Buchstab function ( $w (u)$ )

The continuous Buchstab function satisfies

w (u) = 1/ u

for

1 \leq u \leq 2

and

\frac{d}{d u} (u w (u)) = w (u - 1)

for

u > 2

. It describes the density of rough numbers.

Table of contents
- Oscillation of the Buchstab function Buchstab function
- Buchstab theorem Buchstab function

Oscillation of the Buchstab function

The Buchstab function takes values both above and below

e^{- γ}

on every interval of length one in

[1, \infty)

\frac{d}{d u} (u (w (u) - e^{- γ})) = w (u - 1) - e^{- γ}

propagates any fixed sign forwards; the rapid decay to zero then forces an impossible identically zero tail.

Buchstab theorem

For each fixed

u > 1

Φ (z^{u}, z)

of z-sieved numbers at most

z^{u}

satisfies

Φ (z^{u}, z) \sim z^{u} w (u) / lo g z

z \to \infty

. The fixed-

u

asymptotic does not extend to

u = 1

.

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Buchstab function by

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The Buchstab function is a special arithmetic function used in number theory, particularly in the study of prime numbers and their distribution. It is often denoted by \( B(n) \) or \( b(n) \) and is related to the behavior of the prime counting function and the distribution of prime numbers in relation to composite numbers.

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