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One of the basic general problems in analytic number theory is to understand as much as possible the fluctuations of the Möbius function {\mu(n)}, defined as {(-1)^k} when {n} is the product of {k} distinct primes, and zero otherwise. For instance, as {\mu} takes values in {\{-1,0,1\}}, we have the trivial bound

\displaystyle  |\sum_{n \leq x} \mu(n)| \leq x

and the seemingly slight improvement

\displaystyle  \sum_{n \leq x} \mu(n) = o(x) \ \ \ \ \ (1)

is already equivalent to the prime number theorem, as observed by Landau (see e.g. this previous blog post for a proof), while the much stronger (and still open) improvement

\displaystyle  \sum_{n \leq x} \mu(n) = O(x^{1/2+o(1)})

is equivalent to the notorious Riemann hypothesis.

There is a general Möbius pseudorandomness heuristic that suggests that the sign pattern {\mu} behaves so randomly (or pseudorandomly) that one should expect a substantial amount of cancellation in sums that involve the sign fluctuation of the Möbius function in a nontrivial fashion, with the amount of cancellation present comparable to the amount that an analogous random sum would provide; cf. the probabilistic heuristic discussed in this recent blog post. There are a number of ways to make this heuristic precise. As already mentioned, the Riemann hypothesis can be considered one such manifestation of the heuristic. Another manifestation is the following old conjecture of Chowla:

Conjecture 1 (Chowla’s conjecture) For any fixed integer {m} and exponents {a_1,a_2,\ldots,a_m \geq 0}, with at least one of the {a_i} odd (so as not to completely destroy the sign cancellation), we have

\displaystyle  \sum_{n \leq x} \mu(n+1)^{a_1} \ldots \mu(n+m)^{a_m} = o_{x \rightarrow \infty;m}(x).

Note that as {\mu^a = \mu^{a+2}} for any {a \geq 1}, we can reduce to the case when the {a_i} take values in {0,1,2} here. When only one of the {a_i} are odd, this is essentially the prime number theorem in arithmetic progressions (after some elementary sieving), but with two or more of the {a_i} are odd, the problem becomes completely open. For instance, the estimate

\displaystyle  \sum_{n \leq x} \mu(n) \mu(n+2) = o(x)

is morally very close to the conjectured asymptotic

\displaystyle  \sum_{n \leq x} \Lambda(n) \Lambda(n+2) = 2\Pi_2 x + o(x)

for the von Mangoldt function {\Lambda}, where {\Pi_2 := \prod_{p > 2} (1 - \frac{1}{(p-1)^2}) = 0.66016\ldots} is the twin prime constant; this asymptotic in turn implies the twin prime conjecture. (To formally deduce estimates for von Mangoldt from estimates for Möbius, though, typically requires some better control on the error terms than {o()}, in particular gains of some power of {\log x} are usually needed. See this previous blog post for more discussion.)

Remark 1 The Chowla conjecture resembles an assertion that, for {n} chosen randomly and uniformly from {1} to {x}, the random variables {\mu(n+1),\ldots,\mu(n+k)} become asymptotically independent of each other (in the probabilistic sense) as {x \rightarrow \infty}. However, this is not quite accurate, because some moments (namely those with all exponents {a_i} even) have the “wrong” asymptotic value, leading to some unwanted correlation between the two variables. For instance, the events {\mu(n)=0} and {\mu(n+4)=0} have a strong correlation with each other, basically because they are both strongly correlated with the event of {n} being divisible by {4}. A more accurate interpretation of the Chowla conjecture is that the random variables {\mu(n+1),\ldots,\mu(n+k)} are asymptotically conditionally independent of each other, after conditioning on the zero pattern {\mu(n+1)^2,\ldots,\mu(n+k)^2}; thus, it is the sign of the Möbius function that fluctuates like random noise, rather than the zero pattern. (The situation is a bit cleaner if one works instead with the Liouville function {\lambda} instead of the Möbius function {\mu}, as this function never vanishes, but we will stick to the traditional Möbius function formalism here.)

