Bounded Gaps Between Primes the American Institute of Mathematics

Bounded gaps between primes The American Institute of Mathematics

The following compilation of participant contributions is only intended as a lead-in to the AIM workshop “Bounded gaps between primes.” This material is not for public distribution. Corrections and new material are welcomed and can be sent to [email protected] Version: Fri Nov 14 15:44:34 2014

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Table of Contents A.ParticipantContributions ...... 3 1. Castillo, Abel 2. Fiorilli, Daniel 3. Franze, Craig 4. Freiberg, Tristan 5. Friedlander, John 6. Graham, Sidney 7. Harper, Adam 8. Kaptan, Deniz 9. Maynard, James 10. Milinovich, Micah 11. Pollack, Paul 12. Tao, Terence 13. Thorner, Jesse 14. Turnage-Butterbaugh, Caroline 15. Yildirim, Cem 3

Chapter A: Participant Contributions

A.1 Castillo, Abel I would like to have a better understanding of the limitations of the Maynard-Tao method, as a means of exploring ways in which the method can be improved or generalized. For example, consider the following weakening of the prime k-tuples conjecture: “Fix and admissible linear system L1,L2,...,Lk. There are infinitely many positive integers n for which all of L1(n),L2(n),...,Lk(n) each have o(log log n) distinct prime divisors.” How close can the Maynard-Tao method to proving this conjecture? More vaguely, if we “weaken” the request for primality in different ways, how much closer can we come to using the Maynard- Tao method to make a statement about simultaneous conditions on all of (or a positive proportion of) the linear functions? A.2 Fiorilli, Daniel In this workshop I would like to learn about the recent groundbreaking work on bounded gaps between primes. The GPY sieve has recently been shown to be very flexible, and thus it is apparent that many new important problems can be solved. For example, Erdos’s $10000 problem on large gaps between primes has just been resolved. I hope to discuss such problems in the workshop, and to possibly start a research project involving the Maynard- Tao sieve. More precisely, it is now apparent that one can better understand primes in arithmetic progressions of very large moduli, which has been a central object of study in my research. I hope to go in this direction and apply the new techniques on this problem. A.3 Franze, Craig I am interested in applications of the recent variants of the Selberg sieve to sifting limits and almost-prime k-tuples. It is possible that improvements in these problems could be made, at least when the sieve dimension is small. I would also like to better understand the recent progress on small gaps between primes, and to explore other applications suggested by the participants. A.4 Freiberg, Tristan I am interested not just in small gaps between primes, but in medium and large gaps between primes. Let p denote the nth prime in the sequence of all primes, and d = p p n n n+1− n the nth prime gap. Thus, dn/ log pn 1 on average by the prime number theorem, and Crámer’s model leads us to believe that≈

b 1 −t # n x : dn/ log pn (a,b] e dt. x { ≤ ∈ }∼ Za

Gallagher showed that this would follow from a certain uniform version of the Hardy– Littlewood prime k-tuple conjecture. This is obviously quite out of reach, but W. Banks, J. Maynard and I showed that at least 12.5% of positive real numbers are limit points of the sequence d / log p ≥ . { n n}n 1 The basic philosophy was to combine the Erd˝os–Rankin construction for producing large gaps between primes with the GPY–Zhang–Maynard–Tao–Polymath machinery to get gaps between consecutive primes of the desired size. 4

One does not have to normalize by log pn — the method allows one to consider limit points of dn/f(pn) n≥1 where f(x) is a sufficiently slowly growing “nice” function. (In our work we can{ take f}(x) to grow ever so slightly faster than log x.) Maynard has recently answered a well-known question of Erd˝os by showing that dn/F (pn) can be infinitely often arbitrarily large, where −1 F (x) = (log x)(log2 x)(log4 x)(log3 x) . (This has also been shown using different methods independently by Ford, Green, Konyagin and Tao.) Can one show that a certain percentage of positive reals are limit points of d /f(p ) ≥ , for functions f growing as fast as F ? { n n }n 1 Secondly, Crámer’s model also leads us to expect that 1 e−λλk # n x : π(n + λ log n) π(n)= k . x { ≤ − }∼ k! Again, quite out of reach, but I believe the technology now exists that will allow us to say something interesting here.

