Almost primes in almost all very short intervals

We show that as soon as h→∞$h\rightarrow \infty$ with X→∞$X \rightarrow \infty$ , almost all intervals (x−hlogX,x]$(x-h\log X, x]$ with x∈(X/2,X]$x \in (X/2, X]$ contain a product of at most two primes. In the proof we use Richert's weighted sieve, with the arithmetic information eventually coming from results of Deshouillers and Iwaniec on averages of Kloosterman sums.

In [7, Between Corollary 6.28 and Proposition 6.29] Friedlander and Iwaniec discuss the possibility to improve their result. In particular they mention that using linear sieve theory and estimates for general bilinear forms of exponential sums with Kloosterman fractions from [6], one should be able to improve 19 to 3 . Then they say that 'It would be interesting to get integers with at most two prime divisors'. This is the aim of the current note.
Let us introduce a few notational conventions before stating our main theorem: The letter with or without subscripts always denotes a prime number. We write Ω( ) for the total number of prime factors of and ( ) for the number of distinct prime factors of . Furthermore we write for the indicator function of a claim . Further notational conventions, including our asymptotic notation, are described in Section 1.1. Hence, as soon as ℎ → ∞ with → ∞, almost all intervals of length ℎ log contain 2numbers. Previously it was known, as a consequence of the work of Teräväinen [20] on 2 numbers, that almost all intervals of length (log ) 3.51 contain 2 -numbers. Before Teräväinen's work, the best result was due to Mikawa [17] who showed that as soon as ℎ → ∞ with , almost all intervals ( − ℎ(log ) 5 , ] contain 2 -numbers. On the other hand, by work of Wu [21] it is known that the interval ( − 101∕232 , ] contains 2 numbers for all sufficiently large . The corresponding upper bound ∑ −ℎ log < ⩽ | ⇒ > 1∕8 Ω( )⩽2 = (ℎ) for all ∈ ( ∕2, ] apart from an exceptional set of measure ( ∕ℎ) follows immediately from [7, Corollary 6.28]. We will give an outline of the proof of Theorem 1.1 in Section 1.2.

Notation
We write Λ( ) for the von Mangoldt function, so that Furthermore we write ( ) for the Euler -function so that 1.
For ∶ ℝ → ℂ and g ∶ ℝ → ℝ + , we write ( ) = (g( )) or ( ) ≪ g( ) if there exists a constant > 0 such that | ( )| ⩽ g( ) for every . Similarly, when also takes positive real values, we write ( ) ≫ g( ) if there exist a constant such that ( ) ⩾ g( ) for every . If there is a subscript (e.g. (g( ))), then the implied constant is allowed to depend on the parameter(s) in the subscript.
We say that g ∶ ℝ → ℝ is smooth if it has derivatives of all orders. We will constantly work with smooth compactly supported functions whose support and derivatives are bounded from above independently of our parameters tending to infinity (e.g. ), so that where the implied constant depends only on . For ∈ ℂ we write ( ) ∶= (2 ) and, for any function, we denote byĝ the Fourier transform If g is a smooth and compactly supported function satisfying (1), then one obtains by repeated partial integration thatĝ ( ) ≪ 1 1 + | | for any ∈ ℝ and ⩾ 0.
In summation conditions, we write ∼ for ∈ ( , 2 ]. Furthermore we write ≍ when ≪ ≪ . For ∈ ℤ and ∈ ℕ we write for the inverse of (mod ) (the modulus will be clear from the context, for example, in ( ) the inverse is (mod )).

