One of the basic problems in analytic number theory is to estimate sums of the form
as , where ranges over primes and is some explicit function of interest (e.g. a linear phase function for some real number ). This is essentially the same task as obtaining estimates on the sum
where is the von Mangoldt function. If is bounded, , then from the prime number theorem one has the trivial bound
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 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 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 . For instance, Vaughan’s identity lets one rewrite the sum in (1) as the sum of the Type I sum
the Type I sum
the Type II sum
and the error term , whenever are parameters, and are the sequences
Similarly one can express (4) as the Type I sum
the Type II sum
and the error term , whenever with , and is the sequence
After eliminating troublesome sequences such as via Cauchy-Schwarz or the triangle inequality, one is then faced with the task of estimating Type I sums such as
or Type II sums such as
for various . Here, the trivial bound is , but due to a number of logarithmic inefficiencies in the above method, one has to obtain bounds that are more like for some constant (e.g. ) 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 . 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:
Actually, the Bourgain-Sarnak-Ziegler paper establishes a more quantitative version of this proposition, in which 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 if one has for each fixed non-zero .
As a sample application, Proposition 1 easily gives a proof of the asymptotic
for any irrational . (For rational , 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 exhibits strong correlation with ; by change of variables, we then expect to correlate with and to correlate with , for “typical” at least. On the other hand, since is multiplicative, exhibits strong correlation with . 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 is a “large” but finite set of primes (in the sense that the sum is large), then for a typical large number (much larger than the elements of ), the number of primes in that divide is pretty close to :
In particular, one can sum (7) against and obtain an approximation
that approximates a sum of by a bunch of sparser sums of . Since
we see (heuristically, at least) that in order to establish (4), it would suffice to establish the sparser estimates
for all (or at least for “most” ).
Now we make the change of variables . As the Möbius function is multiplicative, we usually have . (There is an exception when is divisible by , but this will be a rare event and we will be able to ignore it.) So it should suffice to show that
for most . However, by the hypothesis (5), the sequences are asymptotically orthogonal as varies, and this claim will then follow from a Cauchy-Schwarz argument.
— 1. Rigorous proof —
We will need a slowly growing function of , with as , to be chosen later. As the sum of reciprocals of primes diverges, we see that
as . It will also be convenient to eliminate small primes. Note that we may find an even slower growing function of , with as , such that
Although it is not terribly important, we will take and to be powers of two. Thus, if we set to be all the primes between and , the quantity
goes to infinity as .
Proof: We have
On the other hand, we have
and thus (if is sufficiently slowly growing)
Similarly, we have
The expression is equal to when , and when . A brief calculation then shows that
if is sufficiently slowly growing. Inserting these bounds into (8), the claim follows.
From (8) and the Cauchy-Schwarz inequality, one has
which we rearrange as
Since goes to infinity, the term is , so it now suffices to show that
Write . Then we have for all but values of (if is sufficiently slowly growing). The exceptional values contribute at most
which is acceptable. Thus it suffices to show that
Partitioning into dyadic blocks, it suffices to show that
uniformly for , where are the primes between and .
Fix . The left-hand side can be rewritten as
so by the Cauchy-Schwarz inequality it suffices to show that
We can rearrange the left-hand side as
Now if is sufficiently slowly growing as a function of , we see from (5) that for all distinct , we have
uniformly in ; meanwhile, for , we have the crude bound
The claim follows (noting from the prime number theorem that ).
— 2. From Möbius to von Mangoldt? —
It would be great if one could pass from the Möbius asymptotic orthogonality (4) to the von Mangoldt asymptotic orthgonality (1) (or equivalently, to (2)), as this would give some new information about the distribution of primes. Unfortunately, it seems that some additional input is needed to do so. Here is a simple example of a conditional implication that requires an additional input, namely some quantitative control on “Type I” sums:
and thus (by discarding the prime powers and summing by parts)
If is sufficiently slowly growing, then by (9) one has uniformly for all . If is sufficiently slowly growing, this implies that the first term in (12) is also . As for the second term, we dyadically decompose it and bound it in absolute value by
By summation by parts, we can bound
This sum evaluates to , and the claim follows since goes to infinity.
Note that the trivial bound on (10) is , so one needs to gain about two logarithmic factors over the trivial bound in order to use the above proposition. The presence of the supremum is annoying, but it can be removed by a modification of the argument if one improves the bound by an additional logarithm by a variety of methods (e.g. completion of sums), or by smoothing out the constraint . However, I do not know of a way to remove the need to improve the trivial bound by two logarithmic factors.