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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
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) Letbe 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 the singular 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) Letbe 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) Letbe 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
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.
A few months ago I posted a question about analytic functions that I received from a bright high school student, which turned out to be studied and resolved by de Bruijn. Based on this positive resolution, I thought I might try my luck again and list three further questions that this student asked which do not seem to be trivially resolvable.
- Does there exist a smooth function
which is nowhere analytic, but is such that the Taylor series
converges for every
? (Of course, this series would not then converge to
, but instead to some analytic function
for each
.) I have a vague feeling that perhaps the Baire category theorem should be able to resolve this question, but it seems to require a bit of effort. (Update: answered by Alexander Shaposhnikov in comments.)
- Is there a function
which meets every polynomial
to infinite order in the following sense: for every polynomial
, there exists
such that
for all
? Such a function would be rather pathological, perhaps resembling a space-filling curve. (Update: solved for smooth
by Aleksei Kulikov in comments. The situation currently remains unclear in the general case.)
- Is there a power series
that diverges everywhere (except at
), but which becomes pointwise convergent after dividing each of the monomials
into pieces
for some
summing absolutely to
, and then rearranging, i.e., there is some rearrangement
of
that is pointwise convergent for every
? (Update: solved by Jacob Manaker in comments.)
Feel free to post answers or other thoughts on these questions in the comments.
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