Let us call an arithmetic function -bounded if we have for all . In this section we focus on the asymptotic behaviour of -bounded multiplicative functions. Some key examples of such functions include:
- The Möbius function ;
- The Liouville function ;
- “Archimedean” characters (which I call Archimedean because they are pullbacks of a Fourier character on the multiplicative group , which has the Archimedean property);
- Dirichlet characters (or “non-Archimedean” characters) (which are essentially pullbacks of Fourier characters on a multiplicative cyclic group with the discrete (non-Archimedean) metric);
- Hybrid characters .
The space of -bounded multiplicative functions is also closed under multiplication and complex conjugation.
Given a multiplicative function , we are often interested in the asymptotics of long averages such as
for large values of , as well as short sums
where and are both large, but is significantly smaller than . (Throughout these notes we will try to normalise most of the sums and integrals appearing here as averages that are trivially bounded by ; note that other normalisations are preferred in some of the literature cited here.) For instance, as we established in Theorem 58 of Notes 1, the prime number theorem is equivalent to the assertion that
as . The Liouville function behaves almost identically to the Möbius function, in that estimates for one function almost always imply analogous estimates for the other:
Exercise 1 Without using the prime number theorem, show that (1) is also equivalent to
as . (Hint: use the identities and .)
Henceforth we shall focus our discussion more on the Liouville function, and turn our attention to averages on shorter intervals. From (2) one has
as if is such that for some fixed . However it is significantly more difficult to understand what happens when grows much slower than this. By using the techniques based on zero density estimates discussed in Notes 6, it was shown by Motohashi and that one can also establish \eqref. On the Riemann Hypothesis Maier and Montgomery lowered the threshold to for an absolute constant (the bound is more classical, following from Exercise 33 of Notes 2). On the other hand, the randomness heuristics from Supplement 4 suggest that should be able to be taken as small as , and perhaps even if one is particularly optimistic about the accuracy of these probabilistic models. On the other hand, the Chowla conjecture (mentioned for instance in Supplement 4) predicts that cannot be taken arbitrarily slowly growing in , due to the conjectured existence of arbitrarily long strings of consecutive numbers where the Liouville function does not change sign (and in fact one can already show from the known partial results towards the Chowla conjecture that (3) fails for some sequence and some sufficiently slowly growing , by modifying the arguments in these papers of mine).
The situation is better when one asks to understand the mean value on almost all short intervals, rather than all intervals. There are several equivalent ways to formulate this question:
Exercise 2 Let be a function of such that and as . Let be a -bounded function. Show that the following assertions are equivalent:
As it turns out the second moment formulation in (iii) will be the most convenient for us to work with in this set of notes, as it is well suited to Fourier-analytic techniques (and in particular the Plancherel theorem).
Using zero density methods, for instance, it was shown by Ramachandra that
whenever and . With this quality of bound (saving arbitrary powers of over the trivial bound of ), this is still the lowest value of one can reach unconditionally. However, in a striking recent breakthrough, it was shown by Matomaki and Radziwill that as long as one is willing to settle for weaker bounds (saving a small power of or , or just a qualitative decay of ), one can obtain non-trivial estimates on far shorter intervals. For instance, they show
Theorem 3 (Matomaki-Radziwill theorem for Liouville) For any , one has
for some absolute constant .
In fact they prove a slightly more precise result: see Theorem 1 of that paper. In particular, they obtain the asymptotic (4) for any function that goes to infinity as , no matter how slowly! This ability to let grow slowly with is important for several applications; for instance, in order to combine this type of result with the entropy decrement methods from Notes 9, it is essential that be allowed to grow more slowly than . See also this survey of Soundararajan for further discussion.
Exercise 4 In this exercise you may use Theorem 3 freely.
(There is a curious asymmetry to the difficulty level of these bounds; the upper bound in (ii) was established much earlier by Harman, Pintz, and Wolke, but the lower bound in (i) was only established in the Matomaki-Radziwill paper.)
