Mangoldt summatory function is O(x)

Theorem 1

ψ(x)=O(x), in other , ψ(x)x is bounded.


ψ(x)=1xΛ(n)=p primepxlogpxlnp=p primepxlnxlnplnp=p primepxlnxlnplnp+p primex<pxlnp

since 1lnxlnp<2 if p>x. Continuing, we have

p primepxlnxlnplnp+p primex<pxlnpxlnx+π(x)lnxxlnx+8xln2=O(x)

Note that π(x)lnx8xln2 by Chebyshev’s bounds on π(x) (

Title Mangoldt summatory function is O(x)
Canonical name MangoldtSummatoryFunctionIsOx
Date of creation 2013-03-22 17:42:59
Last modified on 2013-03-22 17:42:59
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 5
Author rm50 (10146)
Entry type Theorem
Classification msc 11A41