Joni Teräväinen and I have just uploaded to the arXiv our preprint “The Hardy–Littlewood–Chowla conjecture in the presence of a Siegel zero“. This paper is a development of the theme that certain conjectures in analytic number theory become easier if one makes the hypothesis that Siegel zeroes exist; this places one in a presumably “illusory” universe, since the widely believed Generalised Riemann Hypothesis (GRH) precludes the existence of such zeroes, yet this illusory universe seems remarkably self-consistent and notoriously impossible to eliminate from one’s analysis.

For the purposes of this paper, a Siegel zero is a zero of a Dirichlet -function corresponding to a primitive quadratic character of some conductor , which is close to in the sense that

for some large (which we will call the*quality*) of the Siegel zero. The significance of these zeroes is that they force the Möbius function and the Liouville function to “pretend” to be like the exceptional character for primes of magnitude comparable to . Indeed, if we define an

*exceptional prime*to be a prime in which , then very few primes near will be exceptional; in our paper we use some elementary arguments to establish the bounds for any and , where the sum is over exceptional primes in the indicated range ; this bound is non-trivial for as large as . (See Section 1 of this blog post for some variants of this argument, which were inspired by work of Heath-Brown.) There is also a companion bound (somewhat weaker) that covers a range of a little bit below .

One of the early influential results in this area was the following result of Heath-Brown, which I previously blogged about here:

Theorem 1 (Hardy-Littlewood assuming Siegel zero)Let be a fixed natural number. Suppose one has a Siegel zero associated to some conductor . Then we have for all , where is the von Mangoldt function and is thesingular series

In particular, Heath-Brown showed that if there are infinitely many Siegel zeroes, then there are also infinitely many twin primes, with the correct asymptotic predicted by the Hardy-Littlewood prime tuple conjecture at infinitely many scales.

Very recently, Chinis established an analogous result for the Chowla conjecture (building upon earlier work of Germán and Katai):

Theorem 2 (Chowla assuming Siegel zero)Let be distinct fixed natural numbers. Suppose one has a Siegel zero associated to some conductor . Then one has in the range , where is the Liouville function.

In our paper we unify these results and also improve the quantitative estimates and range of :

Theorem 3 (Hardy-Littlewood-Chowla assuming Siegel zero)Let be distinct fixed natural numbers with . Suppose one has a Siegel zero associated to some conductor . Then one has for for any fixed .

Our argument proceeds by a series of steps in which we replace and by more complicated looking, but also more tractable, approximations, until the correlation is one that can be computed in a tedious but straightforward fashion by known techniques. More precisely, the steps are as follows:

- (i) Replace the Liouville function with an approximant , which is a completely multiplicative function that agrees with at small primes and agrees with at large primes.
- (ii) Replace the von Mangoldt function with an approximant , which is the Dirichlet convolution multiplied by a Selberg sieve weight to essentially restrict that convolution to almost primes.
- (iii) Replace with a more complicated truncation which has the structure of a “Type I sum”, and which agrees with on numbers that have a “typical” factorization.
- (iv) Replace the approximant with a more complicated approximant which has the structure of a “Type I sum”.
- (v) Now that all terms in the correlation have been replaced with tractable Type I sums, use standard Euler product calculations and Fourier analysis, similar in spirit to the proof of the pseudorandomness of the Selberg sieve majorant for the primes in this paper of Ben Green and myself, to evaluate the correlation to high accuracy.

Steps (i), (ii) proceed mainly through estimates such as (1) and standard sieve theory bounds. Step (iii) is based primarily on estimates on the number of smooth numbers of a certain size.