A more recent formulation of the Möbius randomness heuristic is the following conjecture of Sarnak. Given a bounded sequence {f: {\bf N} \rightarrow {\bf C}}, define the topological entropy of the sequence to be the least exponent {\sigma} with the property that for any fixed {\epsilon > 0}, and for {m} going to infinity the set {\{ (f(n+1),\ldots,f(n+m)): n \in {\bf N} \} \subset {\bf C}^m} of {f} can be covered by {O( \exp( \sigma m + o(m) ) )} balls of radius {\epsilon}. (If {f} arises from a minimal topological dynamical system {(X,T)} by {f(n) := f(T^n x)}, the above notion is equivalent to the usual notion of the topological entropy of a dynamical system.) For instance, if the sequence is a bit sequence (i.e. it takes values in {\{0,1\}}), then there are only {\exp(\sigma m + o(m))} {m}-bit patterns that can appear as blocks of {m} consecutive bits in this sequence. As a special case, a Turing machine with bounded memory that had access to a random number generator at the rate of one random bit produced every {T} units of time, but otherwise evolved deterministically, would have an output sequence that had a topological entropy of at most {\frac{1}{T} \log 2}. A bounded sequence is said to be deterministic if its topological entropy is zero. A typical example is a polynomial sequence such as {f(n) := e^{2\pi i \alpha n^2}} for some fixed {\sigma}; the {m}-blocks of such polynomials sequence have covering numbers that only grow polynomially in {m}, rather than exponentially, thus yielding the zero entropy. Unipotent flows, such as the horocycle flow on a compact hyperbolic surface, are another good source of deterministic sequences.

Conjecture 2 (Sarnak’s conjecture) Let {f: {\bf N} \rightarrow {\bf C}} be a deterministic bounded sequence. Then

\displaystyle  \sum_{n \leq x} \mu(n) f(n) = o_{x \rightarrow \infty;f}(x).

This conjecture in general is still quite far from being solved. However, special cases are known:

  • For constant sequences, this is essentially the prime number theorem (1).
  • For periodic sequences, this is essentially the prime number theorem in arithmetic progressions.
  • For quasiperiodic sequences such as {f(n) = F(\alpha n \hbox{ mod } 1)} for some continuous {F}, this follows from the work of Davenport.
  • For nilsequences, this is a result of Ben Green and myself.
  • For horocycle flows, this is a result of Bourgain, Sarnak, and Ziegler.
  • For the Thue-Morse sequence, this is a result of Dartyge-Tenenbaum (with a stronger error term obtained by Maduit-Rivat). A subsequent result of Bourgain handles all bounded rank one sequences (though the Thue-Morse sequence is actually of rank two), and a related result of Green establishes asymptotic orthogonality of the Möbius function to bounded depth circuits, although such functions are not necessarily deterministic in nature.
  • For the Rudin-Shapiro sequence, I sketched out an argument at this MathOverflow post.
  • The Möbius function is known to itself be non-deterministic, because its square {\mu^2(n)} (i.e. the indicator of the square-free functions) is known to be non-deterministic (indeed, its topological entropy is {\frac{6}{\pi^2}\log 2}). (The corresponding question for the Liouville function {\lambda(n)}, however, remains open, as the square {\lambda^2(n)=1} has zero entropy.)
  • In the converse direction, it is easy to construct sequences of arbitrarily small positive entropy that correlate with the Möbius function (a rather silly example is {\mu(n) 1_{k|n}} for some fixed large (squarefree) {k}, which has topological entropy at most {\log 2/k} but clearly correlates with {\mu}).

See this survey of Sarnak for further discussion of this and related topics.

In this post I wanted to give a very nice argument of Sarnak that links the above two conjectures:

Proposition 3 The Chowla conjecture implies the Sarnak conjecture.

