LINEAR CORRELATIONS OF MULTIPLICATIVE FUNCTIONS
LILIAN MATTHIESEN
Abstract. We prove a Green–Tao theorem for multiplicative functions.
Contents
1. Introduction
2. Reduction of the main result to a W -tricked version
3. Majorants for multiplicative functions
4. A majorant for h]
5. Majorants for bounded multiplicative functions
6. The product ν ] ν [ is pseudo-random
7. The average order of ν ] ν [
8. The pretentious W -trick and orthogonality of h with nilsequences
9. Conclusion of the proof of Proposition 2.1
10. Proof of Theorem 1.3 and Corollary 1.4
11. Application to eigenvalues of cusp forms
12. Further applications
References
1
7
10
10
11
13
19
20
25
25
26
29
30
1. Introduction
The purpose of this paper is to establish an asymptotic result for correlations of realvalued multiplicative functions. Throughout this paper we write
X
1 X
q
Sh (x) =
h(n)
and
Sh (x; q, a) =
h(n)
x 16n6x
x 16n6x
n≡a (mod q)
for any q, a ∈ Z, q 6= 0 and x > 1. We begin by describing the class M of multiplicative
functions that our result applies to.
Definition 1.1. Let M be the class of multiplicative functions h : N → R such that:
(i) There is a constant H > 1, depending on h, such that |h(pk )| 6 H k for all primes
p and all integers k > 1.
(ii) |h(n)| ε nε for all n ∈ N and ε > 0,
2010 Mathematics Subject Classification. 11N37 (11B30, 11F30, 11D04).
1
2
LILIAN MATTHIESEN
(iii) There is a positive constant αh such that
1X
|h(p)| log p > αh
x p6x
for all sufficiently large x.
(iv) Stable mean value in arithmetic progressions:
Let C > 0 be any constant, suppose that x0 ∈ (x(log x)−C , x) and that x is sufficiently large, and let A (mod q) be any progression with 1 6 q < (log x)C and
gcd(q, A) = 1. Then
!
Y
q
1
|f
(p)|
Sf (x0 ; q, A) = Sf (x; q, A) + O ϕ(x)
1+
φ(q) log x p6x
p
p-q
for some function ϕ with ϕ(x) → 0 as x → ∞.
Remark. Out of the four conditions above, the last one is certainly the most difficult
to check in any application. If, however αh > 2H/π in (iii), or if f is non-negative and
αh > H/π, then condition (iv) follows from the other three conditions. The way to prove
this is as follows. Let g be the multiplicative functions defined by
(
h(p)/H if k = 1
g(pk ) =
.
(1.1)
0
if k > 1
Then h = g ∗H ∗ g 0 for some multiplicative function g 0 that satisfies g 0 (p) = 0 at all primes,
and for which bounds as in Definition 1.1 (i) hold. Since g is bounded, we may employ the
Lipschitz bounds [8] of Granville and Soundararajan, which will come with an acceptable
error term due to the lower bound on αh /H. Condition (iv) then follows by combining these
Lipschitz bounds with the ‘pretentious large sieve’ (see Proposition 8.4). We will carry out
this approach in detail in Section 11 to show that M contains the functions n 7→ |λf (n)|,
where λf (n) is the normalised Fourier coefficient at n of a primitive cusp form f . Another
interesting example of a function for which αh is sufficiently large is the characteristic
function of sums of two squares for which (iii) clearly holds with αh = 21 . Alternatively,
the Dirichlet series attached to h and its twists by characters can be considered.
The quantities Sh (x; q, a) will play an important role in the statement of our main result.
Since the mean value Sh (x; q, a) is likely to show a very irregular behaviour for small values
of q, we work with a W -trick. To set this up, let w : R>0 → R>0 be any function such that
log log x
< w(x) 6 log log x
log log log x
for all sufficiently large x, and define
W (x) =
Y
p6w(x)
p.
LINEAR CORRELATIONS OF MULTIPLICATIVE FUNCTIONS
3
To avoid the contributions from small prime factors, we will largely work in the progressions
A (mod W (x)) with gcd(A, W ) = 1, and in suitable subprogressions thereof.
Our main result is the following.
Theorem 1.2. Let h1 , . . . , hr ∈ M be multiplicative functions. Let T > 1 be an integer
parameter and let ϕ1 , . . . , ϕr ∈ Z[X1 , . . . Xs ] be linear polynomials whose non-constant parts
are pairwise independent over Q and have coefficients that are independent of T , while
the constant coefficients hi (0) may depend on T with the constraint that hi (0) = O(T ).
Suppose that K ⊂ [−1, 1]s is a convex body with vol(K) 1 such that ϕi (T K) ⊂ [1, T ] for
each 1 6 i 6 r.
f : R>0 → N that takes
Then there are positive constants B1 and B2 and a function W
Q
α(p,x)
f (x) =
f (x) 6
values of the form W
with α(p, x) ∈ N and satisfies the bound W
p<w(x) p
(log x)B1 for all x, and that is such that the following asymptotic holds as T → ∞:
r
X Y
1
hi (ϕi (n)) =
vol T K n∈Zs ∩T K i=1
X
X
w1 ,...,wr
A1 ,...,Ar
p|wi ⇒p<w(T )
f Z)∗
∈(Z/W
B
wi 6(log T ) 2
+o
r
Y
f , Ai
hi (wi )Shi T ; W
i=1
!
1
f )s
(wW
!
r Y
Y
1
|hi (p)|
,
1+
(log T )r j=1 p6T
p
X
r
Y
1ϕj (v)≡wj Aj (mod wj W
f)
v∈
j=1
f Z)s
(Z/wW
(1.2)
f=W
f (T ). The error term dominates as soon as one of
where w = lcm(w1 , . . . , wr ) and W
the functions hi satisfies |Shi (x)| = o(S|hi | (x)).
Unsurprisingly, the above result can be reformulated in terms of character sums. By
means of the pretentious large sieve (see Proposition 8.4), it is moreover possible to restrict
these character sums to a small number of characters that have a large correlation with the
hi . A result of Elliott (Lemma 1.5 below) which will be discussed in the sequel allows one
to even further restrict these character sums. If hi (n) = 1 ∗ χ4 (n) = 41 r(n), the relevant
characters would, for instance, be those induced by 1 and χ4 . The following is such a
reformulation of Theorem 1.2 in terms of finite character sums:
Theorem 1.3. With the assumptions of Theorem 1.2 in place, define for each i ∈ {1, . . . , r}
the integer ki = 1 + dαh−2
e, where αhi is as in Definition 1.1. Consider for each i the set of
i
(i)
(i)
primitive characters of conductor at most (log T )B1 and enumerate them as χ1 , χ2 , . . . in
P
(i)
such way that the averages | n6T χj (n)hi (n)| are in non-increasing order as j increases.
(i)
(i)
f (T ) that are induced from χ(i)
Let Ei be the subset of characters χ modulo W
1 , χ2 , . . . , χki
and that have the property that
X
X
κ(T )
|hi (n)| χ(n)hi (n),
n6T
n6T
4
LILIAN MATTHIESEN
for some fixed function κ : N → R>0 that we are free to choose and which we assume to
satisfy κ(x) = o(1). Then
r
X Y
1
hi (ϕi (n))
vol T K n∈Zs ∩T K i=1
r Y
Y
1
|hi (p)|
βp (χ1 , . . . , χr ) + o
= β∞
1+
(log T )r j=1 p6T
p
χ1 ,...,χr p6T
X Y
!
,
χj ∈Ej
where β∞ =
Qr
i=1
S|hi | (T ), while the local factor at p takes the form
βp (χ1 , . . . , χr ) =
lim
m→∞
1 X
pms
X
r
Y
hj (p
aj
)χ∗j (ϕj (v), pai )
a∈N0 v∈(Z/pm Z)s j=1
vp (ϕi (v))=ai
−1 −1
|hi (p)|
1
+ ...
1+
1−
p
p
with
χ∗j (ϕj (v), pai ) =
Q
(
ϕ (v)
χj,p pjaj
q6=p,
aj
)
χ
(p
j,q
q prime
if p < w(T ),
if p > w(T ).
χj (paj )
Q
f
Here, we have decomposed χj = p χj,p into characters χj,p modulo pvp (W (T )) . If p > w(T ),
the factor at p may be written in the more succinct form:
−1 r Y
|hi (p)|
hi (p)χi (p)
βp (χ1 , . . . , χr ) =
1+
1+
+ Or (H 2 p−2 ).
p
p
j=1
When making additional assumptions on the functions hi , the right hand side of (1.2)
can be significantly simplified. In many cases the quantities Shi (x; q, a) are determined by a
0
product of local densities whenever (a, q) = 1 and when q is small, that is q 6 (log x)C for
any fixed constant C 0 . Given such a local-to-global principle for the Shi (x; q, a), it should
also be possible to reinterpret the right hand side of (1.2) as a product of local densities.
This is, for instance, the case when each hi is χi (n)niti -pretentious for some character χi
and some ti ∈ R. In this case we have #Ei = 1 in Theorem 1.3, as may be deduced from
[1, Theorem 2.1 and Lemma 3.1].
To further illustrate the above, we consider the example of non-negative functions hi
whose values at primes are fairly equidistributed in arithmetic progressions a (mod q) with
gcd(a, q) = 1. The following is an immediate corollary to Theorem 1.2.
