Bombieri's Theorem

Define

$$E(x; q, a)\equiv \psi(x; q,a)-{x\over\phi(q)},$$ (1)

where

$$\psi(x; q, a)=\sum_{\scriptstyle n\leq x\atop\scriptstyle n\equiv a\ \left({{\rm mod\ } {q}}\right)} \Lambda(n)$$ (2)

(Davenport 1980, p. 121), $\Lambda(n)$ is the Mangoldt Function, and $\phi(q)$ is the Totient Function. Now define

$$E(x; q)=\max_{\scriptstyle a\atop\scriptstyle (a,q)=1} \vert E(x; q,a)\vert$$ (3)

where the sum is over $a$ Relatively Prime to $q$, $(a,q)=1$, and

$$E^*(x,q)=\max_{y\leq x} E(y,q).$$ (4)

Bombieri's theorem then says that for $A>0$ fixed,

$$\sum_{q\leq Q}E^*(x,q)\ll \sqrt{x}\,Q(\ln x)^5,$$ (5)

provided that $\sqrt{x}\,(\ln x)^{-4}\leq Q\leq \sqrt{x}$.

References

Bombieri, E. ``On the Large Sieve.'' Mathematika 12, 201-225, 1965.

Davenport, H. ``Bombieri's Theorem.'' Ch. 28 in Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, pp. 161-168, 1980.