aa r X i v : . [ m a t h . N T ] A p r SIGN CHANGES OF HECKE EIGENVALUES
KAISA MATOM ¨AKI AND MAKSYM RADZIWI L L Introduction
Let f be a holomorphic Hecke cusp form or a Maass Hecke cusp form and write λ f ( n ) for its normalized Fourier coefficients, so that f ( z ) = ∞ X n =1 λ f ( n ) n ( κ − / e ( nz )in the holomorphic case, and f ( z ) = √ y X n =0 λ f ( n ) K s − / (2 π | n | y ) e ( nx )in the Maass case.In [14] Kowalski, Lau, Soundararajan and Wu investigate the problem of the firstsign change of λ f ( n ) for holomorphic f . They remark on the similarities with theproblem of the least quadratic residue. This motivates the point of view that thesigns of λ f ( n ) are GL(2) analogues of real characters. The frequency of signs andsign changes and other related questions have been also recently studied in [1, 15,16, 18, 23]. In [5] Ghosh and Sarnak relate sign changes of λ f ( n ) to zeros of f ( z ) onthe imaginary line.The sequence of signs of λ f ( n ) alone is known to determine the form f (see [14]).Because of this we expect a significant amount of randomness in the distributionof the signs of λ f ( n ). It is the finer details of this randomness that we set out toinvestigate in this paper. We start by showing that half of the non-zero λ f ( n ) arepositive and half are negative. This is relatively straightforward and depends only onthe multiplicativity of λ f ( n ). For simplicity we focus on the case of the full modulargroup. Theorem 1.1.
Let f be a holomorphic Hecke cusp form for the full modular groupAsymptotically half of the non-zero λ f ( n ) are positive and half of them are negative. Date : April 23, 2015.The first author was supported by the Academy of Finland grant no. 137883. The second authorwas partially supported by NSF grant DMS-1128155.
Theorem 1.1 was also recently obtained independently (for Maass forms) by Elliottand Kish [2, Theorem 7]. Previously it was known (for holomorphic forms) thata positive proportion of the λ f ( n ) are positive and a positive proportion negative[16, Theorem 1]. If λ f ( n ) and λ f ( n + 1) behave independently, and λ f ( n ) nevervanishes then we expect (1 / o (1)) x sign changes in the sequence ( λ f ( n )) n ≤ x , sinceindividually each λ f ( n ) and λ f ( n + 1) is positive and negative half of the time. Evenif λ f ( n ) happened to vanish we still expect a positive proportion of sign changes onthe relative subset of those n for which λ f ( n ) = 0. Note that we do not consider thesituation where λ f ( n ) < λ f ( n + 1) = 0 to be a sign change, because the signof 0 is undefined. Thus, by number of sign changes of a possibly vanishing sequence λ f ( n ) we mean the number of sign changes of λ f ( n ) on the subset of n at which λ f ( n ) = 0. Our main result is the following. Theorem 1.2.
Let f be a holomorphic Hecke cusp form for the full modular groupor a Hecke Maass cusp form for the full modular group. Then for every large enough x , the number of sign changes in the sequence ( λ f ( n )) n ≤ x is of the order of magnitude (1) { n ≤ x : λ f ( n ) = 0 } ≍ x Y p ≤ xλ f ( p )=0 (cid:16) − p (cid:17) . For holomorphic forms f Serre [21] established that λ f ( n ) = 0 for a positiveproportion of n . This is expected to hold for Hecke Maass forms of eigenvalue > but is not currently known. By comparing the second and fourth moment of | λ f ( n ) | with f a Hecke Maass form, one can show that λ f ( n ) = 0 for at least cX/ log X integers n ≤ X . Slightly better results can be obtained by comparing moments of | λ f ( p ) | .Previously it was only known that there are ≫ x / sign changes in the holomorphiccase (see [16]) and x / − ε sign changes in the Maass case (see [1]). We will prove inSection 7 the following corollary on a variant of Chowla’s conjecture. Corollary 1.3.