Corollary 1.4. Let h1 , . . . , hr ∈ M be non-negative multiplicative functions and suppose
that, given any positive constant C 0 , each of the functions hi satisfies
q
1 Y
hi (p)
Shi (x; q, a) 1+
φ(q) x log x p6x
p
p-q
LINEAR CORRELATIONS OF MULTIPLICATIVE FUNCTIONS
5
0
for all progressions a (mod q) with q 6 (log x)C and (a, q) = 1. Assuming the conditions
of Theorem 1.2, we then have
r
X Y
Y
hi (ϕi (n)) β∞
βp ,
(T → ∞),
n∈Zs ∩T K i=1
where
β∞
p
r
Ts Y Y
hi (p)
= vol(K)
1+
(log T )r j=1 p6T
p
and
βp = lim
m→∞
1 X
pms
X
a∈N0 v∈(Z/pm Z)s
vp (ϕi (v))=ai
r
Y
hj (p)
hj (p ) 1 +
p
j=1
aj
−1
.
The βp satisfy the asymptotic
βp = 1 + O(p−2 ).
Our proof of Theorem 1.2 proceeds via Green and Tao’s nilpotent Hardy–Littlewood
method (see [10]). This method consists of two main parts, one being to establish that a
W -tricked version of any h ∈ M is orthogonal to nilsequences, and the second being to
show that this W -tricked version of h has a majorant function which is pseudo-random in
the sense of [10] and of the ‘correct’ average order in the sense that its average order is
bounded by the average order of the corresponding W -tricked version of |h|. For the first
part we rely on [15], which establishes the required condition for all h ∈ M which satisfy
a certain major arc condition. We will show in Section 8 that this major arc condition is
implied by Definition 1.1 (iv). The second part, that is the construction of correct-order
pseudo-random majorants will take up a large part of this work.
We end this introduction by discussing the implicit condition that |Sh (x)| S|h| (x),
which appears in Theorem 1.2. To start with, we note that Shiu’s [17, Theorem 1] applies
to |h| for every h ∈ M . Thus, if A ∈ (Z/qZ)∗ , then
!
X |h(p)|
q
1
Sh (x; q, A) 6 S|h| (x; q, A) exp
, (x → ∞),
(1.3)
φ(q) log x
p
p6x
p-q
uniformly in A and q, provided that q 6 x1/2 . It will follow from Proposition 2.1 in the
f (x) and
next section that the main term in Theorem 1.2 only dominates if, for q = W
h ∈ {h1 , . . . , hr }, the upper bound (1.3) on Sh (x; q, A) is of the correct order for a positive
proportion of residues A ∈ (Z/qZ)∗ . Let h0 be defined as h0 (n) = h(n)1gcd(n,W (x))=1 . Then
the latter condition certainly holds if
X 0
1
|h (p)|
|Sh0 (x)| S|h0 | (x) exp
.
log x
p
p6x
In this regard, recent work of Elliott shows the following.
6
LILIAN MATTHIESEN
Lemma 1.5 (Elliott, Elliott–Kish). Suppose h is a multiplicative function that satisfies
conditions (i) and (iii) from Definition 1.1 and define h0 : n 7→ h(n)1gcd(n,W (x))=1 .
If f ∈ {h, h0 }, then
X
1
|f (p)|
S|f | (x) exp
,
log x
p
p6x
and |Sf (x)| S|f | (x) holds if and only if there exists th ∈ R such that
X |h(p)| − <(h(p)pith )
< ∞.
p
p prime
More precisely, if there exists th as above, then
x−ith Y 1 + f (p)pith −1 + . . .
Sf (x) = S|f | (x)
+ o(S|f | (x)).
1 − ith p6x 1 + |f (p)|p−1 + . . .
(1.4)
(1.5)
Proof. This lemma is a direct application of Elliott [4, Theorem 4] and either Elliott [4,
Theorem 2] or Elliott–Kish [5, Lemma 21]. Thus, we only need to check P
that f satisfies
ε
the conditions of these results. To start with, since |h(n)| ε n , the sum
|f (q)|/q over
k
all proper prime powers
q
=
p
,
k
>
2,
converges.
Taking
Definition
1.1
(i)
into
account,
P
k
k
−1
we also see that x
pk <x |f (p )| log p H for x > 2. The condition of [5, Lemma 21] is
identical to our condition (iii), the weaker condition from [4, Theorem 2] follows by partial
summation. By [5, Lemma 21] (or [4, Theorem 2]), Shiu’s bound and condition (iii) we
thus have
X
Y x
|h(p)|
|h(n)| 1+
ε x(log x)αh −ε−1 .
(1.6)
log
x
p
n6x
w(x)<p6x
gcd(n,W (x))=1
Related work and an open question. We wish to draw the reader’s attention to some
related work in the context of bounded pretentious multiplicative functions. A bounded
multiplicative function h : N → C is said to be pretentious, if there exists th ∈ R and a
character χh such that
X 1 − <(h(p)χh (p)pith )
< ∞.
(1.7)
p
p prime
The difference between this condition and (1.4) lies in the constant 1 which replaces |h(p)|,
making (1.7) a somewhat stronger condition. The characteristic function of sums of two
squares is one example of a function that is not pretentious because it is 0 at too many
primes. For pretentious h it is known (see [1, 9, 12]) that the quantities Sh (x; q, a) are determined by a product of local densities.
P Klurman [12] recently succeeded in asymptotically
evaluating correlations of the form n6x f1 (P1 (n)) . . . fr (Pr (n)) for bounded pretentious
multiplicative functions f1 , . . . , fr and for arbitrary polynomials P1 , . . . , Pr ∈ Z[x].
Frantzikinakis and Host [7] established a Green–Tao theorem, like Theorem 1.2, for
pretentious multiplicative functions. In [7], the multiplicative functions hi are allowed to
LINEAR CORRELATIONS OF MULTIPLICATIVE FUNCTIONS
7
be complex-valued, they must all be bounded in absolute value by 1 and, in the case where
the asymptotic formula corresponding to (1.2) carries a main term, they have the property
that |Shi (x)| 1. Our result, in contrast, only applies to real-valued hi but it yields
an asymptotic with main term so long as (1.4) applies. This includes, for instance, the
function n 7→ 2−ω(n) , where ω(n) denotes the number of distinct prime factors, or sparse
characteristic functions like that of the set of sums of two squares. While the present paper
is mainly concerned with the construction of correct-order pseudo-random majorants, no
such construction is required in the case of [7] as the trivial majorant given by the all-one
function 1 is a pseudo-random majorant of the correct average order for every pretentious
multiplicative function.
It is an interesting question to investigate if the material in Section 8 can be extended
to complex-valued functions that satisfy Elliott’s condition (1.4). An affirmative answer
would allow one to extend the class M to which Theorem 1.2 applies to all complex-valued
functions that satisfy conditions (i)–(iii), a modified version of (iv), as well as (1.4). This
would yield a natural generalisation of both Theorem 1.2 and the result of Frantzikinakis
and Host.
Acknowledgements. It is a pleasure to thank Stephen Lester for very helpful discussions
and Régis de la Bretèche and Yuri Tschinkel for their questions which led to this work.
2. Reduction of the main result to a W -tricked version
We begin the proof of Theorem 1.2 by a reduction to the following special case which
works in subprogressions whose common difference is a large w(T )-smooth integer. This
removes potential irregularities that occur when working in progressions modulo small q.
Proposition 2.1. Let h1 , . . . , hr ∈ M be multiplicative functions. Let T > 1 be an integer
parameter and let ϕ1 , . . . , ϕr ∈ Z[X1 , . . . Xs ] be linear polynomials as in Theorem 1.2.
Given any constant C > 0, there exist positive constants B1 and B2 > C and, for each
f (T ) 6 (log T )B1 divisible by W (T ) = Q
T , a w(T )-smooth integer W
p6w(T ) p such that the
following holds.
f (T ), and
Let W1 , . . . , Wr ∈ [1, (log T )B1 +B2 ] be w(T )-smooth integers, each divisible by W
0
let W = lcm(W1 , . . . , Wr ). Let A1 , . . . , Ar are integers co-prime to W (T ). And, finally,
T
let K ⊂ [−1, 1]s be a convex body with vol(K) 1 that is such that Wi ϕi ( W
K) + Ai ⊂ [1, T ]
for each 1 6 i 6 r. Then, as T → ∞, we have
X
r
Y
n∈Zs ∩(T /W 0 )K
i=1
hi (Wi ϕi (n) + Ai ) =
r
r
Ts Y
T s Y log w(T )
f
vol(K) 0 s
Sh T ; W (T ), Ai + o
W i=1 i
W 0 s i=1 log T
Y
w(T )<p6T
!
|hi (p)|
1+
.
p
f (T ) take the shape of a fixed power
The fact that the upper bounds on the Wi and W
of log T is essential for a later application of results from [15] which will imply that the
8
LILIAN MATTHIESEN
function n 7→ hi (Wi n+Ai )−Shi (T ; Wi , Ai ) is orthogonal to nilsequences. In order to deduce
Theorem 1.2 from Proposition 2.1, we will therefore need to truncate certain summations
that occur on the way and show that the terms we remove make a negligible contribution.
For this purpose we introduce the following exceptional set.
Definition 2.2 (Exceptional set). Let C > 0, T > 1 and let S 0 C (T ) denote the set all
positive integers less than T that are divisible by the square of an integer d > (log T )C .
To justify that S 0 C (T ) ⊂ {1, . . . , T } is an exceptional set, let ϕ1 , . . . , ϕr ∈ Z[X1 , . . . , Xs ]
be as in Theorem 1.2, let T > 1 and C > 0. Then, first of all:
r
X X
X |Zs ∩ T K|
1ϕi (n)∈S 0 C (T ) |Zs ∩ T K|(log T )−C .
(2.1)
2
d
s
n∈Z ∩T K i=1
d>(log T )C
P
−1+α
If α = min(αh1 , . . . , αhr ), then
for all 1 6 i 6 r. Thus
n6T hi (n) T (log T )
it follows from (2.1) and [3, Lemma 7.9], combined with an application of the CauchySchwarz inequality, that
X Y
X Y
hi (ϕi (n))1ϕi (n)6∈S 0 C (T ) = (1 + o(1))
hi (ϕi (n)),
(2.2)
n∈Zs ∩T K
i
n∈Zs ∩T K
i
provided C is sufficiently large with respect to r, s, the coefficients of the linear forms
ϕi − ϕi (0), and the respective values of H from Definition 1.1 (i) for the hi . Hence,
S 0 C (T ) is indeed exceptional.