The techniques discussed previously were highly complex-analytic in nature, relying in particular on the fact that functions such as or have Dirichlet series , that extend meromorphically into the critical strip. In contrast, the Matomaki-Radziwill theorem does not rely on such meromorphic continuations, and in fact holds for more general classes of -bounded multiplicative functions , for which one typically does not expect any meromorphic continuation into the strip. Instead, one can view the Matomaki-Radziwill theory as following the philosophy of a slightly different approach to multiplicative number theory, namely the pretentious multiplicative number theory of Granville and Soundarajan (as presented for instance in their draft monograph). A basic notion here is the pretentious distance between two -bounded multiplicative functions (at a given scale ), which informally measures the extent to which “pretends” to be like (or vice versa). The precise definition is
Definition 5 (Pretentious distance) Given two -bounded multiplicative functions , and a threshold , the pretentious distance between and up to scale is given by the formula
Note that one can also define an infinite version of this distance by removing the constraint , though in such cases the pretentious distance may then be infinite. The pretentious distance is not quite a metric (because can be non-zero, and furthermore can vanish without being equal), but it is still quite close to behaving like a metric, in particular it obeys the triangle inequality; see Exercise 16 below. The philosophy of pretentious multiplicative number theory is that two -bounded multiplicative functions will exhibit similar behaviour at scale if their pretentious distance is bounded, but will become uncorrelated from each other if this distance becomes large. A simple example of this philosophy is given by the following “weak Halasz theorem”, proven in Section 2:
Proposition 6 (Logarithmically averaged version of Halasz) Let be sufficiently large. Then for any -bounded multiplicative functions , one has
for an absolute constant .
In particular, if does not pretend to be , then the logarithmic average will be small. This condition is basically necessary, since of course .
If one works with non-logarithmic averages , then not pretending to be is insufficient to establish decay, as was already observed in Exercise 11 of Notes 1: if is an Archimedean character for some non-zero real , then goes to zero as (which is consistent with Proposition 6), but does not go to zero. However, this is in some sense the “only” obstruction to these averages decaying to zero, as quantified by the following basic result:
Theorem 7 (Halasz’s theorem) Let be sufficiently large. Then for any -bounded multiplicative function , one has
for an absolute constant and any .
Informally, we refer to a -bounded multiplicative function as “pretentious’; if it pretends to be a character such as , and “non-pretentious” otherwise. The precise distinction is rather malleable, as the precise class of characters that one views as “obstructions” varies from situation to situation. For instance, in Proposition 6 it is just the trivial character which needs to be considered, but in Theorem 7 it is the characters with . In other contexts one may also need to add Dirichlet characters or hybrid characters such as to the list of characters that one might pretend to be. The division into pretentious and non-pretentious functions in multiplicative number theory is faintly analogous to the division into major and minor arcs in the circle method applied to additive number theory problems; see Notes 8. The Möbius and Liouville functions are model examples of non-pretentious functions; see Exercise 24.
In the contrapositive, Halasz’ theorem can be formulated as the assertion that if one has a large mean
for some , then one has the pretentious property
for some . This has the flavour of an “inverse theorem”, of the type often found in arithmetic combinatorics.
Among other things, Halasz’s theorem gives yet another proof of the prime number theorem (1); see Section 2.
We now give a version of the Matomaki-Radziwill theorem for general (non-pretentious) multiplicative functions that is formulated in a similar contrapositive (or “inverse theorem”) fashion, though to simplify the presentation we only state a qualitative version that does not give explicit bounds.
Theorem 8 ((Qualitative) Matomaki-Radziwill theorem) Let , and let , with sufficiently large depending on . Suppose that is a -bounded multiplicative function such that
Then one has
for some .
The condition is basically optimal, as the following example shows:
Exercise 9 Let be a sufficiently small constant, and let be such that . Let be the Archimedean character for some . Show that
Combining Theorem 8 with standard non-pretentiousness facts about the Liouville function (see Exercise 24), we recover Theorem 3 (but with a decay rate of only rather than ). We refer the reader to the original paper of Matomaki-Radziwill (as well as this followup paper with myself) for the quantitative version of Theorem 8 that is strong enough to recover the full version of Theorem 3, and which can also handle real-valued pretentious functions.
With our current state of knowledge, the only arguments that can establish the full strength of Halasz and Matomaki-Radziwill theorems are Fourier analytic in nature, relating sums involving an arithmetic function with its Dirichlet series
which one can view as a discrete Fourier transform of (or more precisely of the measure , if one evaluates the Dirichlet series on the right edge of the critical strip). In this aspect, the techniques resemble the complex-analytic methods from Notes 2, but with the key difference that no analytic or meromorphic continuation into the strip is assumed. The key identity that allows us to pass to Dirichlet series is the following variant of Proposition 7 of Notes 2:
Proposition 10 (Parseval type identity) Let be finitely supported arithmetic functions, and let be a Schwartz function. Then
where is the Fourier transform of . (Note that the finite support of and the Schwartz nature of ensure that both sides of the identity are absolutely convergent.)
