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Kaisa Matomäki, Xuancheng Shao, Joni Teräväinen, and myself have just uploaded to the arXiv our preprint “Higher uniformity of arithmetic functions in short intervals I. All intervals“. This paper investigates the higher order (Gowers) uniformity of standard arithmetic functions in analytic number theory (and specifically, the Möbius function , the von Mangoldt function
, and the generalised divisor functions
) in short intervals
, where
is large and
lies in the range
for a fixed constant
(that one would like to be as small as possible). If we let
denote one of the functions
, then there is extensive literature on the estimation of short sums
Traditionally, asymptotics for such sums are expressed in terms of a “main term” of some arithmetic nature, plus an error term that is estimated in magnitude. For instance, a sum such as would be approximated in terms of a main term that vanished (or is negligible) if
is “minor arc”, but would be expressible in terms of something like a Ramanujan sum if
was “major arc”, together with an error term. We found it convenient to cancel off such main terms by subtracting an approximant
from each of the arithmetic functions
and then getting upper bounds on remainder correlations such as
- For the Möbius function
, we simply set
, as per the Möbius pseudorandomness conjecture. (One could choose a more sophisticated approximant in the presence of a Siegel zero, as I did with Joni in this recent paper, but we do not do so here.)
- For the von Mangoldt function
, we eventually went with the Cramér-Granville approximant
, where
and
.
- For the divisor functions
, we used a somewhat complicated-looking approximant
for some explicit polynomials
, chosen so that
and
have almost exactly the same sums along arithmetic progressions (see the paper for details).
The objective is then to obtain bounds on sums such as (1) that improve upon the “trivial bound” that one can get with the triangle inequality and standard number theory bounds such as the Brun-Titchmarsh inequality. For and
, the Siegel-Walfisz theorem suggests that it is reasonable to expect error terms that have “strongly logarithmic savings” in the sense that they gain a factor of
over the trivial bound for any
; for
, the Dirichlet hyperbola method suggests instead that one has “power savings” in that one should gain a factor of
over the trivial bound for some
. In the case of the Möbius function
, there is an additional trick (introduced by Matomäki and Teräväinen) that allows one to lower the exponent
somewhat at the cost of only obtaining “weakly logarithmic savings” of shape
for some small
.
Our main estimates on sums of the form (1) work in the following ranges:
- For
, one can obtain strongly logarithmic savings on (1) for
, and power savings for
.
- For
, one can obtain weakly logarithmic savings for
.
- For
, one can obtain power savings for
.
- For
, one can obtain power savings for
.
Conjecturally, one should be able to obtain power savings in all cases, and lower down to zero, but the ranges of exponents and savings given here seem to be the limit of current methods unless one assumes additional hypotheses, such as GRH. The
result for correlation against Fourier phases
was established previously by Zhan, and the
result for such phases and
was established previously by by Matomäki and Teräväinen.
By combining these results with tools from additive combinatorics, one can obtain a number of applications:
- Direct insertion of our bounds in the recent work of Kanigowski, Lemanczyk, and Radziwill on the prime number theorem on dynamical systems that are analytic skew products gives some improvements in the exponents there.
- We can obtain a “short interval” version of a multiple ergodic theorem along primes established by Frantzikinakis-Host-Kra and Wooley-Ziegler, in which we average over intervals of the form
rather than
.
- We can obtain a “short interval” version of the “linear equations in primes” asymptotics obtained by Ben Green, Tamar Ziegler, and myself in this sequence of papers, where the variables in these equations lie in short intervals
rather than long intervals such as
.
We now briefly discuss some of the ingredients of proof of our main results. The first step is standard, using combinatorial decompositions (based on the Heath-Brown identity and (for the result) the Ramaré identity) to decompose
into more tractable sums of the following types:
- Type
sums, which are basically of the form
for some weights
of controlled size and some cutoff
that is not too large;
- Type
sums, which are basically of the form
for some weights
,
of controlled size and some cutoffs
that are not too close to
or to
;
- Type
sums, which are basically of the form
for some weights
of controlled size and some cutoff
that is not too large.
The precise ranges of the cutoffs depend on the choice of
; our methods fail once these cutoffs pass a certain threshold, and this is the reason for the exponents
being what they are in our main results.
The Type sums involving nilsequences can be treated by methods similar to those in this previous paper of Ben Green and myself; the main innovations are in the treatment of the Type
and Type
sums.
For the Type sums, one can split into the “abelian” case in which (after some Fourier decomposition) the nilsequence
is basically of the form
, and the “non-abelian” case in which
is non-abelian and
exhibits non-trivial oscillation in a central direction. In the abelian case we can adapt arguments of Matomaki and Shao, which uses Cauchy-Schwarz and the equidistribution properties of polynomials to obtain good bounds unless
is “major arc” in the sense that it resembles (or “pretends to be”)
for some Dirichlet character
and some frequency
, but in this case one can use classical multiplicative methods to control the correlation. It turns out that the non-abelian case can be treated similarly. After applying Cauchy-Schwarz, one ends up analyzing the equidistribution of the four-variable polynomial sequence
For the type sum, a model sum to study is
In a sequel to this paper (currently in preparation), we will obtain analogous results for almost all intervals with
in the range
, in which we will be able to lower
all the way to
.