The following observation is crucial for employing the exceptional set S 0 C (T ) in order
to later discard all cases in which Proposition 2.1 would need to be applied with a value
of W > (log T )C .
Lemma 2.3. Let C > 1 and suppose that T is sufficiently large. Then every integer
w > (log T )3C that is composed only out of primes p 6 w(T ) has a square divisor larger
than (log T )2C and, hence, belongs to S 0 C (T ).
Q
P
Proof. Since W (T ) = p6w(T ) p = exp p6log log T log p ∼ log T . Thus, if T is sufficiently
large, then W (T ) < (log T )C . Factorising w as w1 w22 for a square-free integer w1 , it follows
that w1 6 W (T ) < (log T )C . Hence, w22 > (log T )C .
Proof of Theorem 1.2 assuming Proposition 2.1. Let C > 1 be sufficiently large for (2.2)
to hold, let B2 > 3C be the constant from the statement of Proposition 2.1 with C replaced
by 3C, and define the set
W (T ) = {w ∈ [1, (log T )B2 ] : p|w ⇒ p 6 w(T )}
which, by Lemma 2.3, contains all unexceptional w(T )-smooth integers.
For any given T > 1 and w ∈ W (T ), let us temporarily write wkT n to indicate that
w|n but gcd(W (T ), wn ) = 1, in order to make the following definition: Given any collection
w1 , . . . , wr of positive integers, we define the set of r-tuples
f (T ) − 1}r
v ∈ {0, . . . wW
f
U (w1 , . . . , wr ) =
ϕi (v) − wW (T )ϕi (0)
:
,
wi kT ϕi (v), 1 6 i 6 r
16i6r
LINEAR CORRELATIONS OF MULTIPLICATIVE FUNCTIONS
9
where w = lcm(w1 , . . . , wr ). This definition is made in view of the decomposition
ϕi (M m + v) = M ϕi (m) − M ϕi (0) + ϕi (v)
valid for any v ∈ Zr and M ∈ Z.
By (2.2) and Lemma 2.3, the expression from Theorem 1.2 then satisfies
X
r
Y
n∈Zs ∩T K
i=1
hi (ϕi (n))
X
= (1 + o(1))
X
r
Y
hi (ϕi (n))1wi kT ϕi (n)
w1 ,...,wr n∈Zs ∩T K i=1
∈W (T )
X
= (1 + o(1))
X
X
r
Y
f (T )ϕi (m) + Ui ,
hi wW
w1 ,...,wr (U1 ,...,Ur )
f (T )m i=1
m: wW
∈W (T ) ∈U (w ,...,wr )
1
∈Zs ∩T K
where, again, w = lcm(w1 , . . . , wr ). Note that we made use of the fact that the K is convex
f (T )m ∈ Zs ∩ T K}.
in order to restrict the summation over m to the set {m ∈ Zs : wW
Invoking the multiplicativity of the hi , the above becomes
!
!
r
r
X
Y
X
X
Y
f (T )
Ui
wW
(1 + o(1))
hj (wj )
ϕi (m) +
hi
.
wi
wi
w1 ,...,wr
m∈
j=1
i=1
(U1 ,...,Ur )
f (T ))K
∈U (w1 ,...,wr ) Zs ∩(T /wW
∈W (T )
Since gcd(W (T ), Ui /wi ) = 1 for all i, Proposition 2.1 applies to the inner summation over
m and we deduce that the above equals
!
r
X
Y
(1 + o(1))
hj (wj )
w1 ,...,wr
∈W (T )
X
(U1 ,...,Ur )
∈U (w1 ,...,wr )
j=1
Ts
f (T ))s
(wW
vol(K)
r
Y
i=1
Shi
f (T ), Ui
T;W
wi
Finally, by invoking Shiu’s bound, we obtain
!
r
vol(K)T s
X
X
Y
f , Ai
hi (wi )Shi T ; W
f )s
(wW
w1 ,...,wr A1 ,...,Ar
i=1
∈W (T )
f Z)∗
∈(Z/W
+o
+
r
Y
S|hi |
i=1
X
r
Y
f (T ), Ui
T;W
wi
!
.
1ϕj (v)≡wj Aj (mod wj W
f)
v∈
j=1
f Z)s
(Z/wW
!
r
T r+s Y Y
|hi (p)|
1+
,
(log T )r j=1 p6T
p
which completes the proof.
10
LILIAN MATTHIESEN
3. Majorants for multiplicative functions
For any function h ∈ M , we obtain a simple majorant function h0 : N → R by setting
h (n) := h] (n)h[ (n), where h] and h[ are multiplicatively defined by
0
h] (pk ) = max(1, |h(p)|, . . . , |h(pk )|)
and
h[ (pk ) = min(1, |h(pk )|),
respectively. It is immediate that |h(n)| 6 h0 (n) for all n ∈ N. The function h] belongs to
the class of functions for which pseudo-random majorants were already constructed in [3,
§7]. Our pseudo-random majorant for h will arise as a product of separate majorants for
h] and h[ . Thus, our main task here is to construct general pseudo-random majorants for
bounded multiplicative functions.
4. A majorant for h]
Before we turn to the case of bounded multiplicative functions, let us record the known
pseudo-random majorant for h] . For this purpose, set g = µ ∗ h] and define for any
γ ∈ (0, 1/2) the truncation
log d X
)
1
g(d)χ
h(T
(m)
=
d|m
γ
log T γ
d∈N
of the convolution h] = 1 ∗ g, where χ : R → R>0 is a smooth function with support
in [−1, 1], which is monoton on [−1, 0] and [0, 1] and has the property that χ(x) = 1 for
x ∈ [− 21 , 21 ]. For any fixed value of γ ∈ (0, 1/2), [3, Proposition 7.6] then shows that
h] (n) ν] (n)
for n 6 T , where
[(log log T )3 ] [log2 ((log log T )3 )]
ν] (m) =
X
X
X
κ=4/γ
λ=dlog2 κ−2e
u∈U (λ,κ)
)
H κ 1u|m h] (u)h(T
γ
with an exceptional set


S = n 6 T : ∃p.vp (n) > max{2, C1 logp log T } or

m
Q
vp (m)
p|u p
Y
3
p6T 1/(log log T )
!
+ 1m∈S h] (m),
pvp (m) > T γ/ log log T


,

γ
and where each set U (λ,
l κ) is a sparse
m subset of the integers up to N that is defined as
λ+1
λ
2 κ)
follows. Set ω(λ, κ) = γκ(λ+3−log
and Iλ = [T 1/2 , T 1/2 ]. Then
200

{1},
if κ = 4/γ and λ = log2 κ − 2,



∅,
if κ = 4/γ and λ 6= log2 κ − 2,
U (λ, κ) = 
pi ∈ Iλ distinct primes


, if κ > 4/γ.
 p1 . . . pω(λ,κ) : ]
h (pi ) 6= 1
LINEAR CORRELATIONS OF MULTIPLICATIVE FUNCTIONS
11
3
We note for later reference that p > T 1/(log log T ) for any prime divisor p|u of any u ∈ U (λ, κ)
that appears in the definition of ν] .
5. Majorants for bounded multiplicative functions
Any bounded multiplicative function h[ has the property that whenever n0 |n is a divisor
that is coprime to its cofactor n/n0 , then h[ (n) 6 h[ (n0 ). The key step in turning this simple
observation into the construction of a pseudo-random majorant is to find a systematic way
of assigning to any integer n a suitable divisor n0 . The main property this map must have is
that the pre-image of any divisor n0 should be easily reconstructible, a property which will
allow us to swap the order of summation in later computations. The following assignment
already featured in Erdős’s work [6] on the divisor function:
Given a cut-off parameter x and an integer n ∈ [xγ , x], let Dγ (n) denote the largest
divisor of n that is of the form
Y
pvp (n) , (Q ∈ N)
p6Q
but does not exceed xγ . If m ∈ (0, xγ ) is an integer then its inverse image takes the form
Dγ−1 (m) = {δm ∈ [xγ , x] : P + (m) < P − (δ)},
where P + (m), resp. P − (m), denote the largest, resp. smallest, prime factor of m. For our
purpose it turns out to be of advantage to restrict attention to divisors
m ∈ hP[ i = {m : p|m =⇒ p ∈ P[ },
where
P[ = {p : h(p) < 1}.
Thus, if n ∈ [1, x] is an integer that factorises as n = mm0 with m ∈ hP[ i and m0 6∈ hP[ i,
then we set
(
m
if m 6 xγ
0
Dγ (n) =
.