Let f be a holomorphic Hecke cusp form for the full modular group.There exists a constant c > such that, for all x large enough, (cid:12)(cid:12)(cid:12) X n ≤ x sgn ( λ f ( n )) sgn ( λ f ( n + 1)) (cid:12)(cid:12)(cid:12) ≤ (1 − c ) x. This is non-trivial when all of the λ f ( n ) are non-zero. It would be interesting torule out the possibility of λ f ( n ) regularly flipping its sign and thus to investigate the entropy of the sequence of sign of λ f ( n ).According to the Sato-Tate conjecture (now a theorem), for any I ⊂ [0 ,
2] thereis a positive proportion of primes p such that | λ f ( p ) | ∈ I . Because of this the sizeof | λ f ( n ) | fluctuates wildly, making it hard to detect sign changes. This is similar to IGN CHANGES OF HECKE EIGENVALUES 3 a problem one encounters when studying the zeros of the Riemann zeta-function onthe half-line. In analogy to that problem, we introduce sieve weights w n ≥ w ′ n ≥ (cid:12)(cid:12)(cid:12) X x ≤ n ≤ x + h sgn( λ f ( n )) w n (cid:12)(cid:12)(cid:12) and X x ≤ n ≤ x + h w ′ n with the intent of showing that the first term is frequently less than the second(since this ensures a sign change in the interval [ x, x + h ]). To obtain cancellationsin the first sum in (2) (for almost all x ), we use the work of Harcos on the shiftedconvolution problem [8],(3) X X ≤ m,n ≤ Xam − bn = h λ f ( n ) λ f ( m )(see also [20, Section 3] for the best current result). The choice of sieve weights w n allows us to introduce a bilinear structure, and to bound (2) on average by pickingout the cancellations in (3) coming from the large primes, while ignoring the smallprimes (which are a source of problem due to the size fluctuation of | λ f ( n ) | when n has many small prime factors). To obtain a good lower bound for the second,non-oscillating, sum in (2) for a positive proportion of x we compare the first andsecond moments, X X ≤ n ≤ X w ′ n and X X ≤ n ≤ X w ′ n . The computation of these moments relies on the close resemblence of w ′ n to a multi-plicative function supported on the large primes.Finally Theorem 1.2 does not need to be limited to the full modular group aslong as coefficients of the form are real. For holomorphic forms f with complexmultiplication, Serre has shown that [21] X p ≤ xλ f ( p )=0 p = 12 log log x + O (1) . A variant of Theorem 1.2 could be used to show that, conditioned on the set of those n ≤ x for which λ f ( n ) = 0, there is a positive proportion of sign changes. We endour discussion by posing the following problem. Problem.
Let f be a Hecke Maass cusp form of eigenvalue > . Is it possible toshow that the coefficients λ f ( n ) are not lacunary? Precisely is it possible to showthat X λ f ( p )=0 p < ∞ ? KAISA MATOM ¨AKI AND MAKSYM RADZIWI L L
We remark that one could work out explicitly the proportion of sign changes thatTheorem 1.2 yields, although we expect that doing so is rather hard. It might alsobe possible to apply our techniques to study sign changes of the error term in thenumber of representation of n by a quadratic form in 2 k variables. However to obtain,when applicable, a good result (such as a positive proportion of sign changes) wouldrequire adapting some of the techniques of Selberg designed to study zeros of linearcombinations of L -functions, which would increase the complexity of the proof.The plan of the paper is as follows: In Section 2 we prove Theorem 1.1 using prop-erties of general multiplicative functions. In Section 3 we formulate three propositionsand show how our main result, Theorem 1.2 follows from them. These three propo-sitions are then proved in Sections 4–6. Remark 1.4.
