Bombieri's Theorem

Define

(1)

where

(2)

(Davenport 1980, p. 121),  is the Mangoldt function, and  is the totient function. Now define

(3)

where the sum is over  relatively prime to , , and

(4)

Bombieri's theorem then says that for fixed ,

(5)

provided that .

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.