The argument does not use any number-theoretic properties of the Möbius function; one could replace {\mu} in both conjectures by any other function from the natural numbers to {\{-1,0,+1\}} and obtain the same implication. The argument consists of the following ingredients:

  1. To show that {\sum_{n<x} \mu(n) f(n) = o(x)}, it suffices to show that the expectation of the random variable {\frac{1}{m} (\mu(n+1)f(n+1)+\ldots+\mu(n+m)f(n+m))}, where {n} is drawn uniformly at random from {1} to {x}, can be made arbitrary small by making {m} large (and {n} even larger).
  2. By the union bound and the zero topological entropy of {f}, it suffices to show that for any bounded deterministic coefficients {c_1,\ldots,c_m}, the random variable {\frac{1}{m}(c_1 \mu(n+1) + \ldots + c_m \mu(n+m))} concentrates with exponentially high probability.
  3. Finally, this exponentially high concentration can be achieved by the moment method, using a slight variant of the moment method proof of the large deviation estimates such as the Chernoff inequality or Hoeffding inequality (as discussed in this blog post).

As is often the case, though, while the “top-down” order of steps presented above is perhaps the clearest way to think conceptually about the argument, in order to present the argument formally it is more convenient to present the arguments in the reverse (or “bottom-up”) order. This is the approach taken below the fold.

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One of the basic problems in analytic number theory is to estimate sums of the form

\displaystyle  \sum_{p<x} f(p)

as {x \rightarrow \infty}, where {p} ranges over primes and {f} is some explicit function of interest (e.g. a linear phase function {f(p) = e^{2\pi i \alpha p}} for some real number {\alpha}). This is essentially the same task as obtaining estimates on the sum

\displaystyle  \sum_{n<x} \Lambda(n) f(n)

where {\Lambda} is the von Mangoldt function. If {f} is bounded, {f(n)=O(1)}, then from the prime number theorem one has the trivial bound

\displaystyle  \sum_{n<x} \Lambda(n) f(n) = O(x)

but often (when {f} is somehow “oscillatory” in nature) one is seeking the refinement

\displaystyle  \sum_{n<x} \Lambda(n) f(n) = o(x) \ \ \ \ \ (1)

or equivalently

\displaystyle  \sum_{p<x} f(p) = o(\frac{x}{\log x}). \ \ \ \ \ (2)

Thanks to identities such as

\displaystyle  \Lambda(n) = \sum_{d|n} \mu(d) \log(\frac{n}{d}), \ \ \ \ \ (3)

where {\mu} is the Möbius function, refinements such as (1) are similar in spirit to estimates of the form

\displaystyle  \sum_{n<x} \mu(n) f(n) = o(x). \ \ \ \ \ (4)

Unfortunately, the connection between (1) and (4) is not particularly tight; roughly speaking, one needs to improve the bounds in (4) (and variants thereof) by about two factors of {\log x} before one can use identities such as (3) to recover (1). Still, one generally thinks of (1) and (4) as being “morally” equivalent, even if they are not formally equivalent.

When {f} is oscillating in a sufficiently “irrational” way, then one standard way to proceed is the method of Type I and Type II sums, which uses truncated versions of divisor identities such as (3) to expand out either (1) or (4) into linear (Type I) or bilinear sums (Type II) with which one can exploit the oscillation of {f}. For instance, Vaughan’s identity lets one rewrite the sum in (1) as the sum of the Type I sum

\displaystyle  \sum_{d \leq U} \mu(d) (\sum_{V/d \leq r \leq x/d} (\log r) f(rd)),

the Type I sum

\displaystyle  -\sum_{d \leq UV} a(d) \sum_{V/d \leq r \leq x/d} f(rd),

the Type II sum

\displaystyle  -\sum_{V \leq d \leq x/U} \sum_{U < m \leq x/V} \Lambda(d) b(m) f(dm),

and the error term {\sum_{d \leq V} \Lambda(n) f(n)}, whenever {1 \leq U, V \leq x} are parameters, and {a, b} are the sequences

\displaystyle  a(d) := \sum_{e \leq U, f \leq V: ef = d} \Lambda(d) \mu(e)

and

\displaystyle  b(m) := \sum_{d|m: d \leq U} \mu(d).

Similarly one can express (4) as the Type I sum

\displaystyle  -\sum_{d \leq UV} c(d) \sum_{UV/d \leq r \leq x/d} f(rd),

the Type II sum

\displaystyle  - \sum_{V < d \leq x/U} \sum_{U < m \leq x/d} \mu(m) b(d) f(dm)

and the error term {\sum_{d \leq UV} \mu(n) f(N)}, whenever {1 \leq U,V \leq x} with {UV \leq x}, and {c} is the sequence

\displaystyle  c(d) := \sum_{e \leq U, f \leq V: ef = d} \mu(d) \mu(e).