While this paper was being refereed, we have been able to obtain(by completely different means) a rather general result on multiplicative function inshort intervals [17] which implies among other things that a multiplicative function f : N → R has a positive proportion of sign changes as soon as f ( m ) < m and f ( n ) = 0 for a positive proportion of integers n . This recoversTheorem 1.2 in the holomorphic case, but not in the case of Maass forms, since for aMaass form it is currently not ruled out that λ f ( n ) = 0 for almost all integers n . Inaddition as pointed out by the anonymous referee, the method developed in this paperis general and will work for any multiplicative function satisfying reasonable estimatesfor the associated shifted convolution problem, which is especially interesting whenthe function vanishes on many primes.2. Proof of Theorem 1.1
Theorem 1.1 follows quickly from the following lemma.
Lemma 2.1.
Let
K, L : R + → R + be such that K ( x ) → and L ( x ) → ∞ for x → ∞ . Let g : N → {− , , } be a multiplicative function such that, for every x ≥ , X p ≥ xg ( p )=0 p ≤ K ( x ) and X p ≤ xg ( p )= − p ≥ L ( x ) . Then |{ n ≤ x : g ( n ) = 1 }| = (1 + o (1)) |{ n ≤ x : g ( n ) = − }| = (cid:18)
12 + o (1) (cid:19) x Y p ∈ P (cid:18) − p (cid:19) (cid:18) | g ( p ) | p + | g ( p ) | p + · · · (cid:19) , IGN CHANGES OF HECKE EIGENVALUES 5 where o (1) → when x → ∞ and the rate of convergence depends only on K ( x ) and L ( x ) but not on g . For the proof of this lemma, we use two results concerning averages of multi-plicative functions. The first lemma (see [7] though a good enough result for ourpurposes would also quickly follow from Hal´asz’s theorem) shows that a real-valuedmultiplicative function g : N → [ − ,
1] is small on average unless it “pretends” to bethe constant function 1.
Lemma 2.2.
There exists an absolute positive constant C such that, for any multi-plicative function g with − ≤ g ( n ) ≤ , one has X n ≤ x g ( n ) ≤ C · x exp − X p ≤ x − g ( p ) p ! for all x ≥ . The second lemma concerns average of a positive-valued multiplicative function.
Lemma 2.3.
Let ε > and let K : R + → R + be such that K ( x ) → for x → ∞ .There exists a positive constant x (depending on ε and K ( x ) ) such that if g : N → [0 , is any multiplicative function for which P p>x − g ( p ) p ≤ K ( x ) for all x ≥ , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ≤ x g ( n ) − xM ( g ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < εx for all x ≥ x , where M ( g ) = Y p ∈ P (cid:18) − p (cid:19) (cid:18) g ( p ) p + g ( p ) p + · · · (cid:19) . Proof.
A version which is non-uniform in g can be found in [22, Theorem 11 inSection I.3.8]. However it is easy to see from the proof that the claimed uniformityholds. (cid:3) Proof of Lemma 2.1.
Lemma 2.2 implies that P n ≤ x g ( n ) = o ( x ) and Lemma 2.3 that X n ≤ x | g ( n ) | = (1 + o (1)) x Y p ∈ P (cid:18) − p (cid:19) (cid:18) | g ( p ) | p + | g ( p ) | p + · · · (cid:19) , so the claim follows since g ( n ) only takes values in {− , , } . (cid:3) Lemma 2.1 immediately implies the following result on signs of multiplicativefunctions.
KAISA MATOM ¨AKI AND MAKSYM RADZIWI L L
Lemma 2.4.
Let
K, L : R + → R + be such that K ( x ) → and L ( x ) → ∞ for x → ∞ . Let g : N → R be a multiplicative function such that, for every x ≥ , X p ≥ xg ( p )=0 p ≤ K ( x ) and X p ≤ xg ( p ) < p ≥ L ( x ) . Then |{ n ≤ x : g ( n ) > }| = (1 + o (1)) |{ n ≤ x : g ( n ) < }| = (cid:18)
12 + o (1) (cid:19) x Y p ∈ P (cid:18) − p (cid:19) (cid:18) h ( p ) p + h ( p ) p + · · · (cid:19) , where h ( n ) is the characteristic function of those n for which g ( n ) = 0 , and o (1) → when x → ∞ and the rate of convergence depends only on K ( x ) and L ( x ) but noton g .Proof. Apply Lemma 2.1 to the multiplicative function which is 0 for those n forwhich g ( n ) = 0 and g ( n ) / | g ( n ) | for those n for which g ( n ) = 0. (cid:3) To prove the long interval result we still need the following lemma.
