Joni Teräväinen and myself have just uploaded to the arXiv our preprint “Quantitative bounds for Gowers uniformity of the Möbius and von Mangoldt functions“. This paper makes quantitative the Gowers uniformity estimates on the Möbius function and the von Mangoldt function .

To discuss the results we first discuss the situation of the Möbius function, which is technically simpler in some (though not all) ways. We assume familiarity with Gowers norms and standard notations around these norms, such as the averaging notation and the exponential notation . The prime number theorem in qualitative form asserts that

as . With Vinogradov-Korobov error term, the prime number theorem is strengthened to we refer to such decay bounds (With type factors) as*pseudopolynomial decay*. Equivalently, we obtain pseudopolynomial decay of Gowers seminorm of : As is well known, the Riemann hypothesis would be equivalent to an upgrade of this estimate to polynomial decay of the form for any .

Once one restricts to arithmetic progressions, the situation gets worse: the Siegel-Walfisz theorem gives the bound

for any residue class and any , but with the catch that the implied constant is ineffective in . This ineffectivity cannot be removed without further progress on the notorious Siegel zero problem.In 1937, Davenport was able to show the discorrelation estimate

for any uniformly in , which leads (by standard Fourier arguments) to the Fourier uniformity estimate Again, the implied constant is ineffective. If one insists on effective constants, the best bound currently available is for some small effective constant .For the situation with the norm the previously known results were much weaker. Ben Green and I showed that

uniformly for any , any degree two (filtered) nilmanifold , any polynomial sequence , and any Lipschitz function ; again, the implied constants are ineffective. On the other hand, in a separate paper of Ben Green and myself, we established the following inverse theorem: if for instance we knew that for some , then there exists a degree two nilmanifold of dimension , complexity , a polynomial sequence , and Lipschitz function of Lipschitz constant such that Putting the two assertions together and comparing all the dependencies on parameters, one can establish the qualitative decay bound However the decay rate produced by this argument is*completely*ineffective: obtaining a bound on when this quantity dips below a given threshold depends on the implied constant in (3) for some whose dimension depends on , and the dependence on obtained in this fashion is ineffective in the face of a Siegel zero.

For higher norms , the situation is even worse, because the quantitative inverse theory for these norms is poorer, and indeed it was only with the recent work of Manners that any such bound is available at all (at least for ). Basically, Manners establishes if

then there exists a degree nilmanifold of dimension , complexity , a polynomial sequence , and Lipschitz function of Lipschitz constant such that (We allow all implied constants to depend on .) Meanwhile, the bound (3) was extended to arbitrary nilmanifolds by Ben and myself. Again, the two results when concatenated give the qualitative decay but the decay rate is completely ineffective.Our first result gives an effective decay bound:

Theorem 1For any , we have for some . The implied constants are effective.

This is off by a logarithm from the best effective bound (2) in the case. In the case there is some hope to remove this logarithm based on the improved quantitative inverse theory currently available in this case, but there is a technical obstruction to doing so which we will discuss later in this post. For the above bound is the best one could hope to achieve purely using the quantitative inverse theory of Manners.

We have analogues of all the above results for the von Mangoldt function . Here a complication arises that does not have mean close to zero, and one has to subtract off some suitable approximant to before one would expect good Gowers norms bounds. For the prime number theorem one can just use the approximant , giving

but even for the prime number theorem in arithmetic progressions one needs a more accurate approximant. In our paper it is convenient to use the “Cramér approximant” where and is the quasipolynomial quantity Then one can show from the Siegel-Walfisz theorem and standard bilinear sum methods that and for all and (with an ineffective dependence on ), again regaining effectivity if is replaced by a sufficiently small constant . All the previously stated discorrelation and Gowers uniformity results for then have analogues for , and our main result is similarly analogous:

Theorem 2For any , we have for some . The implied constants are effective.

By standard methods, this result also gives quantitative asymptotics for counting solutions to various systems of linear equations in primes, with error terms that gain a factor of with respect to the main term.

We now discuss the methods of proof, focusing first on the case of the Möbius function. Suppose first that there is no “Siegel zero”, by which we mean a quadratic character of some conductor with a zero with for some small absolute constant . In this case the Siegel-Walfisz bound (1) improves to a quasipolynomial bound

To establish Theorem 1 in this case, it suffices by Manners’ inverse theorem to establish the polylogarithmic bound for all degree nilmanifolds of dimension and complexity , all polynomial sequences , and all Lipschitz functions of norm . If the nilmanifold had bounded dimension, then one could repeat the arguments of Ben and myself more or less verbatim to establish this claim from (5), which relied on the quantitative equidistribution theory on nilmanifolds developed in a separate paper of Ben and myself. Unfortunately, in the latter paper the dependence of the quantitative bounds on the dimension was not explicitly given. In an appendix to the current paper, we go through that paper to account for this dependence, showing that all exponents depend at most doubly exponentially in the dimension , which is barely sufficient to handle the dimension of that arises here.
Now suppose we have a Siegel zero . In this case the bound (5) will *not* hold in general, and hence also (6) will not hold either. Here, the usual way out (while still maintaining effective estimates) is to approximate not by , but rather by a more complicated approximant that takes the Siegel zero into account, and in particular is such that one has the (effective) pseudopolynomial bound

For the analogous problem with the von Mangoldt function (assuming a Siegel zero for sake of discussion), the approximant is simpler; we ended up using

which allows one to state the standard prime number theorem in arithmetic progressions with classical error term and Siegel zero term compactly as Routine modifications of previous arguments also give and The one tricky new step is getting from the discorrelation estimate (8) to the Gowers uniformity estimate One cannot directly apply Manners’ inverse theorem here because and are unbounded. There is a standard tool for getting around this issue, now known as the*dense model theorem*, which is the standard engine powering the

*transference principle*from theorems about bounded functions to theorems about certain types of unbounded functions. However the quantitative versions of the dense model theorem in the literature are expensive and would basically weaken the doubly logarithmic gain here to a triply logarithmic one. Instead, we bypass the dense model theorem and directly transfer the inverse theorem for bounded functions to an inverse theorem for unbounded functions by using the

*densification*approach to transference introduced by Conlon, Fox, and Zhao. This technique turns out to be quantitatively quite efficient (the dependencies of the main parameters in the transference are polynomial in nature), and also has the technical advantage of avoiding the somewhat tricky “correlation condition” present in early transference results which are also not beneficial for quantitative bounds.

In principle, the above results can be improved for due to the stronger quantitative inverse theorems in the setting. However, there is a bottleneck that prevents us from achieving this, namely that the equidistribution theory of two-step nilmanifolds has exponents which are exponential in the dimension rather than polynomial in the dimension, and as a consequence we were unable to improve upon the doubly logarithmic results. Specifically, if one is given a sequence of bracket quadratics such as that fails to be -equidistributed, one would need to establish a nontrivial linear relationship modulo 1 between the (up to errors of ), where the coefficients are of size ; current methods only give coefficient bounds of the form . An old result of Schmidt demonstrates proof of concept that these sorts of polynomial dependencies on exponents is possible in principle, but actually implementing Schmidt’s methods here seems to be a quite non-trivial task. There is also another possible route to removing a logarithm, which is to strengthen the inverse theorem to make the dimension of the nilmanifold logarithmic in the uniformity parameter rather than polynomial. Again, the Freiman-Bilu theorem (see for instance this paper of Ben and myself) demonstrates proof of concept that such an improvement in dimension is possible, but some work would be needed to implement it.

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