The restriction that be finitely supported will be slightly annoying in places, since most multiplicative functions will fail to be finitely supported, but this technicality can usually be overcome by suitably truncating the multiplicative function, and taking limits if necessary.
Proof: By expanding out the Dirichlet series, it suffices to show that
for any natural numbers . But this follows from the Fourier inversion formula applied at .
For applications to Halasz type theorems, one sets equal to the Kronecker delta , producing weighted integrals of of “” type. For applications to Matomaki-Radziwill theorems, one instead sets , and more precisely uses the following corollary of the above proposition, to obtain weighted integrals of of “” type:
Exercise 11 (Plancherel type identity) If is finitely supported, and is a Schwartz function, establish the identity
In contrast, information about the non-pretentious nature of a multiplicative function will give “pointwise” or “” type control on the Dirichlet series , as is suggested from the Euler product factorisation of .
It will be convenient to formalise the notion of , , and control of the Dirichlet series , which as previously mentioned can be viewed as a sort of “Fourier transform” of :
Definition 12 (Fourier norms) Let be finitely supported, and let be a bounded measurable set. We define the Fourier norm
the Fourier norm
and the Fourier norm
One could more generally define norms for other exponents , but we will only need the exponents in this current set of notes. It is clear that all the above norms are in fact (semi-)norms on the space of finitely supported arithmetic functions.
As mentioned above, Halasz’s theorem gives good control on the Fourier norm for restrictions of non-pretentious functions to intervals:
Exercise 13 (Fourier control via Halasz) Let be a -bounded multiplicative function, let be an interval in for some , let , and let be a bounded measurable set. Show that
(Hint: you will need to use summation by parts (or an equivalent device) to deal with a weight.)
Meanwhile, the Plancherel identity in Exercise 11 gives good control on the Fourier norm for functions on long intervals (compare with Exercise 2 from Notes 6):
Exercise 14 ( mean value theorem) Let , and let be finitely supported. Show that
Conclude in particular that if is supported in for some and , then
In the simplest case of the logarithmically averaged Halasz theorem (Proposition 6), Fourier estimates are already sufficient to obtain decent control on the (weighted) Fourier type expressions that show up. However, these estimates are not enough by themselves to establish the full Halasz theorem or the Matomaki-Radziwill theorem. To get from Fourier control to Fourier or control more efficiently, the key trick is use Hölder’s inequality, which when combined with the basic Dirichlet series identity
gives the inequalities
The strategy is then to factor (or approximately factor) the original function as a Dirichlet convolution (or average of convolutions) of various components, each of which enjoys reasonably good Fourier or estimates on various regions , and then combine them using the Hölder inequalities (5), (6) and the triangle inequality. For instance, to prove Halasz’s theorem, we will split into the Dirichlet convolution of three factors, one of which will be estimated in using the non-pretentiousness hypothesis, and the other two being estimated in using Exercise 14. For the Matomaki-Radziwill theorem, one uses a significantly more complicated decomposition of into a variety of Dirichlet convolutions of factors, and also splits up the Fourier domain into several subregions depending on whether the Dirichlet series associated to some of these components are large or small. In each region and for each component of these decompositions, all but one of the factors will be estimated in , and the other in ; but the precise way in which this is done will vary from component to component. For instance, in some regions a key factor will be small in by construction of the region; in other places, the control will come from Exercise 13. Similarly, in some regions, satisfactory control is provided by Exercise 14, but in other regions one must instead use “large value” theorems (in the spirit of Proposition 9 from Notes 6), or amplify the power of the standard mean value theorems by combining the Dirichlet series with other Dirichlet series that are known to be large in this region.
There are several ways to achieve the desired factorisation. In the case of Halasz’s theorem, we can simply work with a crude version of the Euler product factorisation, dividing the primes into three categories (“small”, “medium”, and “large” primes) and expressing as a triple Dirichlet convolution accordingly. For the Matomaki-Radziwill theorem, one instead exploits the Turan-Kubilius phenomenon (Section 5 of Notes 1, or Lemma 2 of Notes 9)) that for various moderately wide ranges of primes, the number of prime divisors of a large number in the range is almost always close to . Thus, if we introduce the arithmetic functions
then we have
and more generally we have a twisted approximation
for multiplicative functions . (Actually, for technical reasons it will be convenient to work with a smoothed out version of these functions; see Section 3.) Informally, these formulas suggest that the “ energy” of a multiplicative function is concentrated in those regions where is extremely large in a sense. Iterations of this formula (or variants of this formula, such as an identity due to Ramaré) will then give the desired (approximate) factorisation of .
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