Lemma 2.5.
One has λ f ( p ) < for a positive proportion of primes.Proof. This is a direct consequence of the Sato-Tate Conjecture, but follows alsowithout such deep information for instance from [19, Theorem 4(iii)] (even with m = 0 there). (cid:3) Proof of Theorem 1.1.
Theorem 1.1 follows immediately from Lemma 2.4 togetherwith non-lacunarity and Lemma 2.5. (cid:3) Outline of the proof of Theorem 1.2 (sign changes)
Let us start by collecting some basic facts about λ f ( n ) which will recur throughthe argument. Lemma 3.1.
Let f be a holomorphic Hecke cusp form for the full modular group ora Maass Hecke cusp form for the full modular group. Write λ f ( n ) for the normalizedFourier coefficients. Then(i) { n ≤ x : λ f ( n ) = 0 } ≍ x Y p ≤ xλ f ( p )=0 (cid:16) − p (cid:17) ≍ x Y p ≤ xλ f ( p )=0 (cid:16) p (cid:17) − . (ii) λ f ( n ) ≪ n / . IGN CHANGES OF HECKE EIGENVALUES 7 (iii) For every z ≥ w ≥ , X w ≤ p ≤ z | λ f ( p ) | p = X w ≤ p ≤ z p + O f (1) . (iv) For every large enough y , X y ≤ p ≤ y | λ f ( p ) |≥ / | λ f ( p ) | ≥ y
10 log y .
Proof. (i) The second asymptotic equality is trivial, so it is enough to prove thefirst one. The upper bound is an immediate consequence of an upper boundsieve and the lower bound follows for instance from [6, Theorem 1] togetherwith Lemma 2.5.(ii) See [12, Appendix 2].(iii) By Hecke relation, and since λ f ( p ) are real, X w ≤ p ≤ z | λ f ( p ) | p = X w ≤ p ≤ z λ f ( p ) + 1 p , and the claim follows since the second symmetric power L -function is cuspidalautomorphic (by [4]) and thus holomorphic and non-vanishing at s = 1 (see e.g.[11, Section 5.12]).(iv) The proof is very similar to [14, Lemma 4.1]. Let us first note that18 (cid:0) x − − ( x − x + 1) (cid:1) = − x x − ≤ ( | x | ≤ / if | x | > / , so that X y ≤ p ≤ y | λ f ( p ) |≥ / | λ f ( p ) | ≥ X y ≤ p ≤ y λ f ( p ) − − ( λ f ( p ) − λ f ( p ) + 1)= 18 X y ≤ p ≤ y λ f ( p ) − λ f ( p ) = (cid:18)
18 + o (1) (cid:19) y log y , by holomorphy and non-vanishing of second and fourth symmetric power L -functions (see [12] and [13, Page 194]). (cid:3) In this section we prove Theorem 1.2 assuming propositions which we will prove inSections 4–6. As described in the introduction, the basic idea is to show incompatiblebounds for mollified short interval sums (2). Let us start by fixing our choices of w n and w ′ n and the associated notation. KAISA MATOM ¨AKI AND MAKSYM RADZIWI L L
Fix X ≥ y = X δ for some small enough δ . Moreover set D + = { d = p · · · p r : p r < · · · < p , p m ≤ y m for all odd m } , where y m = y (1 − γ ) γ m − with a parameter γ ∈ (0 , D + ⊂ [1 , y ]. Now ρ + ( n ) = P d | n,d ∈D + µ ( d ) are Brun’s sieve weights, so that writing P − ( n ) for thesmallest prime factor of n , we have P − ( n ) >y ≤ ρ + ( n ) . In particular ρ + ( n ) ≥ n . We then define w n := X ab = na ≤ y,λ f ( a ) =0( a,b )=1 µ ( a ) ρ + ( b ) | λ f ( b ) | and w ′ n := X ab = na ≤ y,λ f ( a ) =0 P − ( b ) >y µ ( a ) | λ f ( b ) | . Write also k ( x ) = Y p ≤ xλ f ( p )=0 (cid:16) p (cid:17) , so that X n ≤ xλ f ( n ) =0 ≍ X/k ( x )by Lemma 3.1(i).Our goal is to compare the sums(4) X x ≤ n ≤ x + hk ( x ) sgn( λ f ( n )) w n and X x ≤ n ≤ x + hk ( x ) λ f ( n ) =0 w ′ n for a large constant h . Note that w ′ n ≤ w n . Therefore if the first sum is smaller inabsolute value then we have detected a sign change. Note also that w ′ n consists ofonly one term. We first need two results on moments of w ′ n and w n . From these wewill then deduce the behavior of the sums in (4) and the main result will then followshortly. We will prove the propositions below in sections 4 and 5 respectively. Proposition 3.2.