Dγ (m) if m > xγ
Our next aim is to show that a sufficiently smoothed version of the function Dγ0 can be
written
as a truncated divisor sum. To detect whether a given divisor δ|n is of the form
Q
vp (n)
p
for some Q or, equivalently, whether m = nδ has no prime factor p 6 Q,
p6Q
we make use of a sieve majorant similar to the one considered in [10, Appendix D]. The
essential differences are that the parameter corresponding to Q cannot be fixed in our
application and that the divisor sum will be restricted to elements of the set hP[ i. Thus,
let σ[ : R × N → R>0 be defined as
!2
X
log d
,
σ[ (Q; m) =
µ(d)χ
log Q
d|m
d∈hP[ i
12
LILIAN MATTHIESEN
where χ : R → R>0 is a smooth function with support in [−1, 1] and the property χ(x) = 1
for x ∈ [− 21 , 12 ]. This yields a non-negative function with the property that σ[ (Q; m) = 1
if m is free from prime factors p ∈ P[ with p 6 Q. Setting, for 1 6 n 6 x,
X X X
X
n
0
[
γ n
[
k
ν (n) =
+
,
h (m)σ[ x ;
h (mQ )1m<xγ 1Qk m>xγ σ[ Q;
k
m
mQ
γ
k
Q6x
m|n
Q |n
Q∈P[ p|m⇒p<Q
m|n
m∈hP[ i
m<xγ
we obtain a (preliminary) majorant ν 0 : N → R>0 for h[ . The first of two small modifications consists of inserting smooth cut-offs for m and Qk m, leading to the majorant function
ν 00 : N → R>0 defined as
X
log m
00
[
γ n
ν (n) =
h (m)χ
σ x ;
γ log x
m
m|n
m∈hP[ i
+
X
Q∈P[
X
X
m|n
m∈P[
p|m⇒p<Q
Qk |n
[
k
h (mQ )λ
log Qk δ
log x
n
σ Q;
,
δQk
where λ is a smooth cut-off of the interval [γ, 2γ] which is supported in [γ/2, 4γ] and takes
the value 1 on the interval [γ, 2γ]. To carry out the second simplification, we exclude a
3
sparse exceptional set related to the one from Section 4. If Q < T γ/(log log T ) , then an
integer n < T belongs to the exceptional set S from Section 4 if it has a divisor of the
3
4
form Qk m > T γ , where p|m ⇒ p < Q. If Q > T γ/(log log T ) = (log T )γ(log T )/(log log T ) and if
T is sufficiently large, then Q2 > (log T )C1 so that any multiple of Q2 again also belongs
to S . Thus, by defining
X
log m
[
γ n
ν[ (n) =
h (m)χ
σ x ;
γ log x
m
m|n
m∈hP[ i
+
X
X
[
h (Qm)λ
Q|n
m|n
Q prime
p|m⇒p<Q
3
Q>xγ/(log log T )
log Qm
log x
n
σ Q;
,
Qm
we obtain a positive function with the property
|h[ (n)| ν[ (n) + 1S (n)
for any integer n ∈ [1, x], provided x is sufficiently large with respect to the value of C1
from P
the definition of S . By construction (cf. [6] or [13, Lemmas 3.2 and 3.3]) it follows
that n6x 1S (n) x(log x)−C1 /2 . Thus, choosing C1 sufficiently large, we may in view of
(1.6) ensure that
X
X
(1 + |h(n)|)1S (n) = o(
|h(n)|),
n6x
n6x
LINEAR CORRELATIONS OF MULTIPLICATIVE FUNCTIONS
13
i.e., that S is indeed an exceptional set.
6. The product ν ] ν [ is pseudo-random
We will now sketch a proof of the linear forms estimate for ν, which by [10, App. D]
implies that ν is pseudo-random. This linear forms estimate can only be established after
removing the contribution of small prime factors, that is, only when working with a W trick. Recall that w(x) was defined in Section 1.
Proposition 6.1 (Linear forms estimate). Let T > 1 be an integer parameter and B > 1
a constant. For each T and each 1 6 i 6 r, let 1 6 Wi 6 (log T )B be a w(T )-smooth
integer that is divisible by W (T ) and let 0 6 Ai Wi be coprime to W (T ). Let D > 1
be constant, let 1 6 r, s < D be integers and let φ1 , . . . , φr ∈ Z[X1 , . . . Xs ] be linear
polynomials whose non-constant parts are pairwise independent and have coefficients that
are bounded in absolute value by D. The constant coefficients hi (0) may depend on T
provided hi (0) = OD (T ). Suppose that K ⊂ [−1, 1]s is a convex body with vol(K) 1 and
that Wi φi (T K) + Ai ⊂ [1, Wi T ] for each 1 6 i 6 r. Then
!
r
r
Y
X Y
1
1X
νh (Wi φi (n) + Ai ) = (1 + o(1))
νh (Wi n + Ai )
T s vol K n∈Zs ∩T K i=1 i
T n6T i
i=1
as T → ∞, provided γ is sufficiently small.
Proof. Inserting all definition, we obtain
s
X Y
1
νh (φi (n))
T s vol K n∈Zs ∩T K i=1 i
(6.1)
s
X Y
1
ν ] (φi (n))νh[ i (φi (n))
= s
T vol K n∈Zs ∩T K i=1 hi
r
X Y
1
= s
T vol K n∈Zs ∩T K j=1
X
Qi prime
X
[(log log T )3 ] [log2 ((log log T )3 )]
X
κi =4/γ
X
X
X
λi =dlog2 κi −2e uj ∈Uj (λj ,κj )
X
dj :
p|dj ⇒p>w(T )
gcd(dj ,uj )=1
1lcm(δj mj Qj ,δj0 mj Qj ,uj ,dj )|φj (n)
mi :
δj ,δj0 : p|δj δj0
p|mi ⇒
w(T )<p<Qi ⇒p>w(T )
H κj h] (uj )gj] (dj )µ(δj )µ(δj0 )h[ (mj )h[ (Qj )
!
log δj0
log dj
log δj
log Qj mj
λ
χ
χ
χ
.
log x
log xγ
log Qj
log Qj
14
LILIAN MATTHIESEN
Let
c1
cr
αφ (p , . . . , p ) =
1
pms
X
r
Y
u∈(Z/pm Z)s
i=1
1pci |φi (u) ,
where m = max(c1 , . . . , cr ) for any prime p and extend αφ to composite arguments multiplicatively. Writing
∆j = lcm(δj mj Qj , δj0 mj Qj , uj , dj ),
it follows (cf. [10, Appendix A]) that
−1+O(γ) r
X Y
1
x
1∆j |φj (n) = αφ (∆1 , . . . , ∆r ) 1 + O
.
s
x vol K n∈Zs ∩xK j=1
vol K
Thus, the expression above equals
−1+O(γ) X
T
1+O
vol K
κ,λ,u
r
Y
X
αφ (∆1 , . . . , ∆r )
0
Q,m,δ,δ ,d
H κj h] (uj )gj] (dj )µ(δj )µ(δj0 )h[ (mj Qj )
(6.2)
j=1
×λ
log Qj mj
log x
log δj0
log dj
log δj
χ
χ
χ
log xγ
log Qj
log Qj
with the same summation conditions as above. Our aim is to show that the above expression equals
−1+O(γ) Y
r
X
X
H κj h] (uj )gj] (dj )µ(δj )µ(δj0 )h[ (mj Qj )
T
1 + o(1) + O
vol K
∆j
j=1 κj ,λj ,uj Qj ,mj ,δj ,δj0 ,dj
log δj0
log Qj mj
log dj
log δj
×λ
χ
χ
χ
,
log x
log xγ
log Qj
log Qj
(6.3)
again with the summation conditions from (6.1) in place. Note that (6.3) no longer features
the φi , which can only be achieved because we are working with a W -trick.
The proof of the above equality follows the approach of [3, §9] closely, although the
situation here bears a few extra difficulties. All essential tools in this analysis were derived
or developed starting out from material in [10, Appendix D]. The main steps are as follows:
(1) For every fixed choice of κ = (κ1 , . . . , κr ) and λ = (λ1 , . . . λr ) satisfying the summation conditions from (6.1), the sum over {u : ui ∈ U (κi , λi ), 1 6 i 6 r} in (6.2) (resp. (6.3))
X0
X0
can be replaced by (1 + o(1))
, where
indicates that the summation is restricted
u
Q
to tuples (u1 , . . . , ur ) of pairwise coprime integers such that gcd(ui , rj=1 Qj mj dj δj δj0 ) = 1
for every 1 6 i 6 r.
To prove this one shows that the sum over the excluded choices of (u1 , . . . , ur ) makes a
negligible contribution. This follows as in the proof of Claim 2 in [14, §9] by means of the
LINEAR CORRELATIONS OF MULTIPLICATIVE FUNCTIONS
15
−3
Cauchy-Schwarz inequality by combining the lower bound p > T (log log T ) on the prime
factors of any ui ∈ U (κi , λi ) and the observation that
X
X
X
log T
|Zs ∩ T K|
s
Q
r |Z ∩ T K| exp −
1p2 | i φi (n) r
p2
(log log T )3
−3 n∈Zs ∩T K
−3
T (log log Tγ)
6p<T
T (log log Tγ)
6p<T
with the following crude bound on the second moment of (6.2):
H
2r(log log T )3
X
r
Y
3
d(φi (n))6 g ] (φi (n)) H 2r(log log T ) |Zs ∩ T K|(log T )OH,r (1) ,
n∈Zs ∩T K i=1
which follows from [3, Proposition 7.9] and where d denotes the divisor function.
−3
(2) Since the primes Qi satisfy the lower bound Qi > T (log log T ) , we may by proceeding
X
X0
0
P
exactly as in step (1), replace the summation Q by (1 + o(1))
, where
indicates
Q
that theQ
summation is restricted to pairwise distinct primes Q1 , . . . , Qr such that in addition
gcd(Qi , rj=1 mj dj δj δj0 ) = 1.
X0
X0
from (1) and (2) in place, we have in the
and
(3) With the restrictions to
Q
u
argument of (6.2)
αφ (∆1 , . . . , ∆r ) =
˜ 1, . . . , ∆
˜ r)
αφ (∆
u1 . . . ur Q1 . . . Qr
˜ i = ∆i /(Qi ui ) for each i. Observe further that
where ∆
c0 =
r
XXXY
H κj h] (uj )
κ
λ
u
j=1
uj
< ∞.
(6.4)
(4) We replace χ(log m/ log Q) and λ(log m/ log Q)) by multiplicative functions in m,
using the Fourier-type transforms
Z
Z
x
−ixξ
x
e χ(x) =
θ(ξ)e
dξ, e λ(x) =
θ0 (ξ)e−ixξ dξ.