The restriction in our main theorem is needed only to execute step (iv) of this step. Roughly speaking, the Siegel approximant to is a twisted, sieved version of the divisor function , and the types of correlation one is faced with at the start of step (iv) are a more complicated version of the divisor correlation sum

For this sum can be easily controlled by the Dirichlet hyperbola method. For one needs the fact that has a level of distribution greater than ; in fact Kloosterman sum bounds give a level of distribution of , a folklore fact that seems to have first been observed by Linnik and Selberg. We use a (notationally more complicated) version of this argument to treat the sums arising in (iv) for . Unfortunately for there are no known techniques to unconditionally obtain asymptotics, even for the model sum although we do at least have fairly convincing conjectures as to what the asymptotics should be. Because of this, it seems unlikely that one will be able to relax the hypothesis in our main theorem at our current level of understanding of analytic number theory.Step (v) is a tedious but straightforward sieve theoretic computation, similar in many ways to the correlation estimates of Goldston and Yildirim used in their work on small gaps between primes (as discussed for instance here), and then also used by Ben Green and myself to locate arithmetic progressions in primes.

## 29 comments

Comments feed for this article

15 September, 2021 at 11:17 am

asahay22It looks like you missed replacing a \cite with a link in the description of Step (v).

[Corrected, thanks – T.]15 September, 2021 at 11:22 am

gexahedronIf both of the worlds, one where generalized Riemann hypothesis is true, and one where Siegel zeros exist, then maybe they are both correct (“real”)?

15 September, 2021 at 11:57 am

AnonymousThe mere existence of a Siegel zero seems helpful for finding “good approximants”, but is it really necessary ?

15 September, 2021 at 12:11 pm

MicheleThe post and the various references of Prof. Tao concerning this topic are very interesting. With regard the Hardy-Littlewood Conjecture, it is fundamental in several areas of mathematics, mainly in Number Theory. But there are also various applications in some sectors of Theoretical Physics.

https://www.academia.edu/45582699/

https://www.academia.edu/45586191/

.

16 September, 2021 at 12:08 am

MatcIs there a very hand-waving intuition as to why {\lambda} pretending to be like the exceptional character {\chi} helps here? Is there some line of reasoning like “{\lambda} behaves like {\chi} and {\chi} is periodic, so {\lambda} is “periodic” in some vague sense and this prevents some cancellation and thus gives the main term”?

16 September, 2021 at 2:18 am

AnonymousThis is sort of right. The Liouville function is difficult to understand because it is related to the zeros of the zeta function, and we don’t have good unconditional results on zeta zeros. A Siegel zero forces the associated quadratic character to nearly behave like the Liouville function, so one can replace Liouville by a Dirichlet character. As you noted, Dirichlet characters are very nice because they are periodic and you can do Fourier analysis and all sorts of useful things with them.

17 September, 2021 at 8:04 am

Terence TaoOne standard way is to start with the Dirichlet series identity

(valid for ). If has a zero near , this identity suggests that the sum will be large for near , and hence there is not much cancellation in this sum. As such, should be biased to having the same sign as fairly often (i.e., it “pretends” to be like ), which is a phenomenon one can quantify using the concept of “pretentiousness” in “pretentious number theory”.

11 October, 2021 at 3:57 am

HarryThis may lead to a new theory.

16 September, 2021 at 12:31 am

MichaelThe definition of the singular series with the gothic letter, the slash below the second product missed the divide sign

[Corrected, thanks – T]16 September, 2021 at 1:40 am

RevanthaWhy is it surprising that we can derive interesting results after assuming the existence of Siegel zeroes (since it is widely believed that there are no Siegel zeroes) ?

Is the point not about any particular result, but rather that various not-especially-related results are derivable from the same assumption ?

17 September, 2021 at 8:19 am

Terence TaoWell, there are a couple reasons:

1. By exploring this presumably illusory “Siegel zero universe” more extensively, we may eventually be able to hit upon a contradiction, and thus finally resolve the notorious problem of whether Siegel zeroes exist. This could in turn lead to substantial progress more generally towards GRH. (One should caution though that historically every such discovery of an apparent contradiction has fallen apart under more careful scrutiny; the Siegel zero universe seems highly resistant to being destroyed!)

2. There are several results in analytic number theory (e.g., Linnik’s theorem) that are proven unconditionally by splitting into two cases, one where Siegel zereos exist, and one where they don’t, and pursuing a separate argument to handle each case. So this sort of result could end up being one half of a later unconditional result. The non-Siegel-zero case usually comes with effective bounds, so it is of particular interest to see if the Siegel zero case also has effective estimates, as this would lead to effective unconditional estimates.