We have, for all x > large enough, X X ≤ n ≤ X w ′ n ≫ X Y p ≤ Xλ f ( p )=0 (cid:16) − p (cid:17) and X X ≤ n ≤ X w ′ n ≪ X Y p ≤ Xλ f ( p )=0 (cid:16) − p (cid:17) . IGN CHANGES OF HECKE EIGENVALUES 9
Note that the second moment bound is trivial when f is holomorphic since if P − ( b ) > y = X δ then | λ f ( b ) | ≤ τ ( b ) ≪ /δ and consequently w ′ n ≪ /δ . Alsofor holomorphic forms P λ f ( p )=0 /p < ∞ by a result of Serre. However the secondmoment estimate is less trivial for Maass forms. We also establish a bound for thesecond moment of w n which constitutes the technically hardest part of our proof. Proposition 3.3.
We have, X X ≤ n ≤ X w n ≪ X Y p ≤ Xλ f ( p )=0 (cid:16) − p (cid:17) . From Proposition 3.3 and an estimate for the shifted convolution problem of λ f ( n )we deduce the following proposition. Proposition 3.4.
There exist positive constants C and ε such that, uniformly in K > and h ≤ X ε , one has, for at least proportion (1 − /K ) of x ∈ [ X, X ] , (cid:12)(cid:12)(cid:12)(cid:12) X x ≤ n ≤ x + hk ( x ) sgn ( λ f ( n )) w n (cid:12)(cid:12)(cid:12)(cid:12) ≤ CK √ h. On the other hand from Proposition 3.2 we deduce the following complimentaryresult.
Proposition 3.5.
For any h ≥ , there is a positive proportion of x ∈ [ X, X ] suchthat (5) X x ≤ n ≤ x + hk ( X ) w ′ n ≫ h. Proof.
Let ε be a small positive constant and let H be the set of square-free integers n ∈ [ X, X ] for which w ′ n ≥ ε . Then we have by the first part of Proposition 3.2, X n ∈H w ′ n ≥ X X ≤ n ≤ X w ′ n − ε X X ≤ n ≤ Xw ′ n =0 ≫ X Y p ≤ Xλ f ( p )=0 (cid:18) − p (cid:19) . Hence by Cauchy-Schwarz and the second part of Proposition 3.2,(6) |H | ≥ ( P n ∈H w ′ n ) P n w ′ n ≥ δXk ( x ) for some δ >
0. Consider now the following sets U := { X ≤ x ≤ X : | [ x, x + hk ( X )] ∩ H | ≤ δh/ }U := { X ≤ x ≤ X : hδ ≥ | [ x, x + hk ( X )] ∩ H | > δh/ }U := { X ≤ x ≤ X : | [ x, x + hk ( X )] ∩ H | ≥ hδ } . Then by (6) δhX ≤ X X ≤ x ≤ X | [ x, x + hk ( x )) ∩ H | + O (1) ≤ X x ∈U + X x ∈U + X x ∈U ! | [ x, x + hk ( x )] ∩ H | + O (1) ≤ X · δh/ |U | · hδ + X x ∈U | [ x, x + hk ( X )] ∩ H | + O (1)Notice that n ∈ H implies that w ′ n ≥ ε and hence λ f ( n ) = 0. In particular, X X ≤ x ≤ X | [ x, x + hk ( X )] ∩ H | ≪ hk ( X ) X X ≤ n ≤ Xλ f ( n ) =0 hk ( X ) X = | ∆ |≤ hk ( X ) X X ≤ n ≤ Xλ f ( n ) =0 λ f ( n +∆) =0 ≪ h X by a result of Henriot [9, 10], since λ f ( n ) =0 is a multiplicative function of n . Inaddition it follows from this that |U | ≤ Cδ X for some large absolute constant C >