After eliminating troublesome sequences such as {a(), b(), c()} via Cauchy-Schwarz or the triangle inequality, one is then faced with the task of estimating Type I sums such as

\displaystyle  \sum_{r \leq y} f(rd)

or Type II sums such as

\displaystyle  \sum_{r \leq y} f(rd) \overline{f(rd')}

for various {y, d, d' \geq 1}. Here, the trivial bound is {O(y)}, but due to a number of logarithmic inefficiencies in the above method, one has to obtain bounds that are more like {O( \frac{y}{\log^C y})} for some constant {C} (e.g. {C=5}) in order to end up with an asymptotic such as (1) or (4).

However, in a recent paper of Bourgain, Sarnak, and Ziegler, it was observed that as long as one is only seeking the Mobius orthogonality (4) rather than the von Mangoldt orthogonality (1), one can avoid losing any logarithmic factors, and rely purely on qualitative equidistribution properties of {f}. A special case of their orthogonality criterion (which actually dates back to an earlier paper of Katai, as was pointed out to me by Nikos Frantzikinakis) is as follows:

Proposition 1 (Orthogonality criterion) Let {f: {\bf N} \rightarrow {\bf C}} be a bounded function such that

\displaystyle  \sum_{n \leq x} f(pn) \overline{f(qn)} = o(x) \ \ \ \ \ (5)

for any distinct primes {p, q} (where the decay rate of the error term {o(x)} may depend on {p} and {q}). Then

\displaystyle  \sum_{n \leq x} \mu(n) f(n) =o(x). \ \ \ \ \ (6)

Actually, the Bourgain-Sarnak-Ziegler paper establishes a more quantitative version of this proposition, in which {\mu} can be replaced by an arbitrary bounded multiplicative function, but we will content ourselves with the above weaker special case. (See also these notes of Harper, which uses the Katai argument to give a slightly weaker quantitative bound in the same spirit.) This criterion can be viewed as a multiplicative variant of the classical van der Corput lemma, which in our notation asserts that {\sum_{n \leq x} f(n) = o(x)} if one has {\sum_{n \leq x} f(n+h) \overline{f(n)} = o(x)} for each fixed non-zero {h}.

As a sample application, Proposition 1 easily gives a proof of the asymptotic

\displaystyle  \sum_{n \leq x} \mu(n) e^{2\pi i \alpha n} = o(x)

for any irrational {\alpha}. (For rational {\alpha}, this is a little trickier, as it is basically equivalent to the prime number theorem in arithmetic progressions.) The paper of Bourgain, Sarnak, and Ziegler also apply this criterion to nilsequences (obtaining a quick proof of a qualitative version of a result of Ben Green and myself, see these notes of Ziegler for details) and to horocycle flows (for which no Möbius orthogonality result was previously known).

Informally, the connection between (5) and (6) comes from the multiplicative nature of the Möbius function. If (6) failed, then {\mu(n)} exhibits strong correlation with {f(n)}; by change of variables, we then expect {\mu(pn)} to correlate with {f(pn)} and {\mu(pm)} to correlate with {f(qn)}, for “typical” {p,q} at least. On the other hand, since {\mu} is multiplicative, {\mu(pn)} exhibits strong correlation with {\mu(qn)}. Putting all this together (and pretending correlation is transitive), this would give the claim (in the contrapositive). Of course, correlation is not quite transitive, but it turns out that one can use the Cauchy-Schwarz inequality as a substitute for transitivity of correlation in this case.

I will give a proof of Proposition 1 below the fold (which is not quite based on the argument in the above mentioned paper, but on a variant of that argument communicated to me by Tamar Ziegler, and also independently discovered by Adam Harper). The main idea is to exploit the following observation: if {P} is a “large” but finite set of primes (in the sense that the sum {A := \sum_{p \in P} \frac{1}{p}} is large), then for a typical large number {n} (much larger than the elements of {P}), the number of primes in {P} that divide {n} is pretty close to {A = \sum_{p \in P} \frac{1}{p}}:

\displaystyle  \sum_{p \in P: p|n} 1 \approx A. \ \ \ \ \ (7)

A more precise formalisation of this heuristic is provided by the Turan-Kubilius inequality, which is proven by a simple application of the second moment method.