3. Having a good working understanding of the Siegel zero “alternate universe” helps one detect errors (or at least raise serious red flags) in claimed results in analytic number theory that are inconsistent with that universe, such as results that breeze through various “parity barriers” without demonstrating any awareness at all of the potential existence of Siegel zeroes. Sometimes one can narrow down where the specific error occurs in a manuscript by testing individual assertions in that manuscript under the hypothesis of a Siegel zero to find out the first place where an inconsistency occurs. In a similar spirit, the Siegel zero universe places barriers to improving the constants in various bounds (e.g., the constants in the large sieve inequality) and allows one to assert that certain results are “optimal at the current level of technology” by demonstrating that any further improvement in the bounds would be incompatible with the existence of Siegel zeroes.

4. In some cases, such as in our paper, the Siegel zero universe is *more* computationally tractable than the GRH universe that one expects to be the real one; GRH does not let us prove either the Hardy-Littlewood or Chowla conjectures at our current level of technology! This allows us to develop calculations to a level of precision that is not available on GRH, which can help develop technology that will one day also be useful for applications outside the Siegel zero universe. For instance, the techniques in this paper seem to be adaptable to allow for relatively straightforward calculations for the predicted asymptotics for divisor correlations such as , even if we are still missing the technology to rigorously establish these correlations in most cases (right now we can basically only handle the cases when , or when and ).

5. Related to 4., if one can prove a conjecture such as the Hardy-Littlewood or Chowla conjecture in the Siegel zero universe, it provides some heuristic support for this conjecture also being true in the real universe: it shows that such conjectures are provable “in principle”, albeit not in the universe that we want them to be true in. Nevertheless, some of the ideas, calculations, methods, or insights used to resolve the conjecture in the Siegel zero universe might also be of use attacking the same conjecture in the real universe.

6. It may end up that the Siegel zero universe is indicative of some other alternate, exotic form of number theory which is actually self-consistent. An analogy here is with the parallel postulate, which was widely believed to be true in the real world. A systematic exploration of the alternate universe where the parallel postulate failed eventually led to the discovery of self-consistent non-Euclidean geometries. Perhaps there is some sort of nonstandard model of some sort of “generalized number theory” in which Siegel zeroes do exist? (One could argue for instance that elliptic curves of high (analytic) rank sort of count as a weak Siegel zero, in that their associated L-function have a zero on the real axis, though so deep inside the critical strip that their influence is rather weak.)

17 September, 2021 at 2:07 pm

RevanthaThanks, that clarifies. It is interesting to encounter these persuasive reasons given that the approach is initially unexpected from a logic perspective.

17 September, 2021 at 5:02 pm

SiegelvsgrhCan we conclude a statement is unconditionally valid if it is valid conditionally on siegel zero and valid conditionally on grh? Is it a possible reason 7?

18 September, 2021 at 8:32 am

Terence TaoAt a heuristic level perhaps, but at a fully rigorous level there is of course a large middle ground in which Dirichlet L-functions have zeroes that are neither close to 1 nor on the critical line. There is a sort of “unified universe” in which one makes instead the assumption that all zeroes of L-functions lie on either the critical line or the real axis, and there are some interesting conclusions to be drawn there; in particular, Sarnak and Zaharescu showed in this case that there are strong limits to the strength of a Siegel zero in this universe.

18 September, 2021 at 2:03 pm

SiegelvsgrhEssentially the zero free region improvement and error bound are related. So are you saying there is an error bound (above the optimal GRH bound) below which Sigel zeros can be ruled out?

30 September, 2021 at 8:40 pm

duck_masterRandom thought: Do you know if it’s possible to use the assumed existence of the Siegel zero of to obtain any nontrivial amount of control on sums like (especially, I presume, in the regime , where the currently known Siegel-zero-based results operate)? Since we’re assuming the (presumably illusory) existence of a Siegel zero, we may as well make full use of it.