Finally, one can show that pn pn+1 a mod q for infinitely many n, for any given arithmetic progression a mod q ((q,a≡) = 1).≡ Can one show that for infinitely many n, p a mod q and p b mod q, given (q,ab) = 1? And can one show that there are n ≡ n+1 ≡ infinitely many pairs of consecutive primes pn and pn+1 that both split completely in a given non-abelian extension of the rationals? A.5 Friedlander, John Am interested in prime numbers. Am one of the organizers. A.6 Graham, Sidney A. The simplicity and efficacy of the Maynard weights is amazing. It would be nice to have a good heuristic explanation why they work so well. This could lead to further applications or suggest similar ideas that would be effective in sieve applications. B. There has been little progress on general weighted sieves since the work of Greaves in the 1980s. Could Maynard’s construction or similar ideas be used to develop weighted either simpler or more powerful than the currently known weighted sieves? A.7 Harper, Adam I am interested in the various results that have been proved giving (“restricted”) levels of distribution beyond 1/2 for the primes and other sequences. I would like to understand more systematically what the trade offs are in the results, between assuming well factorability of the sequence under study (or its combinatorial decompositions), assuming well factorability of the moduli q that we sum over, or both, and in inserting absolute values or allowing the residue class a to vary. I am also interested in the structure and further applications of the Maynard–Tao “multi-dimensional” sieve weights. For example, the classical Selberg sieve weights are es- sentially the same weights that arise when mollifying the Riemann zeta function or when proving log-free zero-density estimates. I would be interested to understand whether the new sieve weights might have any uses in these contexts. 5 A.8 Kaptan, Deniz I am a Ph.D. student under the supervision of János Pintz, working on applications of the GPY and Maynard-Tao methods. My main goal in attending the workshop is to learn from the experts of the field. The following are some questions that I was led to while following the recent breakthroughs.

