Two of the most famous open problems in additive prime number theory are the twin prime conjecture and the binary Goldbach conjecture. They have quite similar forms:
- Twin prime conjecture The equation has infinitely many solutions with prime.
- Binary Goldbach conjecture The equation has at least one solution with prime for any given even .
In view of this similarity, it is not surprising that the partial progress on these two conjectures have tracked each other fairly closely; the twin prime conjecture is generally considered slightly easier than the binary Goldbach conjecture, but broadly speaking any progress made on one of the conjectures has also led to a comparable amount of progress on the other. (For instance, Chen’s theorem has a version for the twin prime conjecture, and a version for the binary Goldbach conjecture.) Also, the notorious parity obstruction is present in both problems, preventing a solution to either conjecture by almost all known methods (see this previous blog post for more discussion).
In this post, I would like to note a divergence from this general principle, with regards to bounded error versions of these two conjectures:
- Twin prime with bounded error The inequalities has infinitely many solutions with prime for some absolute constant .
- Binary Goldbach with bounded error The inequalities has at least one solution with prime for any sufficiently large and some absolute constant .
The first of these statements is now a well-known theorem of Zhang, and the Polymath8b project hosted on this blog has managed to lower to unconditionally, and to assuming the generalised Elliott-Halberstam conjecture. However, the second statement remains open; the best result that the Polymath8b project could manage in this direction is that (assuming GEH) at least one of the binary Goldbach conjecture with bounded error, or the twin prime conjecture with no error, had to be true.
All the known proofs of Zhang’s theorem proceed through sieve-theoretic means. Basically, they take as input equidistribution results that control the size of discrepancies such as
for various congruence classes and various arithmetic functions , e.g. (or more generaly for various ). After taking some carefully chosen linear combinations of these discrepancies, and using the trivial positivity lower bound
one eventually obtains (for suitable ) a non-trivial lower bound of the form
where is some weight function, and is the set of such that there are at least two primes in the interval . This implies at least one solution to the inequalities with , and Zhang’s theorem follows.
In a similar vein, one could hope to use bounds on discrepancies such as (1) (for comparable to ), together with the trivial lower bound (2), to obtain (for sufficiently large , and suitable ) a non-trivial lower bound of the form
for some weight function , where is the set of such that there is at least one prime in each of the intervals and . This would imply the binary Goldbach conjecture with bounded error.
However, the parity obstruction blocks such a strategy from working (for much the same reason that it blocks any bound of the form in Zhang’s theorem, as discussed in the Polymath8b paper.) The reason is as follows. The sieve-theoretic arguments are linear with respect to the summation, and as such, any such sieve-theoretic argument would automatically also work in a weighted setting in which the summation is weighted by some non-negative weight . More precisely, if one could control the weighted discrepancies
to essentially the same accuracy as the unweighted discrepancies (1), then thanks to the trivial weighted version
of (2), any sieve-theoretic argument that was capable of proving (3) would also be capable of proving the weighted estimate
However, (4) may be defeated by a suitable choice of weight , namely
where is the Liouville function, which counts the parity of the number of prime factors of a given number . Since , one can expand out as the sum of and a finite number of other terms, each of which consists of the product of two or more translates (or reflections) of . But from the Möbius randomness principle (or its analogue for the Liouville function), such products of are widely expected to be essentially orthogonal to any arithmetic function that is arising from a single multiplicative function such as , even on very short arithmetic progressions. As such, replacing by in (1) should have a negligible effect on the discrepancy. On the other hand, in order for to be non-zero, has to have the same sign as and hence the opposite sign to cannot simultaneously be prime for any , and so vanishes identically, contradicting (4). This indirectly rules out any modification of the Goldston-Pintz-Yildirim/Zhang method for establishing the binary Goldbach conjecture with bounded error.
The above argument is not watertight, and one could envisage some ways around this problem. One of them is that the Möbius randomness principle could simply be false, in which case the parity obstruction vanishes. A good example of this is the result of Heath-Brown that shows that if there are infinitely many Siegel zeroes (which is a strong violation of the Möbius randomness principle), then the twin prime conjecture holds. Another way around the obstruction is to start controlling the discrepancy (1) for functions that are combinations of more than one multiplicative function, e.g. . However, controlling such functions looks to be at least as difficult as the twin prime conjecture (which is morally equivalent to obtaining non-trivial lower-bounds for ). A third option is not to use a sieve-theoretic argument, but to try a different method (e.g. the circle method). However, most other known methods also exhibit linearity in the “” variable and I would suspect they would be vulnerable to a similar obstruction. (In any case, the circle method specifically has some other difficulties in tackling binary problems, as discussed in this previous post.)
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10 July, 2014 at 4:57 am
Stijn Hanson
Really interesting article. What current evidence is there for the randomness of the Möbius function? I.e. is there anything which would lead us to believe that the Möbius randomness principle should fail?
10 July, 2014 at 10:45 pm
Terence Tao
We have a number of results that assert that exhibits non-trivial cancellation for many reasonable functions ; see e.g. this recent paper of Liu and Sarnak for some examples. We also have smallness of the Mobius function in the Gowers uniformity norms, thanks to the work of Ben Green, Tamar Ziegler, and myself (see http://arxiv.org/abs/1009.3998 ).