Outline of the proof
In this section we provide a simplified outline of the proof of Theorem 1.1. We start by applying Richert's weighted sieve (see, e.g. [7,Chapter 25]) which is tailored to finding numbers. More precisely, writing = ℎ log , = 5∕36 and ( ) = ∏ < , we show in Section 2 that, for almost all ∈ ( ∕2, ], Such sums can be estimated using the work of Deshouillers and Iwaniec [4] and its consequences. To apply these results, one needs some factorability properties of the coefficients − . Here we can utilize the well-factorability of the linear sieve coefficients. The claim (6) can be proved similarly, except in this case well-factorability is not so useful as is smaller. However, we can decompose the prime by Vaughan's identity and again finish by applying suitable bounds for averages of Kloosterman fractions. Also we will need to argue somewhat more carefully to avoid + , depending on , making it to depend only on a dyadictype interval to which belongs.
In the above-mentioned work Mikawa [17] also used weighted sieve and estimates for Kloosterman sums but he did not take advantage of cancellations among the sieve weights for which reason he needed longer intervals (see Remark 2.2 below for more information about [17]).
Remark 2.1. The reason that we do not use only the linear sieve is that using -sieve (with, e.g. = 30) to sieve out primes < makes getting certain mean square estimates (like (49) below) easier. However, it is suggested in [7, between Corollary 6.28 and Proposition 6.29] that one could prove such mean square estimates also for the linear sieve alone.
On the other hand, the reason that we do not use only the -sieve with = 30 is that the linear sieve leads to superior sieving results -in particular our lower bound for (13) would be negative if we only used the -sieve. Let = 30, let > 0 be small and take = and = 1∕1000 . Write also ( , ) = ∏ ⩽ < and define Now define the upper and lower bound linear sieve weights ± = ( ) ∈ ± and the upper and lower bound -sieve weights ± = ( ) ∈ ± , so that, for any ∈ ℕ, (see, e.g.
We cannot obtain a lower bound for ( , ( )) directly by multiplying the lower bounds for ( , ( , ))=1 and ( , ( ))=1 since for some both lower bounds might be negative. However, we can use (14) to derive a lower bound for ( , ( ))=1 that is familiar from the vector sieve (see, e.g.
say. Hence Note that − are supported on ⩽ , so they are lower bound sieve weights with level . Let us now turn to obtaining an upper bound for (( ) , ). If we can obtain level of distribution for ( ), we can typically apply a sieve of level ∕ to ( ) . However, it will be technically convenient if the level is more stable when varies and if has a smooth weight.
( 1 9 ) Consequently, writing we have, for any ∈ ℙ, Hence Note that, for ∈ , the smooth weight ( ∕ and Define the upper bound linear sieve weights + , = ( ) ∈ + , so that, for any ∈  and ∈ ℕ, Recalling also (14) we see that, for any ∈  and ∈ ℕ, Note that + , are supported on ⩽ = ∕ √ 2 +2 . Combining (13) and (22) and then using (16) and (24) we obtain Writing, for ∈ { , }, we see that, for every ∈ ( ∕2, ], Hence, in order to establish that (12) holds for all ∈ ( ∕2, ] apart from an exceptional set of measure ( ∕ℎ), it suffices to show that and that We will establish (26) in Section 3. In Section 4 we collect some lemmas needed in establishing (27). Then we will do some preliminary work on type I sums in almost all very short intervals in Section 5 before establishing (27) in Section 6.
Remark 2.2. We have not optimized the level of distribution or the sieve weights as the current set-up suffices for obtaining 2 -numbers. As pointed out to the author by James Maynard and Maksym Radziwiłł, it might be possible to alternatively use Greaves' most sophisticated weighted sieve [10] together with Bettin-Chandee [1] estimates for Kloosterman sums. In this alternative approach the estimation of 2 from Proposition 5.1 below would be simpler, whereas the sieve weights and thereby the estimation of 1 would become more complicated.
On the other hand, after the completion of this work, the author realized, thanks to a comment by Andrew Granville, that it would probably suffice to use Kloosterman sum estimates based on the Weil bound as Mikawa [17] does. This would again simplify the treatment of 2 . However, our results in Section 5 give better bilinear level of distribution in almost all short intervals, which might be of benefit for other applications, so we have decided to keep the current approach.
Plugging in the values from (29) and evaluating the sums over by the prime number theorem and then substituting = , we get Evaluating the integral, we obtain By a numerical calculation and (32) below we see that indeed ( , ) ≫ 1∕ log once ′ is small enough.

AUXILIARY RESULTS
Before turning to proving (27) we collect here some known auxiliary results. We will use some standard estimates for multiplicative functions. Note first that, for any divisor-bounded (i.e. a function bounded by ( ) for some ) multiplicative function ∶ ℕ → ℂ, we have Furthermore, for ∈ ℝ and ⩾ ⩾ 2, The following consequence of Shiu's [19] bound allows us to estimate divisor sums.
and the claim follows immediately from (32). □ Next we record Vaughan's identity in a form that is convenient for us.

