One week until organizational meeting of the learning seminar; a brief overview of the works of GPY, Zhang, and Maynard

This is a reminder to everyone that wants to participate in the learning seminar for bounded gaps between primes is next Thursday, 05 Dec 2013, in room MC 5045 at 2:30 PM. The flyer is located here.

Further, I just want to give a quick overview of the basic set-up in the series of papers due to Goldston, Pintz, and Yildirim as well as those of Zhang and Maynard. The basic counting function is something that looks like the following

S = \displaystyle \sum_{n=N+1}^{2N} \left(\sum_{i=1}^k \theta(n + h_i) - \rho \log(3N)\right)w_n,

where \theta is the function that is equal to \log n when n is a prime, and zero otherwise. w_n are positive weights that differ from paper to paper. Also, h_i \in \mathcal{H} is a set of admissible tuples, such that the h_i‘s do not take on all residue classes modulo p for any prime p. The basic strategy is to show that S > 0 for all large N. This would show that for infinitely many integers N > 0, there exists N < n \leq 2N such that the inner sum \displaystyle \sum_{i=1}^k \theta(n + h_i) - \rho \log(3N) is positive, and so n+h_1, n+h_2, \cdots, n + h_k contains at least \rho primes.

The main approach is to consider the two pieces of S, namely

\displaystyle S_1 = \sum_{n=N+1}^{2N} \left(\sum_{i=1}^k \theta(n + h_i)\right) w_n, and

\displaystyle S_2 = \rho \log(3N) \sum_{n=N+1}^{2N} w_n

separately and obtain a lower bound for S_1 and simultaneously an upper bound for S_2. The estimates for these sums will rely on some information about the level of distribution of the primes. We say that the primes have level of distribution \nu if the following inequality holds

\displaystyle \sum_{q \leq Q} \max_{\substack{a \\ \gcd(a,q) = 1}} \left \lvert \theta(N; q, a) - \frac{N}{\phi(q)} \right \rvert \ll \frac{N}{(\log N)^A}

for any Q \leq N^{\nu - \epsilon} and any A > 0. Here \displaystyle \theta(N; q, a) = \sum_{\substack{n \leq N \\ n \equiv a \pmod{q}}} \theta(n). The famous Bombieri-Vinogradov theorem, which among other results led to Enrico Bombieri winning the Fields Medal in 1974, confirms that the primes have level of distribution at least 1/2. It is conjectured by the Elliott-Halberstam conjecture that the primes have level of distribution 1.

In the GPY paper “Primes in tuples I”, Goldston, Pintz, and Yildirim remarked that if it can be proved that the primes have level of distribution strictly larger than 1/2, then one would have bounded gaps between primes. This is exactly the train of thought followed by Zhang in his seminal paper. Zhang was not able to show that the primes have a level of distribution greater than 1/2 but rather the following weaker result

\displaystyle \sum_{\substack{q < D^2 \\ d | \mathcal{P}}} \sum_{a \in \mathcal{C}_i(q) } \left \lvert \theta(N; q, a) - \frac{N}{\phi(q)} \right \rvert \ll \frac{N}{(\log N)^A}.

Here \mathcal{P} = \displaystyle \prod_{p \leq N^{\omega}} p, D = N^{1/4 + \omega}, \mathcal{C}_i(q) = \{a : 1 \leq a \leq q, \gcd(a,q) = 1, P(a - h_i) \equiv 0 \pmod{q}\},  P(n) = \displaystyle \prod_{i=1}^k (n + h_i), and \omega > 0 is a small positive number.

Unwinding the notation this basically means that if the range of summation in the outermost sum in the definition of level of distribution of prime is limited to those numbers q such that q is square-free and whose prime divisors are all at most N^\omega, then the primes can be exhibited to have “level of distribution” of at least 1/2 + 2\omega. This turns out to be sufficient after Zhang modified the weight function w_n from the GPY paper so that this weaker notion of level of distribution applies.

Turning to Maynard’s argument, the innovation there is completely different. Instead of seeking to improve on the level of distribution of primes, modified or not, Maynard instead sought to directly attack the counting function. In particular, he chose the weight functions w_n completely differently from before. In Maynard’s paper, the w_n‘s are defined in an intricate way in terms of piecewise smooth functions. This extra flexibility turned out to be extraordinarily powerful. It should be remarked that Maynard’s approach is not entirely new; in particular, Goldston, Pintz, and Yildirim have been able to obtain other results using the same basic idea.

In any case, I hope that this explanation was helpful to you and hope to see you on next Thursday.

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