0. Applying Cauchy-Schwarz and the previous two bounds, we get X x ∈U | [ x, x + hk ( X )] ∩ H | ≤ |U | / · X X ≤ x ≤ X | [ x, x + hk ( X )] ∩ H | ! / ≤ C δ hX for some large absolute constant C . Combining the above inequalities it follows that δhX ≤ Xδh/ |U | · hδ + C δ X + O (1)Taking δ small enough but fixed, we conclude that |U | ≫ X . The claim now followssince the desired lower bound (5) holds for every x ∈ U by the definitions of U and H (cid:3) We are now ready to prove Theorem 1.2 which follows from combining Proposition3.4 and Proposition 3.5.
IGN CHANGES OF HECKE EIGENVALUES 11
Proof of Theorem 1.2.
According to Proposition 3.5 we have X x ≤ n ≤ x + hk ( x ) w ′ n ≥ ch for a positive proportion δ of x ∈ [ X, X ], with c > (cid:12)(cid:12)(cid:12)(cid:12) X x ≤ n ≤ x + hk ( x ) sgn( λ f ( n )) w n (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cδ √ h for a proportion of at least 1 − δ > − δ of x ∈ [ X, X ]. Therefore once h islarge enough but fixed (larger than ( C/ ( δc )) ), we will have a positive proportion of x ∈ [ X, X ] for which (cid:12)(cid:12)(cid:12)(cid:12) X x ≤ n ≤ x + hk ( x ) sgn( λ f ( n )) w n (cid:12)(cid:12)(cid:12)(cid:12) < X x ≤ n ≤ x + hk ( x ) w ′ n . For every such x a sign change of λ f ( n ) must occur in the interval [ x, x + hk ( x )] since w n ≥ w ′ n ≥ n . Hence there are ≫ X/k ( X ) sign changes, and the claimfollows from Lemma 3.1(i). (cid:3) Proof of Proposition 3.2
We will use the following general bound for mean-values of multiplicative functions.
Lemma 4.1.
Let g : N → [0 , ∞ ) be a multiplicative function. Let A, B be constantssuch that for all y ≥ , X p ≤ y g ( p ) log p ≤ Ay and X p ∈ P X ν ≥ g ( p ν ) p ν log p ν ≤ B. Then, for x ≥ , x X n ≤ x g ( n ) ≪ ( A + B + 1) Y p ≤ x − p + X ν ≥ g ( p ν ) p ν ! , where the implied constant is absolute.Proof. See [22, Theorem III.3.5]. (cid:3)
We will start by proving the upper bound stated in Proposition 3.2. We note that w ′ n ≤ g ( n ) where g ( n ) is a multiplicative function such that g ( p ν ) = p ≤ y, ν = 1 and λ f ( p ) = 00 if p ≤ y and ν > λ f ( p ) = 0 | λ f ( p ν ) | if p > y By Lemma 4.1 and Lemma 3.1(ii)-(iii), X X ≤ n ≤ X w ′ n ≪ X Y p ≤ yλ f ( p )=0 (cid:16) − p (cid:17) Y y