In particular, one can sum (7) against {\mu(n) f(n)} and obtain an approximation

\displaystyle  \sum_{n \leq x} \mu(n) f(n) \approx \frac{1}{A} \sum_{p \in P} \sum_{n \leq x: p|n} \mu(n) f(n)

that approximates a sum of {\mu(n) f(n)} by a bunch of sparser sums of {\mu(n) f(n)}. Since

\displaystyle  x = \frac{1}{A} \sum_{p \in P} \frac{x}{p},

we see (heuristically, at least) that in order to establish (4), it would suffice to establish the sparser estimates

\displaystyle  \sum_{n \leq x: p|n} \mu(n) f(n) = o(\frac{x}{p})

for all {p \in P} (or at least for “most” {p \in P}).

Now we make the change of variables {n = pm}. As the Möbius function is multiplicative, we usually have {\mu(n) = \mu(p) \mu(m) = - \mu(m)}. (There is an exception when {n} is divisible by {p^2}, but this will be a rare event and we will be able to ignore it.) So it should suffice to show that

\displaystyle  \sum_{m \leq x/p} \mu(m) f(pm) = o( x/p )

for most {p \in P}. However, by the hypothesis (5), the sequences {m \mapsto f(pm)} are asymptotically orthogonal as {p} varies, and this claim will then follow from a Cauchy-Schwarz argument.

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I’ve just uploaded to the arXiv the paper A remark on partial sums involving the Möbius function, submitted to Bull. Aust. Math. Soc..

The Möbius function {\mu(n)} is defined to equal {(-1)^k} when {n} is the product of {k} distinct primes, and equal to zero otherwise; it is closely connected to the distribution of the primes. In 1906, Landau observed that one could show using purely elementary means that the prime number theorem

\displaystyle  \sum_{p \leq x} 1 = (1+o(1)) \frac{x}{\log x} \ \ \ \ \ (1)

(where {o(1)} denotes a quantity that goes to zero as {x \rightarrow \infty}) was logically equivalent to the partial sum estimates

\displaystyle  \sum_{n \leq x} \mu(n) = o(x) \ \ \ \ \ (2)

and

\displaystyle  \sum_{n \leq x} \frac{\mu(n)}{n} = o(1); \ \ \ \ \ (3)

we give a sketch of the proof of these equivalences below the fold.

On the other hand, these three inequalities are all easy to prove if the {o()} terms are replaced by their {O()} counterparts. For instance, by observing that the binomial coefficient {\binom{2n}{n}} is bounded by {4^n} on the one hand (by Pascal’s triangle or the binomial theorem), and is divisible by every prime between {n} and {2n} on the other hand, we conclude that

\displaystyle  \sum_{n < p \leq 2n} \log p \leq n \log 4

from which it is not difficult to show that

\displaystyle  \sum_{p \leq x} 1 = O( \frac{x}{\log x} ). \ \ \ \ \ (4)

Also, since {|\mu(n)| \leq 1}, we clearly have

\displaystyle  |\sum_{n \leq x} \mu(n)| \leq x.

Finally, one can also show that

\displaystyle  |\sum_{n \leq x} \frac{\mu(n)}{n}| \leq 1. \ \ \ \ \ (5)

Indeed, assuming without loss of generality that {x} is a positive integer, and summing the inversion formula {1_{n=1} = \sum_{d|n} \mu(d)} over all {n \leq x} one sees that

\displaystyle  1 = \sum_{d \leq x} \mu(d) \left\lfloor \frac{x}{d}\right \rfloor = \sum_{d \leq x} \mu(d) \frac{x}{d} - \sum_{d<x} \mu(d) \left \{ \frac{x}{d} \right\} \ \ \ \ \ (6)

and the claim follows by bounding {|\mu(d) \left \{ \frac{x}{d} \right\}|} by {1}.

