Mathematics Lecture Notes

Theorem6.2Bombieri-Vinogradov Theorem

For any $A > 0$ , there exists $B = B(A) > 0$ such that: $\sum_{q \leq Q} \max_{(a,q)=1} \max_{y \leq x} \left|\psi(y; q, a) - \frac{y}{\phi(q)}\right| \ll \frac{x}{(\log x)^A}$ where $Q = x^{1/2}(\log x)^{-B}$ . In words: the PNT for arithmetic progressions holds on average over moduli $q \leq x^{1/2-\varepsilon}$ , with an error as strong as what GRH would give individually.

Proof

The proof combines the large sieve inequality with estimates for character sums and L-functions.

Step 1: Reduce to characters. The error $E(x; q, a) = \psi(x; q, a) - x/\phi(q)$ is bounded by $\frac{1}{\phi(q)}\sum_{\chi \neq \chi_0} |\psi(x, \chi)|$ plus the contribution from $\chi_0$ (which is $O(\log q)$ ).

Step 2: Large sieve inequality. The Bombieri-Davenport large sieve: $\sum_{q \leq Q} \frac{q}{\phi(q)} \sum_{\chi \bmod q}^{*} |\psi(x, \chi)|^2 \ll (x + Q^2) x \log x$ where $\sum^*$ denotes sum over primitive characters.

Step 3: From $L^2$ to $L^1$ . By Cauchy-Schwarz and the large sieve, the sum over $q$ of $\max_a |E(x;q,a)|$ is bounded. The key technical step uses Vaughan's identity to decompose $\Lambda(n)$ into bilinear sums: $\Lambda(n) = \Lambda_1(n) - \Lambda_2(n) + \Lambda_3(n)$ where each piece is a convolution that can be estimated by the large sieve.

Step 4: Putting it together. The bilinear structure of each piece allows application of the large sieve with $Q \leq x^{1/2}/(\log x)^B$ , giving the bound $O(x/(\log x)^A)$ for the total error. $\square$

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ExampleApplication: Almost-All Results for Primes

The Bombieri-Vinogradov theorem shows: for almost all $q \leq x^{1/2-\varepsilon}$ , $\pi(x;q,a) = \frac{\mathrm{Li}(x)}{\phi(q)} + O\!\left(\frac{x}{\phi(q)(\log x)^A}\right)$ . This replaces GRH in many applications. Combined with the Selberg sieve, it gives: every large even number is the sum of a prime and a product of at most 2 primes (Chen's theorem).

RemarkBeyond Bombieri-Vinogradov

The Elliott-Halberstam conjecture posits that the theorem holds with $Q = x^{1-\varepsilon}$ . Partial progress: Bombieri-Friedlander-Iwaniec proved a version with $Q = x^{4/7-\varepsilon}$ for specific arithmetic functions. The Maynard-Tao theorem on bounded gaps between primes uses a variant of BV with a weaker exponent but additional averaging.

Math Notes

Bombieri-Vinogradov Theorem

Proof Ideas