y µ ( a ) | λ f ( b ) | ≥ X X/y ≤ p ≤ X | λ f ( p ) | X X/p ≤ a ≤ X/pλ f ( a ) =0 | µ ( a ) | ≫ X Y p ≤ Xλ f ( p )=0 (cid:16) − p (cid:17) X X/y ≤ p ≤ X | λ f ( p ) | p ≫ X Y p ≤ Xλ f ( p )=0 (cid:16) − p (cid:17) X X/y ≤ p ≤ X p by Lemma 3.1(iv). Hence X X ≤ n ≤ X w ′ n ≫ X Y p ≤ Xλ f ( p )=0 (cid:16) − p (cid:17) log log 2 X log(2 X/y ) ≫ X Y p ≤ Xλ f ( p )=0 (cid:16) − p (cid:17) . Proof of Proposition 3.3
Recall that ρ + ( n ) are upper bound Brun sieve weights, so that (compare with [3,Section 6.2]) ρ + ( n ) − P − ( n ) >y = X r ≥ X n = p ··· p r +1 dp > ··· >p r +1 >y r +1 p ℓ +1 ≤ y ℓ +1 ( ∀ ℓ
Note that w n ≤ w ′ n + 2 w ′′ n . By Proposition 3.2 it is enough to consider P w ′′ n . Inthe sum over divisors of n in the definition of w ′′ n we write a = cd with P + ( c ) ≤ y r +1 and P − ( d ) > y r +1 . This allows us to re-write the sum over ab = n as follows X cdb = ncd ≤ y,λ f ( cd ) =0 P − ( bd ) >y r +1 P + ( c ) ≤ y r +1 µ ( cd ) | λ f ( b ) | ω ( b ) ≤ X cℓ = nP + ( c ) ≤ y r +1 λ f ( c ) =0 µ ( c ) g r ( ℓ ) := G r ( n )(7)where g r ( ℓ ) is a multiplicative function such that g r ( p ν ) := X p ν = dbP − ( bd ) >y r +1 µ ( d ) | λ f ( b ) | ω ( b ) = | λ f ( p ) | if p ≥ y r +1 and ν = 14 | λ f ( p ν ) | + 4 | λ f ( p ν − ) | if p ≥ y r +1 and ν ≥
20 otherwise . By Cauchy-Schwarz and (7), X X ≤ n ≤ X w ′′ n ≪ X r ≥ − r X X ≤ n ≤ X G r ( n ) Note that G r ( n ) is also multiplicative, and G r ( p ) = (1 + 4 | λ f ( p ) | ) ≤ | λ f ( p ) | if p > y r +1 ;0 if p ≤ y r +1 and λ f ( p ) = 0;1 if p ≤ y r +1 and λ f ( p ) = 1,and G r ( p ν ) ≤ p ν/ by Lemma 3.1(ii). By Lemma 4.1 and Lemma 3.1(iii) we get X X ≤ n ≤ X G r ( n ) ≪ X Y p ≤ y r +1 λ f ( p )=0 (cid:16) − p (cid:17) · Y y r +1 ≤ p ≤ X (cid:16) | λ f ( p ) | p (cid:17) ≪ X Y p ≤ Xλ f ( p )=0 (cid:16) − p (cid:17) · (cid:16) log X log y r +1 (cid:17) Combining the above equations and recalling that y r = X δ/ − γ ) γ m − we obtain X X ≤ n ≤ X w ′′ n ≪ X Y p ≤ Xλ f ( p )=0 (cid:16) − p (cid:17) X r ≥ − r (cid:16) − γ ) δγ r (cid:17) Picking γ = 2 − / we see that the sum over r is O (1). Combining the previousbounds then leads to X X ≤ n ≤ X w n ≪ X Y p ≤ Xλ f ( p )=0 (cid:16) − p (cid:17) as was claimed. 6. Proof of Proposition 3.4
An important role in the proof of Proposition 3.4 is played by estimates for shiftedconvolution sums.
Lemma 6.1.
There exists a small δ > such that, uniformly in a, b, A, B, h ≤ x δ , (8) X x ≤ aAm,bBn ≤ xaAm − bBn = h λ f ( Am ) λ f ( Bn ) ≪ x − δ . Proof.