In this paper I extend these observations to more general multiplicative subsemigroups of the natural numbers. More precisely, if {P} is any set of primes (finite or infinite), I show that

\displaystyle  |\sum_{n \in \langle P \rangle: n \leq x} \frac{\mu(n)}{n}| \leq 1 \ \ \ \ \ (7)

and that

\displaystyle  \sum_{n \in \langle P \rangle: n \leq x} \frac{\mu(n)}{n} = \prod_{p \in P} (1-\frac{1}{p}) + o(1), \ \ \ \ \ (8)

where {\langle P \rangle} is the multiplicative semigroup generated by {P}, i.e. the set of natural numbers whose prime factors lie in {P}.

Actually the methods are completely elementary (the paper is just six pages long), and I can give the proof of (7) in full here. Again we may take {x} to be a positive integer. Clearly we may assume that

\displaystyle  \sum_{n \in \langle P \rangle: n \leq x} \frac{1}{n} > 1, \ \ \ \ \ (9)

as the claim is trivial otherwise.

If {P'} denotes the primes that are not in {P}, then Möbius inversion gives us

\displaystyle  1_{n \in \langle P' \rangle} = \sum_{d|n; d \in \langle P \rangle} \mu(d).

Summing this for {1 \leq n \leq x} gives

\displaystyle  \sum_{n \in \langle P' \rangle: n \leq x} 1 = \sum_{d \in \langle P \rangle: d \leq x} \mu(d) \frac{x}{d} - \sum_{d \in \langle P \rangle: d \leq x} \mu(d) \left \{ \frac{x}{d} \right \}.

We can bound {|\mu(d) \left \{ \frac{x}{d} \right \}| \leq 1 - \frac{1}{d}} and so

\displaystyle  |\sum_{d \in \langle P \rangle: d \leq x} \mu(d) \frac{x}{d}| \leq \sum_{n \in \langle P' \rangle: n \leq x} 1 + \sum_{n \in \langle P \rangle: n \leq x} 1 - \sum_{n \in \langle P \rangle: n \leq x} \frac{1}{n}.

The claim now follows from (9), since {\langle P \rangle} and {\langle P' \rangle} overlap only at {1}.

As special cases of (7) we see that

\displaystyle  |\sum_{d \leq x: d|m} \frac{\mu(d)}{d}| \leq 1

and

\displaystyle  |\sum_{n \leq x: (m,n)=1} \frac{\mu(n)}{n}| \leq 1

for all {m,x}. Since {\mu(mn) = \mu(m) \mu(n) 1_{(m,n)=1}}, we also have

\displaystyle  |\sum_{n \leq x} \frac{\mu(mn)}{n}| \leq 1.

One might hope that these inequalities (which gain a factor of {\log x} over the trivial bound) might be useful when performing effective sieve theory, or effective estimates on various sums involving the primes or arithmetic functions.

This inequality (7) is so simple to state and prove that I must think that it was known to, say, Landau or Chebyshev, but I can’t find any reference to it in the literature. [Update, Sep 4: I have learned that similar results have been obtained in a paper by Granville and Soundararajan, and have updated the paper appropriately.] The proof of (8) is a simple variant of that used to prove (7) but I will not detail it here.

Curiously, this is one place in number theory where the elementary methods seem superior to the analytic ones; there is a zeta function {\zeta_P(s) = \sum_{n \in \langle P \rangle} \frac{1}{n^s} = \prod_{p \in P} (1-\frac{1}{p^s})^{-1}} associated to this problem, but it need not have a meromorphic continuation beyond the region {\{ \hbox{Re}(s) > 1 \}}, and it turns out to be remarkably difficult to use this function to establish the above results. (I do have a proof of this form, which I in fact found before I stumbled on the elementary proof, but it is far longer and messier.)