R
R
Setting I = [−(log T γ )1/2 , (log T γ )1/2 ], one obtains
Z
1+iξ
1
log m
χ
= m− log Q θ(ξ)dξ + OE m− log Q (log T γ )−E
log Q
I
and an analogous expression for λ.
Inserting the new expressions for χ and λ at all instances in (6.2) (resp. (6.3)) and
multiplying out, we obtain a main term and an error term. Any integral occurring in the
main term runs over the compact interval I and, thanks to the factor αφ (∆1 , . . . , ∆r ), the
summation over κ, λ, u, Q, m, δ, δ 0 , d is absolutely convergent. Thus, in the main term we
16
LILIAN MATTHIESEN
can swap sums and integrals. Taking into account steps (1)–(3), (6.2) then becomes
(1 + o(1))
r
XXXY
H κj h] (uj )
κ
λ
u
uj
j=1
!
Z X X
r
Y
˜
˜
αφ (∆1 , . . . , ∆r )
× ...
Jj∗ θ0 (ξ3,j )θ(ξ4,j )θ(ξ5,j )θ(ξ6,j ) dξ
Q1 , . . . , Qr
I
I Q
j=1
m,δ,δ 0 ,d


!
r
Ω(di )
X
Y
˜
˜
1
H
αφ (∆1 , . . . , ∆r ) 
+ OE 
,
(6.5)
0 1/ log T γ
E
(log T )
(Q
m
d
δ
δ
Q
,
.
.
.
,
Q
)
i
i
i
i
1
r
i
0
i=1
Z
Qm,δ,δ ,d
where
1+iξ
− γ log 3x
Jj∗ = h(Qj )Qj
Jj (Qj )
and
1+iξ
Jj (Qj ) =
1+iξ3
1+iξ4
− log Q5
−
−
j
gj] (dj )h(mj )mj γ log x dj γ log x µ(δj )µ(δj0 )δj
1+iξ
0
− log Q6
j
δj
.
To proceed further, the following two lemmas are required
Lemma 6.2. Suppose κ, λ and ui ∈ U (κi , λi ) for 1 6 i 6 r are fixed. Then we have
!
r
r
X
Y
Y
X Jj (Qj )
˜
˜
αφ (∆1 , . . . , ∆r )
Jj (Qj ) = (1 + o(1))
˜j
∆
j=1
j=1 m ,δ ,δ 0 ,d
m,δ,δ 0 ,d
j
j
j
j
−3
for each choice of primes Q1 , . . . , Qr > T (log log T ) .
Lemma 6.3. For each 1 6 j 6 r, we have
Z
Z ∗
X
Jj 0
. . . θ (ξ3,j )θ(ξ4,j )θ(ξ5,j )θ(ξ6,j ) dξ j
˜
I
I Q ,m ,δ ,δ 0 ,d Qj ∆j
j
j j j j
log w(x) Y
|h(p)|
1+
.
log x
p
γ
w(x)<p<x
The two lemmas above correspond to [3, Lemmas 9.4 and 9.3]. The proof of Lemma 6.2
is very similar to that of [3, Lemmas 9.4] rather fairly lengthy so that we omit it here. The
proof of Lemma 6.3, in contrast, needs to work with significantly weaker assumptions on
the Dirichlet series involved than the corresponding one from [3]. We will therefore carry
out this proof below. Before we do so, let us, however, show how to deduce the equality of
(6.2) and (6.3) and, thus, complete the proof of Proposition 6.1.
LINEAR CORRELATIONS OF MULTIPLICATIVE FUNCTIONS
17
By applying first Lemma 6.2 and then Lemma 6.3, we obtain
Z X X
r
˜ 1, . . . , ∆
˜ r) Y
αφ (∆
...
Jj∗
Q1 , . . . , Qr j=1
I
I Q
0
Z
m,δ,δ ,d
r
Y
1+iξ
3
−1−
h(Qj )Qj γ log x
−
mj ,δj ,δj0 ,dj
j=1
Z Y
=o
I
j
X
Qj ,mj ,δj ,δj0 ,dj

= o
log w(x)
log x
X
Jj∗ 0
θ (ξ3,j ) . . . θ(ξ6,j ) dξ
˜j
Qj ∆
Y
w(x)<p<xγ
Jj (Qj )
˜j
∆
Y
r
θ0 (ξ3,j 0 ) . . . θ(ξ6,j 0 )dξ
j 0 =1
!

h(p) 
1+
.
p
This completes the the proof of Proposition 6.1, assuming Lemma 6.3.
It remains to prove Lemma 6.3. We emphasise that, although this lemma corresponds
to [3, Lemma 9.3], the proof below is significantly stronger in that it does not rely on any
assumption on the behaviour close to s = 1 of the Dirichlet series attached to h] or to h[ .
Proof of Lemma 6.3. We proceed by decomposing the sum in the integrand into Euler
products, keeping in mind that the contribution of higher prime powers only affects the
implied constant. We have
∗
X
Jj
0 )
Q
m
d
lcm(δ
,
δ
j
j
j
j
j Qj ,mj ,δj ,δj0 ,dj
!
Y
g(p0 )
1+
×
1+iξ4
0
p0 1+ γ log x p >w(x)
p0 6∈P[
(
Y
p>w(x)
p∈P[
1+
h[ (p)
1+iξ3
p1+ γ log x
−
1
1+iξ5
p1+ γ log x
−
1
1+iξ6
p1+ γ log x
!
+
2+iξ5 +iξ6
1+ γ log
x
p
1
! )
X h[ (Q) Y
h[ (p)1p<Q
1
1
1
+
1
+
−
−
+
.
1+iξ3
1+iξ3
1+iξ5
1+iξ6
2+iξ5 +iξ6
1+ γ log x
1+ γ log x
1+ log Q
1+ log Q
1+ log Q
Q
p
p
p
p
Q∈P[
p>w(x)
p∈P[
a4,p
After splitting the above factors of the form (1 + ap1,p
s + · · · + ps ) into individual factors
Q
a
ai,p
(1 + pi,p
s ), we make use of the following bounds to analyse the products
p (1 + ps ) with
18
LILIAN MATTHIESEN
negative coefficients ai,p = −1. We have
Y
!
1−
1+ γ1+iξ
p log x 1
p>w(x)
p∈P[
Y
1 + iξ
−1
1+
ζ
γ log x
1
1+
p
p>w(x)
p6∈P[
!
!
Y
1+ γ1+iξ
log x
q<w(x)
1 + iξ Y
γ log x 1+
1 + iξ γ log x Y 1
1
1+
1+
p
q
γ
p>w(x)
p6∈P[
Y
1
1
p1+ γ log x
|1 + iξ|
Y 1
1+
q
q<w(x)
q<w(x)
w(x)<p<x
p6∈P[
Y
!
1
1+
1+ γ1+iξ
log x
q
1
,
1−
p
γ
w(x)<p<x
p∈P[
and similarly for γ log x replaced by log Q. These bounds yield
∗
X
Jj
0 Q
m
d
lcm(δ
,
δ
)
j j Qj ,mj ,δj ,δj0 ,dj j j j
Y
|(1 + iξ5 )(1 + iξ6 )|
1+
p0 >w(x)
!
g(p0 )
1
p0 1+ γ log x
p0 6∈P[
(
Y
p>w(x)
p∈P[
+
!
[
1+
h (p)
X
h[ (Q)
Q∈P[
Q1+ γ log x
1
1+
1
p1+ γ log x
Y
1
!
Y
2
p1+ γ log x
1+
w(x)<p<Q
p∈P[
w(x)<p<xγ
p∈P[
h[ (p)
!
1
p1+ γ log x
1
1−
p
1
1−
p
2 Y
w(x)<p00
p00 ∈P[
Note that
Y
1+
p∈A
−2
ap
c
p1+ log y
Y
p∈A
p6y
ap
1+
p
1+
!)
1
2
p00 1+ log Q
.
LINEAR CORRELATIONS OF MULTIPLICATIVE FUNCTIONS
19
whenever c > 1 and the ap , p ∈ A , are non-negative and bounded. Thus, the above is
bounded by
( Y Y
g(p0 )
1
h[ (p)
|(1 + iξ5 )(1 + iξ6 )|
1+ 0
1−
1+
p
p
p
γ
0
γ
w(x)<p<x
p∈P[
w(x)<p <x
p0 6∈P[
+
|(1 + iξ5 )(1 + iξ6 )|
X
h[ (Q)
Q∈P[
Q1+ γ log x
Y
1+
1
Y
w(x)<p0 <xγ
p0 6∈P[
g(p0 )
1+ 0
p
h[ (p)
1
p1+ γ log x
w(x)<p<Q
p∈P[
Y
w(x)<p<xγ
p∈P[
h[ (p)
1+
p
!
1−
!)
1
1
p1+ γ log x
1
1−
p
1
|h(p)|
1−
|(1 + iξ5 )(1 + iξ6 )|
1+
p
p
w(x)<p<xγ
log w(x) Y
|h(p)|
γ |1 + iξ5 ||1 + iξ6 |
1+
.
log x
p
Y
w(x)<p<x
Integrating and taking the decay properties of the function θ and θ0 into account yields the
result.
7. The average order of ν ] ν [
As a consequence of the proof of Proposition 6.1, we obtain the following lemma which
shows that ν ] ν [ is of the correct average order on a positive proportion of the progressions
f (T )) with gcd(A, W (T )) = 1 provided (cf. the discussion at the end of Section 1)
A (mod W
that |Sh (T )| S|h| (T ).
Lemma 7.1. We have
f (T ), A)| Sν ] ν [ (T ; W
f (T ), A) log w(T )
|Sh (T ; W
log T
Y
w(T )<p<T
|h(p)|
1+
p
whenever gcd(A, W (T )) = 1.
