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Bombieri's Theorem

Define

E(x;q,a)=psi(x;q,a)-x/(phi(q)),
(1)

where

psi(x;q,a)=sum_(n<=x; n=a (mod q))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_(a; (a,q)=1)|E(x;q,a)|
(3)

where the sum is over a relatively prime to q, (a,q)=1, and

E^*(x,q)=max_(y<=x)E(y,q).
(4)

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

sum_(q<=Q)E^*(x,q)<<sqrt(x)Q(lnx)^5,
(5)

provided that sqrt(x)(lnx)^(-A)<=Q<=sqrt(x).

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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.

Referenced on Wolfram|Alpha

Bombieri's Theorem

Cite this as:

Weisstein, Eric W. "Bombieri's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BombierisTheorem.html

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