While we do not have any reason to disbelieve the Mobius pseudorandomness heuristic over the integers, the situation is more subtle over function fields, particularly in low characteristic. For instance there is an observation of Swan that shows that if is any polynomial in , then is always squarefree with an even number of irreducible factors, thus is identically , in violation of the function field Mobius pseudorandomness heuristic. Conrad and Conrad have some further examples of this type. However, this phenomenon seems to be localised to low characteristic (and high degree) and does not seem to impact the situation over the integers.
10 July, 2014 at 6:44 am
Anonymous
Good work on the conjecture Golbach however no has siloved absolutely.
The reason for not giving solution is:
First will demonstrate the prime numbers (such as are, where they are and how many there) this the has solved the mathematical ANDRI LOPEZ with two equations.
All number whose origin is (5 + 6[5a + (1;2;3;4)]) or (7 + 6[7a + (1;2;3;4;5;6;)]) that no is multiple of any of the following numbers (3;5;7;13;17;19;23;29;31;37;43) is a prime number absolute.
see: Universal Journal of Computational Mathematics Vol 1 (3); 2013
title: Equation for all prime number.
10 July, 2014 at 11:46 am
Eytan Paldi
the parity problem implies (as already remarked) for that for any admissible region (i.e. satisfies the conditions on ) its M-value can’t exceed 2. This application for the Selberg variational problem is quite remarkable in the sense that it seems to be quite difficult to prove that for any admissible by direct analytic method, and the only known proof is via the parity problem.
10 July, 2014 at 12:57 pm
gowers
After such an interesting post, I hesitate to make such a trivial remark, but I may as well: there’s something odd about your last sentence (the one in brackets), which makes it seem as though there’s a couple of words accidentally missed out.
[Corrected, thanks – T.]
11 July, 2014 at 5:46 am
0_Lesh_a
And if you take information about the equations for 3,4,5,6 … and to prove that they differ in some law. Do you think it will be harder than finding a solution to the exact equation 2?
12 July, 2014 at 9:49 am
Gil Kalai
Does the parity problem also applies for the assertion that there are infinitely many solutions for p-q = T where p and q are primes and T is some fixed number other than 2?
13 July, 2014 at 9:31 pm
Terence Tao
Yes. As a general rule of thumb, whenever one is trying to find a prime pair in some graph over the natural numbers (or over “almost primes”), the parity problem arises as an obstruction whenever the graph is 2-colourable (or even “mostly 2-colourable”, in that the proportion of monochromatic edges is tiny), assuming that the 2-colouring is “non-arithmetic” in some sense (so that it does not exhibit correlation with things like the Mobius function). This is for instance the case with the graph with edges with , which can be 2-coloured by a simple colouring of congruence classes, as well as the Goldbach-with-bounded error graph , which can be 2-coloured after deleting the numbers near (assuming for sake of argument that there are no primes of the form ). In contrast, the graph of edges with is not subject to this obstruction, even if one restricts to integers coprime to both 2 and 3, due to the presence of triangles. The situation with is more interesting; initially it appears that there are triangles, giving hope that the parity problem is not present here, but after deleting multiples of 2 and 3, all triangles are removed, and the parity problem is now evident.
12 July, 2014 at 12:00 pm
Polymath 8 – a Success! | Combinatorics and more
[…] barrier indeed shows up causes this attempt to fail. (Update July 14:) This is further explained in this new post over Tao’s […]
17 July, 2014 at 2:29 am
Anonymous
Professor Tao.
Constant (H) in: 0 < p' – p < H
N < p' + p < N + H ; is H = 44
The solution you have in your email from last month (June)..
5 October, 2014 at 1:46 pm
sylvainjulien
You might be interested in taking a glance at my blog ideasfornumbertheory.wordpress.com that hopefully could shed a new light on these two conjectures.
4 March, 2015 at 4:30 pm
Anthony Vossman
I would think that the only way around parity obstruction would have to begin by permitting the identity elements, 0,1, to be considered “conditionally prime” within the trivial range <4, thereby generalizing the binary G conjecture to a full set Z. From there, considering that the identity elements are also perfect squares, the binary G could be restated as, "Every perfect square is the sum of a semiprime and another perfect square", thus fully relating the additive and multiplicative universes, and which would make for an easier conjecture to address. These "tough" NT problems are really only related to the fact that axiomatic systems in general, are simply constructive edifices honoring a much deeper bilateral symmetry condition that is a pre-conditional for all arithmetic axioms, terms and relations. The only way you will find 'easy' proofs for these problems is to generalize them over the entire Z. Unfortunately, the symmetric beauty of the primes is spoiled by the conventionalities required of "uniqueness". Mathematics has to realize that the human brain is a prisoner of the bilateral symmetry that defines it.
14 March, 2015 at 5:48 am
gninrepoli
Perhaps some molifer can help solve the problem, such as use in the work of Yitang Zhang:
27 March, 2015 at 6:29 am
gninrepoli
– Euler’s totient function
I’m not sure, but if Mobius inversion possible for the Euler function, you can use the Selberg sieve for finding suitable mollifier to smooth small arcs and possibly remove zero, which occurs in the evaluation of large arcs.