Proof. The first bound is immediate since ν ] ν [ is, outside of the sparse set S , a majorant
for |h|. The second bound follows from the upper bound (6.5) on
s
X Y
1
νh (φi (n)),
T s vol K n∈Zs ∩T K i=1 i
where we are only interested in the case s = 1. To see this, we apply first Lemma 6.2 and
then Lemma 6.3 to the integrand, and finally recall that the outer sum (6.4) converges. 20
LILIAN MATTHIESEN
8. The pretentious W -trick and orthogonality of h with nilsequences
In [15], the non-correlation of (W -tricked) multiplicative functions with nilsequences is
proved for multiplicative functions which admit the W -trick recorded in Definition 8.1
below. In this section we establish this W -trick for all functions f ∈ M .
It is an interesting question to investigate whether a suitably adapted version of this
W -trick can be established for functions f that satisfy the condition (1.4) for h = f . Such
a W -trick would need to take into account the rotation arising from the factor pit .
Definition 8.1. Let κ(x) and θ(x) be non-negative functions, which will be bounded in
all applications. Let FH (κ(x), θ(x)) denote the class of all multiplicative functions f ∈ M
for which there exist functions ϕ0 : N → R and q ∗ : N → N with the following properties:
(1) ϕ0 (x) → 0 as x → ∞,
(2) q ∗ (x) is w(x)-smooth and q ∗ (x) 6 (log x)κ(x) for all x ∈ N,
f (x) = q ∗ (x)W (x), then the estimate
(3) if x ∈ N and if we set W
f (x)
q0 W
|I|
X
f (x), A)
f (m) − Sf (x; W
m∈I
f (x))
m≡A (q0 W
Y 1
W (x)q
|f (p)|
= O ϕ (x)
1+
log x φ(W (x)q) p<x
p
0
(8.1)
p-W (x)q
holds uniformly for all intervals I ⊆ {1, . . . , x} with |I| > x(log x)−θ(x) , for all
f (x)Z)∗ .
integers 0 < q0 6 (log x)θ(x) , and for all A ∈ (Z/q0 W
f is controlled
The set FH (κ(x), θ(x)) contains functions that have a W -trick where W
by κ and where θ(x) is a measure for the quality of the major arc estimate.
8.1. The elements of M admit a W -trick. We begin with the two lemmas that will
reduce the task of proving (8.1) to that of bounding a restricted character sum. The first
lemma is the following.
Lemma 8.2. If x > 1, f ∈ M and E > H > 1, where H is as in Definition 1.1(i), then
(8.1) follows for θ(x) = E if there is ϕ00 = o(1) such that
|f (p)|
Wq Y
00
f
f
1+
(8.2)
Sf (y; q0 W (x), A) = Sf (y; W (x), A) + O ϕ (x)
φ(W q) p<x
p
p-W q
f (x)Z)∗ . More precisely, we may take
for all x > y > x(log x)−E and for all A ∈ (Z/q0 W
0
00
−E
ϕ (x) = ϕ(x) + ϕ (x) + (log x) in (8.1), where ϕ is as in Definition 1.1 (iv).
LINEAR CORRELATIONS OF MULTIPLICATIVE FUNCTIONS
21
Proof. Suppose that I = [y1 , y1 + y2 ] ⊂ [1, x] with y2 > x(log x)−E and note that condition
(iv) of the Definition 1.1 implies that
Wq Y
|f (p)|
f
f
Sf (y1 + y2 ; W (x), A) = Sf (x; W (x), A) + O ϕ(x)
1+
.
φ(W q) p<x
p
p-W q
Another application of Definition 1.1 (iv) shows that the left hand side of (8.1) satisfies
f (x)
q0 W
|I|
X
f (m)
(8.3)
m∈I
f (x))
m≡A (q0 W
y
y1 + y2 1
f
f
=
Sf y1 + y2 ; q0 W (x), A − Sf y1 ; q0 W (x), A
y2
y2
y
y1 + y2 1
f (x), A − Sf y1 ; q0 W
f (x), A
Sf x; q0 W
=
y2
y2
|f (p)|
Wq Y
+ O ϕ(x)
1+
.
φ(W q) p<x
p
p-W q
We now split into two cases. If, on the one hand, x > y1 > y2 (log x)−2E > x(log x)−3E ,
then Definition 1.1 (iv) shows that (8.3) is equal to
!!
Y
W
q
|f
(p)|
f (x), A + O ϕ(x)
Sf x; q0 W
.
1+
φ(W q) p<x
p
p-W q
In this case, (8.1) follows from (8.2) and the above with ϕ0 = ϕ + ϕ00 .
−2E
2
) and it follows
If, on the other hand, y1 6 y2 (log x)−2E , then y1y+y
=
(1
+
O
(log
x)
2
from E > H that
y1 f (x), A 6 y1 (log x)H−1 6 (log x)−E−1
Sf y1 ; q0 W
y2
y2
Wq Y
|f (p)|
−E
6 (log x)
1+
.
φ(W q) p<x
p
p-W q
Thus, (8.3) equals in this case
f (x), A) + O
Sf (x; q0 W
!
Y
|f
(p)|
W
q
1+
ϕ(x) + (log x)−E
φ(W q) p<x
p
p-W q
and an application of (8.2) yields (8.1) with ϕ0 (x) = ϕ(x) + ϕ00 (x) + (log x)−E .
Following the above reduction, we now proceed to analyse the difference of the two mean
values that appear in (8.2).
22
LILIAN MATTHIESEN
Lemma 8.3 (Restricted character sum). Suppose that x > y > x(log x)−E , let q0 > 1 be
f (x)) = 1. Then
an integer and suppose gcd(A, q0 W
f
1
f , A) − Sf (y; q0 W
f , A) = q0 W
Sf (y; W
f)
x φ(q0 W
X∗
χ(A)
f (n)χ(n),
(8.4)
n6x
f)
χ (mod q0 W
where
X
X∗
indicates the restriction of the sum to characters that are not induced from
f ).
characters (mod W
Proof. We have
f , A) − Sf (y; q0 W
f , A)
Sf (y; W
!
f
W
=
y
X
f (n) − q0
n6y
f)
n≡A (mod W
f
1 W
=
f)
y φ(q0 W
f
1 W
=
f)
y φ(q0 W
f (n)
n6y
f)
n≡A (mod q0 W
!
X
f)
χ (mod q0 W
X
χ(A0 ) − q0 χ(A)
X
f (n)χ(n)
n6y
A0
f)
(mod q0 W
f)
A≡A0 (mod W
!
X∗
f)
χ (mod q0 W
where
X
X
χ(A0 ) − q0 χ(A)
X
f (n)χ(n),
(8.5)
n6y
A0
f)
(mod q0 W
f)
A≡A0 (mod W
X∗
indicates the restriction of the sum to characters that are not induced from
f ); for all other characters we have χ(A0 ) = χ(A) and the difference
characters (mod W
in the brackets above is zero. It remains to show that the sum over A0 in (8.5) vanishes.
However,
X
X
X
1
χ(A0 ) =
χ0 (A)
χ(A0 )χ0 (A0 ) = 0,
f
φ(W ) χ0 (mod W
f)
f)
f)
A0 (mod q W
A0 (mod q W
0
0
f)
A≡A0 (mod W
f . Thus the lemma follows.
since χχ0 is a non-trivial character modulo q0 W
We aim to exploit the fact that the character sum on the right hand side of (8.4) is
restricted by means of the following consequence of the ‘pretentious large sieve’:
Proposition 8.4 (Granville and Soundararajan [9]). Let C > 0 be fixed and let f be a
bounded multiplicative function. For any given x, consider the set of primitive characters of
conductor at most (log x)C and enumerate them as χ1 , χ2 , . . . in such a way that |Sf χ1 (x)| >
|Sf χ2 (x)| > . . . . If x is sufficiently large, then the following holds for all x1/2 6 X 6 x
LINEAR CORRELATIONS OF MULTIPLICATIVE FUNCTIONS
23
and q 6 (log x)C . Let C be any set of characters modulo q, q 6 (log x)C , which does not
contain characters induced by χ1 , . . . , χk , where k > 2. Then
1 X
X
χ(a)
f (n)χ(n)
φ(q)
n6X
χ∈C
√
eOC ( k) X
C
q
log log x
log x
1− √1
k
log
log x
log log x
Y p6q,p-q
|f (p)| − 1
1+
p
.
Equipped with these tools, we are now ready to state and prove the two main results of
this section.
Proposition 8.5. Suppose that f ∈ M and let H be the parameter from Definition 1.1 (i).
Let d, mG > 1 be integers and let G/Γ be a nilmanifold of dimension mG , equipped with a
filtration of length d. Then there exist, for any positive x, positive integers κ(x) and E(x)
such that κ(x), E(x) d,mG ,H 1, such that E(x) is sufficiently large with respect to d, mG , H
and κ(x) for [15, Theorem 5.1] to apply to G/Γ, and such that f ∈ FH (κ(x), E(x)). More
precisely, (8.1) holds with ϕ0 (x) = ϕ(x) + (log x)−αf /(3H) + (log x)−E(x) , where ϕ is as in
(iv) of Definition 1.1.