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Ben Green and I have just uploaded to the arXiv our paper, “The Möbius function is asymptotically orthogonal to nilsequences“, which is a sequel to our earlier paper “The quantitative behaviour of polynomial orbits on nilmanifolds“, which I talked about in this post.  In this paper, we apply our previous results on quantitative equidistribution polynomial orbits in nilmanifolds to settle the Möbius and nilsequences conjecture from our earlier paper, as part of our program to detect and count solutions to linear equations in primes.  (The other major plank of that program, namely the inverse conjecture for the Gowers norm, remains partially unresolved at present.)  Roughly speaking, this conjecture asserts the asymptotic orthogonality

\displaystyle |\frac{1}{N} \sum_{n=1}^N \mu(n) f(n)| \ll_A \log^{-A} N (1)

between the Möbius function \mu(n) and any Lipschitz nilsequence f(n), by which we mean a sequence of the form f(n) = F(g^n x) for some orbit g^n x in a nilmanifold G/\Gamma, and some Lipschitz function F: G/\Gamma \to {\Bbb C} on that nilmanifold.  (The implied constant can depend on the nilmanifold and on the Lipschitz constant of F, but it is important that it be independent of the generator g of the orbit or the base point x.)  The case when f is constant is essentially the prime number theorem; the case when f is periodic is essentially the prime number theorem in arithmetic progressions.  The case when f is almost periodic (e.g. f(n) = e^{2\pi i \alpha n} for some irrational \alpha) was established by Davenport, using the method of Vinogradov.  The case when f was a 2-step nilsequence (such as the quadratic phase f(n) = e^{2\pi i \alpha n^2}; bracket quadratic phases such as f(n) = e^{2\pi \lfloor \alpha n \rfloor \beta n} can also be covered by an approximation argument, though the logarithmic decay in (1) is weakened as a consequence) was done by Ben and myself a few years ago, by a rather ad hoc adaptation of Vinogradov’s method.  By using the equidistribution theory of nilmanifolds, we were able to apply Vinogradov’s method more systematically, and in fact the proof is relatively short (20 pages), although it relies on the 64-page predecessor paper on equidistribution.  I’ll talk a little bit more about the proof after the fold.

There is an amusing way to interpret the conjecture (using the close relationship between nilsequences and bracket polynomials) as an assertion of the pseudorandomness of the Liouville function from a computational complexity perspective.    Suppose you possess a calculator with the wonderful property of being infinite precision: it can accept arbitrarily large real numbers as input, manipulate them precisely, and also store them in memory.  However, this calculator has two limitations.  Firstly, the only operations available are addition, subtraction, multiplication, integer part x \mapsto [x], fractional part x \mapsto \{x\}, memory store (into one of O(1) registers), and memory recall (from one of these O(1) registers).  In particular, there is no ability to perform division.  Secondly, the calculator only has a finite display screen, and when it shows a real number, it only shows O(1) digits before and after the decimal point.  (Thus, for instance, the real number 1234.56789 might be displayed only as \ldots 34.56\ldots.)

Now suppose you play the following game with an opponent.

  1. The opponent specifies a large integer d.
  2. You get to enter in O(1) real constants of your choice into your calculator.  These can be absolute constants such as \sqrt{2} and \pi, or they can depend on d (e.g. you can enter in 10^{-d}).
  3. The opponent randomly selects an d-digit integer n, and enters n into one of the registers of your calculator.
  4. You are allowed to perform O(1) operations on your calculator and record what is displayed on the calculator’s viewscreen.
  5. After this, you have to guess whether the opponent’s number n had an odd or even number of prime factors (i.e. you guess \lambda(n).)
  6. If you guess correctly, you win $1; otherwise, you lose $1.

For instance, using your calculator you can work out the first few digits of \{ \sqrt{2}n \lfloor \sqrt{3} n \rfloor \}, provided of course that you entered the constants \sqrt{2} and \sqrt{3} in advance.  You can also work out the leading digits of n by storing 10^{-d} in advance, and computing the first few digits of 10^{-d} n.

Our theorem is equivalent to the assertion that as d goes to infinity (keeping the O(1) constants fixed), your probability of winning this game converges to 1/2; in other words, your calculator becomes asymptotically useless to you for the purposes of guessing whether n has an odd or even number of prime factors, and you may as well just guess randomly.

[I should mention a recent result in a similar spirit by Mauduit and Rivat; in this language, their result asserts that knowing the last few digits of the digit-sum of n does not increase your odds of guessing \lambda(n) correctly.]

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