TYPE I SUMS IN ALMOST ALL SHORT INTERVALS
We shall use the following general result as a starting point for showing (27).  (1).
Let ∈ ℝ be bounded for all . Let 2 ⩽ 0 ⩽ 1− for some fixed ∈ (0, 1). Then Remark 5.2. This can be compared with [7, Proposition 6.25] which is non-trivial for 0 < 1∕2 (log ) − . For our choices of we will be able to estimate successfully for a wider range of 0 .
Proof of Proposition 5.1. We start by squaring out, obtaining Here ) . (36) Applying the Poisson summation (Lemma 4.3(i)) and (2) we get, for every 2 ⩽ 0 ⩽ 1− , ∑ g Using this in (36) we see that Substituting this into (35), we obtain Squaring out, the first term equals The error term here is by the inequality | | ⩽ 2 + 2 and the Shiu bound (Lemma 4.1) Consequently, subtracting and adding the expected main term, The ≠ 0 summands of the first line contribute 2 whereas the = 0 summand equals Hence it suffices to show that the main term on the second line of (37) contributes 1 , that is, Here ) . ( 1 , 2 ) 2 1 , 2 (1 − 1 , 2 ) = 1 .
It will suffice to study 2 with replaced by a type II sequence (i.e. = ∑ = for some complex coefficients , supported on certain ranges) and with replaced by a type I sequence (i.e. = ∑ = with supported on small and medium sized ). In type II case we shall use the following lemma. Lemma 5.3. Let 2 ⩽ ⩽ 1∕60 , and let g be a smooth compactly supported function satisfying (1). Let , and be bounded complex coefficients and , , ⩾ 1. Assume that ⩽ ≪ 21∕50 and max{ , } ≪ 14∕25 . Then This will readily follow from the following bound.
The sums over and clearly contribute ( ∕100 ). Hence the above is , as claimed. □ In the type I case (i.e. when studying 2 from Proposition 5.1 with replaced by a type I sequence = ∑ = with supported on small and medium sized ) we shall use the following lemma.

6.1
Showing that ± ≪ Noticing that ,ℎ log ≪ ℎ log , it suffices to show that and Splitting the sum over in (50) according to whether | or not and applying the inequality ( + ) 2 ⩽ 2 2 + 2 2 , we see that the left-hand side of (50) is say. Let us first consider + 1,1 . Applying the Cauchy-Schwarz inequality, we obtain Recalling the support of we see that and Using (51) and rearranging, we see that Substituting = ′ and applying (52), we obtain Let us now turn to + 1,2 . Applying the Cauchy-Schwarz inequality, we see that Using (52), rearranging and using (52) again, we see that Combining this with (53) we see that (50) reduces to showing that a claim very similar to (49). A similar more general claim will be encountered also in [16]. † Let us consider (54). Note first that the definition of + , in (25) implies that, for any ∈ , Here the sum over 2 is We can argue similarly with (49), and thus, noting that the support of ± is contained in [1, ], it suffices to show that A similar claim was shown in [9] and also in [7, Lemma 6.18] though there is a slight mistake in the latter proof. For completeness, we provide a detailed proof here. The starting point for proving (55) is the following lemma.
. † In the first arXiv version of [16] we used different sieve weights and utilized an incorrect version of Lemma 6.1 below (see Remark 6.2 below), so one should look at a more recent version (which is not yet on arXiv) .
(56) Remark 6.2. In [7, Proof of Lemma 6.18] it is claimed that However, there is a mistake in the proof on the second line of the second display of [7, p. 76], where the condition ( , ) = 1 is missing. Taking this condition into account leads to non-diagonal contribution as in our lemma though in applications the non-diagonal contribution is easy to handle. Our proof actually shows the exact formula .
The summand in the -sum is multiplicative, so, looking at the Euler product factors, the -sum equals .

Showing that ± ≪
It suffices to establish that, for some small > 0 and any bounded , These will follow from Lemmas 5.3 and 5.5.

A C K N O W L E D G E M E N T S
The author is grateful to John Friedlander and Henryk Iwaniec for discussions concerning [7,Chapter 6], to James Maynard and Maksym Radziwiłł for pointing out the possible alternative approach described in Remark 2.2, and to Andrew Granville for pointing out Mikawa's work [17].
The author wishes to thank the referee for comments that helped to greatly improve the exposition of the paper. The author was supported by Academy of Finland grant no. 285894.

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