Another random thought: Since we expect in some kind of statistical pointwise sense for inputs of size , not only should we expect “exceptional primes” to be rare, but we should also expect them to prefer being at places modulo such that is closer to -1, rather than at places where the exceptional character gives a value far away from -1.

3 October, 2021 at 4:49 pm

Terence TaoThe divisor functions are not sensitive to zeroes of the zeta function or Dirichlet L-functions, since their Dirichlet series involves zeta functions in the numerator rather than the denominator. So it does not appear that Siegel zeroes are particularly helpful in understanding these functions.

Note that a quadratic character already only takes on two values, and , and half of the residue classes will take on the value and half will take on the value . The exceptional primes are those in the first category (as well as the small number of primes that divide ) and the non-exceptional ones are the second category.

16 September, 2021 at 9:17 pm

Lior SilbermanIn the statement of Conjecture 1.3 in the paper, I think should be : any Liouville factors should cause cancellation.

[Thanks, this will be fixed in the next revision of the ms. -T]17 September, 2021 at 12:29 pm

AnonymousIn theorem 3, the estimate is dependent on $latek k$ but not on (also there is a restriction on but not on ) – this asymmetry between and seems incorrect.

18 September, 2021 at 8:26 am

Terence TaoActually, this asymmetry seems inherent to our current state of understanding; under the assumption of a Siegel zero, the Liouville function behaves like the exceptional character which is relatively easy to control, while the von Mangoldt function behaves like a twisted version of the divisor function which is moderately difficult to control. In particular we can only handle up to copies of the latter, but an unlimited number of copies of the former, and our bounds worsen (and hypotheses tighten) as increases from zero to two.

17 September, 2021 at 2:13 pm

AnonymousIf there is no Siegel zero, is it still possible to use Dobner’s proof of a generalized Newman’s conjecture for a large class of Dirichlet Series (containing all Dirichlet L-functions associated to primitive characters) by using suitable deformations of these L-functions with positive deformation parameter – thereby ensuring the existence of a Siegel zero for – which can be used to find “good approximants” and corresponding estimates as described in this post ?

It seems that the deformation parameter should be optimized (as a function of ) to have a “sufficiently good” quality parameter in a corresponding estimate (analogous to theorem 3).

18 September, 2021 at 8:29 am

Terence TaoUnfortunately there does not seem to be much interaction between Siegel zeroes and heat flow type deformations of zeta functions (or L-functions). Note for instance that such deformations fail to have an Euler product, so their number-theoretic impact is sharply limited (indeed, in practice, the only way number theory interacts with the deformations is through the connection between number-theoretic objects such as the primes, and the original Riemann zeta function .)

18 September, 2021 at 3:36 am

AnonymousIn theorem 3, it seems that (in the exponent of the lower bound for x) should be , and the estimate error term should have its implied constant be dependent on .

[Corrected, thanks – T.]19 September, 2021 at 1:27 am

QuillNot a relevant question, but do the news and social media affect your productivity these days, and how do you manage your consumption of them? I found it interesting that Timothy Gowers said on his Twitter feed that Twitter is affecting his work, and that he is taking a break from it this month of September. He seems pretty concerned about the climate crisis and also comments on current events and social issues.

19 September, 2021 at 4:56 am

RaphaelWhat about the ‘other direction’ if it comes to GRH violations, i. e. the assumption of zeros very close to x=1/2? Can we expect anything similar fruitful?

19 September, 2021 at 5:31 am

Anonymousand what about allowing almost zeros? From the general heuristics it seems the result should be very similar.

19 September, 2021 at 5:48 am

AnonymousFor example an arithmetic progression of Siegel zero like arguments of the L-function converging to zero?

1 October, 2021 at 11:32 am

AlexSiegel zeros do not exist.

1 October, 2021 at 11:52 am

AnonymousIs it possible to use these results on the Chowla conjecture (in the presence of a siegel zero) to get new information on the distribution of such zeros?