(i) Would it be possible to select a suitable family of tuples i and employ counting arguments or results of additive combinatorics so that the primes detectedH by their translates ( N/(log N)k for each tuple) in [N, 2N] would have to intersect in a way that would yield additional≫ information about the distribution of primes? (ii) Since we find less than k primes in a k-tuple, what could we salvage from the method if we relaxed admissibility (by requiring, say, that the k-tuple fail to cover all residue classes only for primes ℓ for a parameter ℓ)? We would lose at least by leaving out the bad primes from the sifting≤ procedure but on the other hand gain in tuple length if the method does go through. (iii) How small can N be relative to k or the diameter of in the Maynard-Tao method to guarantee the existence of primes in tuples in [N, 2N]? H A.9 Maynard, James I have been involved in some of the sieve-theoretic side of recent work on bounded gaps between primes. Natural questions which I am interested in in this topic include: 1. There are many arithmetic conjectures which follow trivially from the prime k-tuples conjecture (Carmichael numbers with 3 prime factors, Artin’s primitive root conjecture, etc.) Are there interesting arithmetic applications of the weak forms of the prime k-tuples conjecture which are now provable? 2. There are parity-related barriers to obtaining several natural strengthenings of the bounded gap results. Are there situations where one can incorporate extra arithmetic information into the method to break the parity barrier? A toy problem might be to attempt to show a positive proportion of primes lie in a different residue class (modulo 4) to the preceeding prime. 3. What is the weakest set of conditions one can put on the moduli for Zhang’s equidistribution result to hold with an exponent larger than 1/2? Can one say more if one weakens the statement (e.g. correct order of magnitude for most moduli rather than asymptotic)? Is there any chance to use ‘Kloostermania’-type bounds at some part of the argument? A.10 Milinovich, Micah I am analytic number theorist whose research thus far has been focused on the distribution of zeros of zeta and L-functions using explicit formulae, mean-value estimates for L-functions, and other Dirichlet polynomial techniques. I would like to pursue a wider vari- ety of research problems in number theory, including those in sieve theory. Thus, my primary interest in this workshop is to be introduced to open problems that might be accessible to either to me or to a graduate student under my supervision. 6 A.11 Pollack, Paul I am interested in extensions of the Maynard–Tao method that allow one to treat problems in elementary number theory previously only attackable under the prime k-tuples conjecture. An example of work already done in this direction is the resolution of some conjectures of Erdos and Turán by Banks, Freiberg, and Turnage–Butterbaugh, as well as their very simple re-proof of Shiu’s theorem. Another example is the recent work by Banks, Freiberg, and Maynard on the limit points of the sequence of normalized prime gaps. I am also interested in exploring which sets of primes can be shown to have this “bounded gaps” property, and whether this can be shown in the strong sense that the primes in the bounded length interval are actually consecutive. Some questions that may be open: Consider the set of primes p 1 (mod 3) for which 2 is a cube mod p. Are there infinitely many pairs of consecutive primes≡ like this lying in an interval of bounded length? Are there infinitely many pairs of primes p dn to hold for infinitely many n. (Claudia Spiro has some− partial progress on this.) A.12 Tao, Terence I was involved in the recent Polymath8 efforts to improve Zhang’s (and later, May- nard’s) results on bounded gaps between primes, and can give talks on the methods used in these efforts. A.13 Thorner, Jesse I am interested in applications of the Maynard-Tao construction. I have two applications in mind. 1. Let f(z) = ∞ a(n)e2πinz be a newform of even weight k 2 and level N 1. n=1 ≥ ≥ Suppose further thatPf has trivial nebentypus and does not have complex multiplication. Then a(n) is real for all n, and when p is prime, we have the Deligne bound a(p) 2p(k−1)/2. a(p) | |≤ Defining cos(θ )= − , Sato and Tate conjectured that the sequence cos(θ ) as p runs p 2p(k 1)/2 { p } through the primes is equidistributed in the interval [ 1, 1] with respect to the measure 2 √ 2 − dµST = π 1 t dt. Barnet-Lamb, Gehrarty, Harris, and Taylor proved that this conjecture is true. It follows− from Weyl’s criterion that if I [ 1, 1] is a subinterval, then # p x : cos(θ ) I µ (I)# p x . ⊂ − { ≤ p ∈ }∼ ST { ≤ } For a given newform f and a given subinterval I [ 1, 1], it is not known whether ⊂ − the primes in p x : cos(θp) I have a positive level of distribution (possibly omitting arithmetic progressions{ ≤ whose moduli∈ } are not coprime to N). If a positive level of distribution can be established, I think it would be interesting to see if bounded gaps between these primes can be established. Establishing a positive level of distribution would require some assumptions on the symmetric power L-functions of f, including the Generalized Riemann Hypothesis. 2. In my prior work, I showed that for any Galois extension of number fields L/Q, there exist bounded gaps between primes in Q with a given splitting condition in L, i.e. 7 bounded gaps between primes in Chebotarev sets. By the work of Abel Castillo, Chris Hall, Robert Lemke Oliver, Paul Pollack, and Lola Thompson, we know that there are bounded gaps between primes in number fields. I think it would be interesting to combine the two to obtain bounded gaps between primes in Chebotarev sets where the splitting condition is in a Galois extension L/K where K = Q. 6 A.14 Turnage-Butterbaugh, Caroline My main goal of the AIM workshop on bounded gaps between primes is to deepen my understanding of the techniques and arguments used by Goldston-Pintz-Yildirim, Zhang, and Maynard. As a relative newcomer to sieve theory, I plan to use the workshop as a venue to develop and add skills to my repertoire. I also plan to take advantage of the opportunity to brainstorm ideas for new projects and form new collaborative relationships. A.15 Yildirim, Cem There has been spectacular recent developments in the subject of ‘gaps between primes’. In the AIM Workshop on Bounded Gaps Between Primes, I hope to benefit from the insights of the mathematicians who will be there, in particular those who have realized the developments on bounded gaps and on large gaps between primes. We naturally wonder how these methods and results can be taken further. The questions include: (i) How can the bound 246 obtianed in the Maynard-Tao method be improved ? (ii) How can Zhang’s method be improved and/or simplified ? (iii) How can the conditional (upon Generalized Elliott-Halberstam conjecture) bound 6 of the Maynard-Tao method be reduced to 2 ? What hypothesis (presumably on the behaviour of some arithmetic functions) would that require ? (iv) Can the recent results on large gaps between primes be taken further in the sense that a function growing faster than that in Rankin’s classical result be explicitly given ? (v) Are there any implications on the distribution of zeros of the Riemann-ζ and Dirich- let L-functions ? The above are questions about pushing the theory ahead of its current limits. There are a few other research possibilities which are I think more practicable: (vi) Applying the weights in the Maynard-Tao method to the problem of ‘positive proportion of small gaps between primes’ with the aim of getting closer to the prediction from Hardy-Littlewood prime tuples conjecture, i.e. improving the results of the article GPY-4 (‘Primes in tuples IV: Density of small gaps between primes’, by Goldston, Pintz and Yıldırım). In GPY-4, for any η > 0, arbitrarily small but fixed, the lower bound for the proportion of gaps of size <η log N between consecutive primes located in [N, 2N] was found as involving η exponentially. Now we have hope of improving this to a polynomial function of η. (vii) The approach of the article GGPY-1 (‘Small gaps between primes or almost primes’, by Goldston, Graham, Pintz and Yıldırım) proved to be very fruitful for the de- velopment of Maynard’s breakthrough work. Can we improve upon the bound 6 given in GGPY-2 (Small gaps between products of two primes’, by Goldston, Graham, Pintz and Yıldırım) on the gaps between consecutive E2-numbers by using the weights in Maynard’s work ? Same question for blocks of E2-numbers. 8

(viii) A set of problems continuing and generalizing the results of the paper GGPY-3 (‘Small gaps between almost primes, the parity problem, and some conjectures of Erd¨os on consecutive integers’, by Goldston, Graham, Pintz and Yıldırım).