Maier's theorem

In

The theorem states that if π is the prime-counting function and λ is greater than 1 then

${\displaystyle {\frac {\pi (x+(\log x)^{\lambda })-\pi (x)}{(\log x)^{\lambda -1}}}}$

does not have a limit as x tends to infinity; more precisely the

Maier proved his theorem using Buchstab's equivalent for the counting function of quasi-primes (set of numbers without prime factors lower to bound ${\displaystyle z=x^{1/u}}$, ${\displaystyle u}$ fixed). He also used an equivalent of the number of primes in arithmetic progressions of sufficient length due to Gallagher.

${\displaystyle \int _{2}^{Y}\left(\sum _{2<p\leq x}\log p-\sum _{2<n\leq x}1\right)^{2}\,dx}$

of one version of the prime number theorem.

See also

• Maier's matrix method

References

• Zbl 0569.10023
• Zbl 1226.11096
• Zbl 1141.11043