The following corollary summarises an immediate consequence of the above proposition
and [15, Theorem 5.1]. This is the result that will be used in the proof of Proposition 2.1
in the final section.
f (x) be as guaranteed
Corollary 8.6. Let f and G/Γ be as in Proposition 8.5 and let W
by the above conclusion that f ∈ FH (κ(x), E(x)). Suppose that G is r − 2-step nilpotent,
let G• be the filtration of length d and suppose that G/Γ has a Q0 -rational Mal’cev basis
adapted to this filtration. Then the following estimate holds for all g ∈ poly(Z, G• ) and for
all 1-bounded Lipschitz functions F : G/Γ → C
W
X
f
f
f
(f
(
W
n
+
A)
−
S
(T
;
W
,
A))F
(g(n)Γ)
(8.6)
f
T
f
n6T /W
(
)
Od,m (1)
Y 1
Q0 G
1
W
|f (p)|
kF kLip ϕ(N ) +
+
,
1+
log w(N ) (log log T )1/(2r+2 dim G) log T φ(W ) p<T
p
p-W (N )
where ϕ is as in Definition 1.1 (iv). The implied constants depend, in addition to kF kLip ,
also on H, on the step and dimension of G, on Q0 and on the degree of the filtration G• .
We now turn to the proof of the main result of this section.
Proof of Proposition 8.5. To start with, assume that |f (p)| 6 1 for all primes p and let αf
be as in Definition 1.1 (iii). Setting ε := 21 min(1, αf /2) and k := dε−2 e > 2, we aim to
apply Proposition 8.4 with this value of k to bound (8.4). That is, our task is to show that
f (x) and E so that E is large enough for [15, Theorem 5.1]
there are suitable choices of W
to apply and so that Proposition 8.4 applies to all sets of characters that appear in (8.4).
24
LILIAN MATTHIESEN
For this purpose, we define two sequences of integers, H0 , H1 , . . . , Hk and E0 , E1 , . . . , Ek ,
as follows. Let H0 = 1. If Hi is defined, let Ei d,mG ,Hi 1 be such that Ei is sufficiently
large for [15, Theorem 5.1] to apply for any r −2-step nilmanifold. If Hi and Ei are defined,
set Hi+1 = 2Hi + Ei + 1.
Let C = 2Hk + Ek + 2 and let χ1 , . . . , χk be the characters of conductor at most (log x)C
that are defined in the statement of Proposition 8.4. We now define k + 1 disjoint intervals
[2ri W (x), 2ri W (x)(log x)Ei ], 0 6 i 6 k, where ri = dlog2 ((log x)Hi )e. Since W (x) <
(log x)2 , these are all contained in [1, (log x)C ]. Thus, there exists an index j ∈ {0, . . . , k}
such that the interval [2rj W (x), 2rj W (x)(log x)Ej ] does not contain the conductor of any
of the characters χ1 , . . . , χk .
Setting q ∗ (x) = 2rj and E = Ej , we note that the conductor of each character in the
sum (8.4) belongs to [q ∗ (x)W (x), q ∗ (x)W (x)(log x)E ]. Hence Proposition 8.4 shows that
f , A) − Sf (x; q0 W
f , A) =
Sf (x; W
f
1 q0 W
f)
x φ(q0 W
X∗
χ(A)
d,mG ,H,αf
log log x
log x
f (n)χ(n)
n6x
f)
χ (mod q0 W
X
1− √1
k
log
log x
log log x
d,mG ,H,αf (log x)−1+αf /2 (log log x)2
d,mG ,H,αf
Wq
(log log x)2 1
exp
α
/2
(log x) f log x φ(W q)
X
p6x
p-W q
|f (p)|
.
p
To handle the general case, we decompose f as f = g ∗H ∗ g 0 , with g as in (1.1) for h = f .
Let εH = min(1, αf /H)/2 so that the above applies to g with ε replaced by εH and with αf
replaced by αf /H. Keeping in mind that g 0 is supported in square-full numbers only, we
obtain the following estimate by combining the above with an application of the hyperbola
method:
f
1
q0 W
f)
x φ(q0 W
X∗
χ(A)
f)
χ (mod q0 W
X
f (n)χ(n)
n6x
H
H
X
X
D6x1−1/H d1 ...dH−1
f D 1 q0 W
f) x φ(q0 W
d,mG ,H,αf
g(d1 ) . . . g(dH−1 )|χ(D)| X
D
i=1
=D
X∗
f)
χ (mod q0 W
χ(A)
X
n:
x1−1/H max(d1 ,...,di−1 )
6Dn6x
(log log x)2 1
Wq
exp
α
/(2H)
f
log x φ(W q)
(log x)
X
p6x
p-W q
g(n)χ(n)
|f (p)|
.
p
LINEAR CORRELATIONS OF MULTIPLICATIVE FUNCTIONS
This yields (8.2) with ϕ00 (x) = (log x)−αf /(3H) . Hence, Lemma 8.2 implies the result.
25
9. Conclusion of the proof of Proposition 2.1
We now have everything in place to complete the proof of Proposition 2.1 and, thus,
Theorem 1.2. We begin by noting that Proposition 2.1 holds trivially unless
log w(T ) Y
|hi (p)|
f
|Shi (T, W (T ), Ai )| 1+
(9.1)
log T
p
w(T )<p6T
for all 1 6 i 6 r. Assuming (9.1) from now on, it follows from Corollary 8.6 and the inverse
theorem for uniformity norms from [11] that the function
h̃i : n 7→
f (T )n + Ai ) − Sh (T, W
f (T ), Ai )
hi (W
i
,
f (T ), Ai )
S|h | (T, W
i
f (T ), satisfies
defined for positive integers n 6 T /W
kh̃i kU r−1 = o(1).
By Lemma 7.1, the majorant νi[ νi] for hi is of the correct order. Thus,
!
νi[ νi]
1
ν̃i =
+1
f (T ), Ai )
2 Sh (T, W
i
is a correct-order majorant for h̃i . Furthermore, Proposition 6.1 shows that this majorant
is pseudo-random. Proposition 2.1 now follows from the generalised von Neumann theorem
[10, Proposition 7.1].
10. Proof of Theorem 1.3 and Corollary 1.4
To deduce Theorem 1.3 from Theorem 1.2, we first observe that Proposition 8.4 and
Lemma 1.5 imply that
−1 !
X
Y f X
1
1
W
f (T ), Ai ) =
Shi (T, W
χi (Ai )
hi (n)χi (n) + o S|h0i | (T )
1−
f)
T n6T
p
φ(W
χi ∈Ei
p6w(T )
X
Y χi,p (Ai ) Y 1 + χi (p0 )hi (p0 )p0−1 + . . . =
S|h0i | (T )
1 − p−1
1 + |hi (p0 )|p0−1 + . . .
χi ∈Ei
p6w(T )
w(T )6p0 6T
−1 !
Y 1
+ o S|h0i | (T )
1−
,
p
p6w(T )
where h0i (n) = hi (n)1gcd(n,W (T ))=1 . When combined with a truncation argument as in §2,
the stability condition (iv) from Definition 1.1 yields furthermore
−1
Y |hi (p)|
S|h0i | (T ) = (1 + o(1))S|hi | (T )
1+
+ ...
,
p
p6w(T )
26
LILIAN MATTHIESEN
which can be inserted into the previous expression.
f
If ϕ(v) ≡ Ai wi (mod pvp (wi W ) ) for all p 6 w(T ), then
Y
Y
ϕi (v)
ϕi (v)
ϕi (v) Y
=
=
χi (Ai ) = χi
χi,p
χi,p
χi,p (pvp (wi ) ).
wi
wi
pvp (wi ) q6=p
p6w(T )
p6w(T )
Thus, with the help of the Chinese Remainder Theorem, we obtain
!
r
r
Y
X
X
X Y
1
hi (wi )χi (Ai )
1ϕj (v)≡wj Aj (mod wj W
f)
f )s
(wW
w1 ,...,wr
v∈
i=1
j=1
A1 ,...,Ar
p|wi ⇒p<w(T )
f Z)∗
∈(Z/W
wi 6(log T )B2
= (1 + o(1))
Y
f Z)s
(Z/wW
lim
m→∞
p<w(T )
1 X
pms
X
r
Y
aj
hj (p )χi,p
a∈N0 v∈(Z/pm Z)s j=1
vp (ϕi (v))=ai
ϕi (v)
pvp (wi )
Y
χi,p (pvp (wi ) ).
q6=p
Finally, we observe that for p > w(T ) with T sufficiently large:
lim
m→∞
1 X
pms
X
r
Y
hj (paj )χi (paj )
a∈N0 v∈(Z/pm Z)s j=1
vp (ϕi (v))=ai
r Y
1
hj (p)χi (p)
=
1−
1+
+ Or (p−2 H 2 ),
p
p
j=1
(10.1)
which follows from [3, (5.6)] with M = 1 by arguing as in [3, §5.2]. Theorem 1.3 now
follows by inserting all the above information into the expression (1.2).
Corollary 1.4 follows fairly straightforwardly from (1.2) by inserting the assumptions of
the corollary and by applying an identity of the form (10.1) but with the factors χj (pai )
and χj (p) all removed, i.e. replaced by 1.
11. Application to eigenvalues of cusp forms
In this section we show that Theorem 1.2 applies to the normalised eigenvalues of holomorphic cusp forms. Let f be a primitive cusp form of weight k > 2 and level N > 1.
Then f has a Fourier expansion of the form
f (z) =
∞
X
λf (n)n(k−1)/2 e(nz),
n=1
where λf is multiplicative and satisfies Deligne’s bound
|λf (n)| 6 d(n),
where d is the divisor function.
Lemma 11.1. |λf (n)| ∈ M . Thus, hi = |λfi (n)| is a permissible choice in Theorem 1.2.
LINEAR CORRELATIONS OF MULTIPLICATIVE FUNCTIONS
27
Proof. Conditions (i) and (ii) of Definition 1.1 follow directly from the properties summarised above. Condition (iii) follows from Rankin [16, Theorem 2] since
X
1X
x
|λf (p)| log p >
λf (p)2 log p ∼ .
