A basic estimate in multiplicative number theory (particularly if one is using the Granville-Soundararajan “pretentious” approach to this subject) is the following inequality of Halasz (formulated here in a quantitative form introduced by Montgomery and Tenenbaum).
As a qualitative corollary, we conclude (by standard compactness arguments) that if
as . In the more recent work of this paper of Granville and Soundararajan, the sharper bound
is obtained (with a more precise description of the term).
The usual proofs of Halasz’s theorem are somewhat lengthy (though there has been a recent simplification, in forthcoming work of Granville, Harper, and Soundarajan). Below the fold I would like to give a relatively short proof of the following “cheap” version of the inequality, which has slightly weaker quantitative bounds, but still suffices to give qualitative conclusions such as (2).
Theorem 2 (Cheap Halasz inequality) Let be a multiplicative function bounded in magnitude by . Let and , and suppose that is sufficiently large depending on . If (1) holds for all , then
The non-optimal exponent can probably be improved a bit by being more careful with the exponents, but I did not try to optimise it here. A similar bound appears in the first paper of Halasz on this topic.
The idea of the argument is to split as a Dirichlet convolution where is the portion of coming from “small”, “medium”, and “large” primes respectively (with the dividing line between the three types of primes being given by various powers of ). Using a Perron-type formula, one can express this convolution in terms of the product of the Dirichlet series of respectively at various complex numbers with . One can use based estimates to control the Dirichlet series of , while using the hypothesis (1) one can get estimates on the Dirichlet series of . (This is similar to the Fourier-analytic approach to ternary additive problems, such as Vinogradov’s theorem on representing large odd numbers as the sum of three primes.) This idea was inspired by a similar device used in the work of Granville, Harper, and Soundarajan. A variant of this argument also appears in unpublished work of Adam Harper.
I thank Andrew Granville for helpful comments which led to significant simplifications of the argument.
— 1. Basic estimates —
We need the following basic tools from analytic number theory. We begin with a variant of the classical Perron formula.
Proof: By telescoping series (and treating the contribution of trivially), it suffices to show that
of , where
and is a fixed smooth function supported on that equals at the origin. Basic Fourier analysis then tells us that is a Schwartz function with total mass one. This gives the crude bound
for any . For or , we use the bound (say) to arrive at the bound
for we again use and write
and use the Lipschitz bound for to obtain
for such . Putting all these bounds together, we see that
for all . In particular, we can write (3) as
The expression is bounded by when , is bounded by when , is bounded by when or , and is bounded by otherwise. From these bounds, a routine calculation (using the hypothesis ) shows that
and so it remains to show that
we see from the triangle inequality and the support of that
But from integration by parts we see that , and the claim follows.
Next, we recall a standard mean value estimate for Dirichlet series:
Proof: This follows from Lemma 7.1 of Iwaniec-Kowalski; for the convenience of the reader we reproduce the short proof here. Introducing the normalised sinc function , we have
But a standard Fourier-analytic computation shows that vanishes unless , in which case the integral is , and the claim follows.
Now we recall a basic sieve estimate:
Proposition 5 (Sieve bound) Let , let be an interval of length , and let be a set of primes up to . If we remove one residue class mod from for every , the number of remaining natural numbers in is at most .
Finally, we record a standard estimate on the number of smooth numbers:
Proof: See Corollary 1.3 of this paper of Hildebrand and Tenenbaum. (The result also follows from the more classical work of Dickman.) We sketch a short proof here due to Kevin Ford. Let denote the set of numbers that are “smooth” in the sense that they have no prime factor larger than . It then suffices to prove the bound
By the prime number theorem, the contribution to of those is , and the contribution of those with consists only of prime powers, which contribute . Combining these estimates, we can get a bound of the form
where is a quantity to be chosen later. Thus we can bound the left-hand side of (4) by
which by Euler products can be bounded by
By the mean value theorem applied to the function , we can bound by for . By Mertens’ theorem, we thus get a bound of
If we make the choice , we obtain the required bound (4).
— 2. Proof of theorem —
By increasing as necessary we may assume that (say). Let be small parameters (depending on ) to be optimised later; we assume to be sufficiently large depending on . Call a prime small if , medium if , and large if . Observe that for any we can factorise as a Dirichlet convolution
- is the restriction of to those natural numbers whose prime factors are all small;
- is the restriction of to those natural numbers whose prime factors are all medium;
- is the restriction of to those natural numbers whose prime factors are all large.
Note that is the restriction of to those numbers whose prime factors are all small or medium. By Proposition 6, the number of such can certainly be bounded by if is sufficienty large. Thus the contribution of this term to (5) is .
Similarly, is the restriction of to those numbers which contain at least one large prime factor, but no medium prime factors. By Proposition 5 the number of such is bounded by if is sufficiently large. Thus the contribution of this term to (5) is , and hence
Note that is only supported on numbers whose prime factors do not exceed , so the Dirichlet series of is absolutely convergent for and is equal to , where are the Dirichlet series of respectively. Since is bounded in magnitude by (being a restriction of ), we may apply Proposition 3 and conclude (for large enough, and discarding the denominator) that
We now record some estimates:
Lemma 7 For sufficiently large , we have
Proof: We just prove the former inequality, as the latter is similar. By Proposition 4, we have
The term vanishes unless , and we have , so we can bound the right-hand side by
The inner summand is bounded by and supported on those that are not divisible by any small primes. From Proposition 5 and Mertens’ theorem we conclude that
We also have an estimate:
Lemma 8 For sufficiently large , we have
for all .
Proof: From Euler products, Mertens’ theorem, and (1) we have
Applying Hölder’s inequality, we conclude that
Setting and we obtain the claim.