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
Once one restricts to arithmetic progressions, the situation gets worse: the Siegel-Walfisz theorem gives the bound
for any residue classIn 1937, Davenport was able to show the discorrelation estimate
For the situation with the norm the previously known results were much weaker. Ben Green and I showed that
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
Our first result gives an effective decay bound:
Theorem 1 For 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
Theorem 2 For 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
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
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.
In analytic number theory, an arithmetic function is simply a function from the natural numbers
to the real or complex numbers. (One occasionally also considers arithmetic functions taking values in more general rings than
or
, as in this previous blog post, but we will restrict attention here to the classical situation of real or complex arithmetic functions.) Experience has shown that a particularly tractable and relevant class of arithmetic functions for analytic number theory are the multiplicative functions, which are arithmetic functions
with the additional property that
whenever are coprime. (One also considers arithmetic functions, such as the logarithm function
or the von Mangoldt function, that are not genuinely multiplicative, but interact closely with multiplicative functions, and can be viewed as “derived” versions of multiplicative functions; see this previous post.) A typical example of a multiplicative function is the divisor function
that counts the number of divisors of a natural number . (The divisor function
is also denoted
in the literature.) The study of asymptotic behaviour of multiplicative functions (and their relatives) is known as multiplicative number theory, and is a basic cornerstone of modern analytic number theory.
There are various approaches to multiplicative number theory, each of which focuses on different asymptotic statistics of arithmetic functions . In elementary multiplicative number theory, which is the focus of this set of notes, particular emphasis is given on the following two statistics of a given arithmetic function
:
- The summatory functions
of an arithmetic function
, as well as the associated natural density
(if it exists).
- The logarithmic sums
of an arithmetic function
, as well as the associated logarithmic density
(if it exists).
Here, we are normalising the arithmetic function being studied to be of roughly unit size up to logarithms, obeying bounds such as
,
, or at worst
A classical case of interest is when is an indicator function
of some set
of natural numbers, in which case we also refer to the natural or logarithmic density of
as the natural or logarithmic density of
respectively. However, in analytic number theory it is usually more convenient to replace such indicator functions with other related functions that have better multiplicative properties. For instance, the indicator function
of the primes is often replaced with the von Mangoldt function
.
Typically, the logarithmic sums are relatively easy to control, but the summatory functions require more effort in order to obtain satisfactory estimates; see Exercise 7 below.
If an arithmetic function is multiplicative (or closely related to a multiplicative function), then there is an important further statistic on an arithmetic function
beyond the summatory function and the logarithmic sum, namely the Dirichlet series
for various real or complex numbers . Under the hypothesis (3), this series is absolutely convergent for real numbers
, or more generally for complex numbers
with
. As we will see below the fold, when
is multiplicative then the Dirichlet series enjoys an important Euler product factorisation which has many consequences for analytic number theory.
In the elementary approach to multiplicative number theory presented in this set of notes, we consider Dirichlet series only for real numbers (and focusing particularly on the asymptotic behaviour as
); in later notes we will focus instead on the important complex-analytic approach to multiplicative number theory, in which the Dirichlet series (4) play a central role, and are defined not only for complex numbers with large real part, but are often extended analytically or meromorphically to the rest of the complex plane as well.
Remark 1 The elementary and complex-analytic approaches to multiplicative number theory are the two classical approaches to the subject. One could also consider a more “Fourier-analytic” approach, in which one studies convolution-type statistics such as
as
for various cutoff functions
, such as smooth, compactly supported functions. See this previous blog post for an instance of such an approach. Another related approach is the “pretentious” approach to multiplicative number theory currently being developed by Granville-Soundararajan and their collaborators. We will occasionally make reference to these more modern approaches in these notes, but will primarily focus on the classical approaches.
To reverse the process and derive control on summatory functions or logarithmic sums starting from control of Dirichlet series is trickier, and usually requires one to allow to be complex-valued rather than real-valued if one wants to obtain really accurate estimates; we will return to this point in subsequent notes. However, there is a cheap way to get upper bounds on such sums, known as Rankin’s trick, which we will discuss later in these notes.
The basic strategy of elementary multiplicative theory is to first gather useful estimates on the statistics of “smooth” or “non-oscillatory” functions, such as the constant function , the harmonic function
, or the logarithm function
; one also considers the statistics of periodic functions such as Dirichlet characters. These functions can be understood without any multiplicative number theory, using basic tools from real analysis such as the (quantitative version of the) integral test or summation by parts. Once one understands the statistics of these basic functions, one can then move on to statistics of more arithmetically interesting functions, such as the divisor function (2) or the von Mangoldt function
that we will discuss below. A key tool to relate these functions to each other is that of Dirichlet convolution, which is an operation that interacts well with summatory functions, logarithmic sums, and particularly well with Dirichlet series.
This is only an introduction to elementary multiplicative number theory techniques. More in-depth treatments may be found in this text of Montgomery-Vaughan, or this text of Bateman-Diamond.
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