2 p6x
2
p6x
for large x. Thus, it remains to show that condition (iv) holds too. For this purpose, we
will first deduce from the Sato–Tate law that (iii) in fact holds for some α|λf | > 2/π. This
will allow us later to employ the Lipschitz bounds [9] for multiplicative functions. Note
that the Sato–Tate conjecture [2] implies:
#{p 6 x : 0 6 |λp | 6 α} log x ∼ xµ(α),
where
µ(α) =
for α ∈ [0, 2]. Thus,
X
|λf (p)| log p >
p6x
2
1
arcsin(α/2) + sin(2 arcsin(α/2)).
π
π
X
(1 − ε)|λf (p)| log x
x1−ε <p6x
> (1 − ε)
N
X
n=1
X
x1−ε <p6x
2(n−1)/N 6|λf (p)|62n/N
n−1
log x
N
N
X
2(n − 1) µ(2(n − 1)/N ) − µ(2n/N ) .
∼ x(1 − ε)
N
n=1
Choosing ε > 0 sufficiently small and N sufficiently large, the claimed lower bound on α|λf |
now follows, since
Z 2
Z 2
2
1
8
2
α dµ(α) = 2µ(2) −
arcsin(α/2) + sin(2 arcsin(α/2)) dα =
> .
π
π
3π
π
0
0
If g is the multiplicative functions defined by
(
|λf (p)|/2 if k = 1
,
g(pk ) =
0
if k > 1
P
then (iii) holds for g with some αg > π1 , and, hence, n6x g(n) x(log x)−1+1/π+c for
some c > 0. Thus, if x0 ∈ (x(log x)−C , x], then the Lipschitz bounds [8] of Granville and
Soundararajan imply that
1
1X
1 X
0
g(n) = 0
g(n) + OC,ε0 (log x)−1+ π +ε
(11.1)
x n6x
x n6x0
!
1 X
1X
= 0
g(n) + o
g(n) .
x n6x0
x n6x
28
LILIAN MATTHIESEN
If χ (mod q) is a character for which there exists t = tχ ∈ R such that (1.4) holds for
h(n) = g(n)χ(n), then Elliott’s asymptotic formula (1.5), combined with (11.1), shows
that
1X
g(n)(χ(A)χ(n) + χ(A)χ(n))
(11.2)
x n6x
!
χ(A) −it Y 1 + g(p)χ(p)pit−1 + . . .
x
+ o(Sg (x))
= Sg (x) · 2<
1 − it
1 + g(p)p−1 + . . .
p6x
!
Y 1 + g(p)χ(p)pit−1 + . . . χ(A)
= Sg (x0 ) · 2<
x−it
+ o(Sg (x))
−1 + . . .
1 − it
1
+
g(p)p
p6x
1 X
= 0
g(n)(χ(A)χ(n) + χ(A)χ(n)) + o(Sg (x)).
x n6x0
If no such t exists for χ, then Lemma 1.5 yields
1 X
g(n)(χ(A)χ(n) + χ(A)χ(n)) = o(Sg (x)).
x0 n6x0
(11.3)
for all x0 as above. In order to employ these facts, as well as Proposition 8.4, which is also
a statement about bounded multiplicative functions, we shall now decompose S|λf | (x; q, A)
into character sums involving the bounded function g. Since |λf (n)| = g ∗ g ∗ g 0 , where
g 0 (n) is zero unless n is square-full and satisfies |g 0 (pk )| 6 C 0 k for some constant C 0 , we
have
S|λf | (x; q, A)
X
1
=
φ(q)
(11.4)
χ(A)
χ (mod q)
X
=
X
6x1/2
1X
|λf (n)|χ(n)
x n6x
g 0 (n0 )g(n1 )
)1/2
n0
n1 6(x/n0
(n0 ,q)=1 (n1 ,q)=1
+
X
6x1/2
X
1
φ(q)
g 0 (n0 )g(n1 )
)1/2
n0
n1 <(x/n0
(n0 ,q)=1 (n1 ,q)=1
X
χ(An0 n1 )χ(A)
χ (mod q)
1
φ(q)
X
χ (mod q)
χ(An0 n1 )
1
x
1
x
X
g(n2 )χ(n2 )
n2 6x/(n0 n1 )
(n2 ,q)=1
X
(x/n0
g(n2 )χ(n2 )
)1/2 <n
2 6x/(n0 n1 )
(n2 ,q)=1
+ O(x−1/12 ),
where n denotes the inverse modulo q, and where the truncation of the summation in n0 is
justified as follows. Observe that the bound |λf (n)| 6 d(n) ε nε implies that g 0 (n) ε nε .
This follows inductively for n = pk , since |λf (pk )| = g 0 (pk ) + 2g 0 (pk−1 )g(p) + g 0 (pk−2 )g(p)2 .
Observing further that for any square-full integer n0 > y its largest square divisor q 2
LINEAR CORRELATIONS OF MULTIPLICATIVE FUNCTIONS
satisfies q > y 1/3 , we have
X g 0 (n0 )
x
Sg∗g
; q, An0 n0
n0
1/2
x
X
x1/2 <n0 6x
(n0 ,q)=1
<n0 6x
(n0 ,q)=1
Y
1
X
x1/6 <q6x1/2
q 2−2ε
1+
p|q
1
p1−ε
g 0 (n0 )
ε
n0
X
29
1
n1−ε
x1/2 <n0 6x 0
X d(q)
X 1
1
(1−3ε)/6 ,
ε
2−2ε
2−3ε
q
q
x
1/6
1/6
q>x
q>x
which is seen to be O(x−1/12 ) if ε was chosen sufficiently small.
Proposition 8.4, applied with k = dπ 1/2 e and q 6 (log x)C , furthermore shows that, if
Eq ⊂ {χ1 , . . . , χk , χ1 , . . . , χk } is the subset of all characters with conductor q, then (11.4)
equals
S|λf | (x; q, A)
X
X
=
g 0 (n0 )g(n1 )
n0 6x1/2 n1 6(x/n0 )1/2
(n0 ,q)=1 (n1 ,q)=1
+
X
6x1/2
X
1 X
1
χ(An0 n1 )χ(A)
φ(q) χ∈E
x
q
g 0 (n0 )g(n1 )
)1/2
n0
n1 <(x/n0
(n0 ,q)=1 (n1 ,q)=1

Y
 q
1
+ o
 φ(q) log x
p6x
1 X
1
χ(An0 n1 )
φ(q) χ∈E
x
q
X
g(n2 )χ(n2 )
n2 6z/(n0 n1 )
(n2 ,q)=1
X
(x/n0
g(n2 )χ(n2 )
)1/2 <n
2 6z/(n0 n1 )
(n2 ,q)=1

|λf (p)| 
,
1+

p
p-q
where z = x in the sums over n2 . Note that #Eq 6 2k 1 and that we always have
z/(n0 n1 ) > zx−3/4 in the expression above. Thus, by splitting the second sum over n2 into
a difference of two sums, and applying afterwards (11.2) or (11.3), repectively, to those
two sums that run up to z/(n0 n1 ), we deduce that the above continues to hold for every
z ∈ (x(log x)−C , x), with an error term of the exact same shape. In view of (11.4), the
above equals, however,
!
q
1 Y
|λf (p)|
S|λf | (z; q, A) + o
,
1+
φ(q) log x p6x
p
p-q
which is the content of property (iv) of Definition 1.1 and shows that |λf | ∈ M .
12. Further applications
In this section we sketch the proofs of two applications that follow from the Selberg–
Delange method.
Lemma 12.1. Let δ ∈ (0, 1) and let h(n) = δ ω(n) , where ω(n) denotes the number of
distinct prime factors of n. Then h ∈ M .
30
LILIAN MATTHIESEN
Proof. Conditions (i)–(iii) are immediate. Thus, let x, x0 and q be as in Definition 1.1 (iv).
The Selberg–Delange method allows us to deduce an asymptotic formula for
X
h(n)χ(n)
n6x
P
for any character χ (mod q), by relating the Dirichlet series n>1 h(n)χ(n)n−s to L(s, χ)δ .
The shape of this asymptotic formula implies that either both
√
√
1X
1 X
−c log x
h(n)χ(n) = O(e−c log x ) and
h(n)χ(n)
=
O(e
)
x n6x
x0 n6x0
are small, or
1X
1 X
h(n)χ(n) ∼ 0
h(n)χ(n).
x n6x
x n6x0
Hence, Sh (x; q, a) ∼ Sh (x0 ; q, a), as required.
Lemma 12.2. Let K/Q be a finite Galois extension and let h(n) denote the characteristic
function of integers that are composed of primes p which split completely over K. Then
h ∈ M.
Proof. Conditions (i) and (ii) are immediate. Condition (iii) follows from the Chebotarev
density theorem. To verify (iv), suppose that [K : Q] = d, and let P denote the set of
rational primes that split completely in K. Let
Y
−1
F (s) =
1 − dp−s
p∈P
for <s > 1. Then there is a function G(s) that is holomorphic and non-zero in <s > 12
and such that ζK (s) = F (s)G(s), thus F (s) has a meromorphic continuation to <s > 21 .
Condition (iv) then follows as above from the Selberg–Delange method. In this case, we
Q
−1
relate p∈P (1 − p−s ) to F (s)1/d , that is, one needs to analyse
Y
−1
1/d Y
−1
1 − p−s
1 − dp−s
=
1 − p−s
G(s)1/d ζK (s)−1/d ,
p∈P
p∈P
or, if χ (mod q) is a Dirichlet character, then we consider instead
Y
−1
1/d Y
−1
1 − χ(p)p−s
1 − dχ(p)p−s
=
1 − χ(p)p−s
Gχ (s)1/d L(ξ, s)−1/d ,
p∈P
p∈P
where Gχ is the twist of G by χ and where ξ is a suitable Hecke character.
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KTH, Department of Mathematics, Lindstedtsvgen 25, 10044 Stockholm, Sweden
E-mail address: [email protected]