σ > 0

, then

\sum_{n \leq x} a_{n} lo g (x / n) = \frac{1}{2 πi} \int_{σ - i \infty}^{σ + i \infty} \frac{F ( s ) x ^{s}}{s ^{2}} d s = \frac{1}{2 π} \int_{R} \frac{F ( σ + i t ) x ^{σ + i t}}{( σ + i t ) ^{2}} d t .

The scalar kernel is

(2 π)^{- 1} \int_{R} e^{v (σ + i t)} (σ + i t)^{- 2} d t = v_{+}

. This follows by Fourier inversion applied to

u e^{- σ u} 1_{u \geq 0}

, whose Fourier transform is

(σ + i t)^{- 2}

. Absolute convergence of the series and of

\int ∣ σ + i t ∣^{- 2} d t

justifies interchange. The factor

d s = i d t

is essential.

Unsmoothing a logarithmically weighted sum ( $∣ S (x) ∣ ≪ x (lo g x)^{(β + k) /2}$ )

Suppose

∣ a_{n} ∣ \leq (lo g (2 n))^{k}

and

A (x) = \sum_{n \leq x} a_{n} lo g (x / n) = O (x (lo g x)^{β})

, with

0 \leq β \leq k

. For

0 < h \leq 1

, direct subtraction gives

A (x e^{h}) - A (x) = h \sum_{n \leq x} a_{n} + \sum_{x < n \leq x e^{h}} a_{n} lo g (x e^{h} / n) .

The last term has absolute value

O ((x h^{2} + h) (lo g x)^{k})

. Choose

h = (lo g x)^{(β - k) /2}

to obtain the displayed bound, with an additional

O ((lo g x)^{k})

term harmless as

x \to \infty

for fixed

k

. This elementary finite-difference argument converts a logarithmically smoothed Perron formula bound to cancellation in its original partial sum.

Truncated Perron formula

A truncated Perron integral over

c - i T

to

c + i T

recovers a summatory function with an error controlled by

\sum_{n} ∣ a_{n} ∣ (\frac{x}{n})^{c} min (1, \frac{1}{T ∣ l o g ( x / n ) ∣}) .

Short-interval Perron bound for the second Chebyshev function ( $ψ (x + y) - ψ (x) ≪ y \int_{- T}^{T} ∣ ζ^{'} (c + i t) / ζ (c + i t) ∣ d t + x lo g^{2} x / T$ )

For

c = 1 + 1/ lo g x

,

1 \leq y \leq x

and

2 \leq T \leq x

, apply the truncated Perron formula with Von Mangoldt function coefficients at both endpoints on the same vertical line. The difference kernel is

((x + y)^{s} - x^{s}) / s = \int_{x}^{x + y} u^{s - 1} d u

O (y)

. The usual near-integer error bound contributes

O (x lo g^{2} x / T)

. The logarithmic derivative on this line is finite because of the Euler product.

Truncated Perron kernel estimate ( $K_{T} (v) = (2 πi)^{- 1} \int_{c - i T}^{c + i T} v^{s} d s / s$ )

For

v \neq = 1

,

K_{T} (v)

differs from

1_{v > 1}

by

O (v^{c} min (1, (T ∣ lo g v ∣)^{- 1}))

. Close the contour to the left for

v > 1

and to the right for

v < 1

; horizontal sides give the logarithmic denominator and the nearby transition is bounded. At

v = 1

, direct integration gives

K_{T} (1) = π^{- 1} arctan (T / c) = 1/2 + O (c / T)

. This explains the half-weight endpoint in the truncated Perron formula. The constants may be taken uniform for

1 \leq c \leq 2

and

T \geq 2

.

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Perron's formula is a result in analytic number theory that provides a way to express the sum of the count of integer solutions to certain equations involving prime numbers. It specifically relates to the distribution of prime numbers and is often applied in studies of prime power distributions. The formula is closely associated with the theory of Dirichlet series and often comes up in the context of additive number theory.

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