MULTIPLICATIVE FUNCTIONS IN ARITHMETIC PROGRESSIONS

MULTIPLICATIVE FUNCTIONS IN ARITHMETIC PROGRESSIONS

ANTALBALOG, ANDREWGRANVILLEANDKANNANSOUNDARARAJAN

Dedicated to Paulo Ribenboim on the occasion of his 80th birthday.

RÉSUMÉ. Nous développons une théorie des fonctions multiplicatives (avec les valeurs sur ou à l’intérieur du cercle d’unité) en progressions arithmétiques analogues à la théorie bien connue des nombres premiers en progressions arithmétiques.

ABSTRACT. We develop a theory of multiplicative functions (with values inside or on the unit circle) in arithmetic progressions analogous to the well-known theory of primes in arithmetic progressions.

1. Introduction

1.1. Historical preliminaries

A central focus of multiplicative number theory has been the question of how primes are distributed in arithmetic progressions (seee.g. [4]): There areπ(x)primes up tox, of whichπ(x;q, a)belong to the arithmetic progressiona (modq), and

π(x;q, a)∼ π(x) (1.1) φ(q)

asx → ∞, whenever (a, q) = 1. Hereφ(q) stands for the Euler φ–function. In other words, the primes are eventually equidistributed amongst the plausible arithmetic progressions(modq). In applications it is important to know how bigxmust be, as a function ofq, for (1.1) to hold. Unconditionally we can only prove (1.1) for allqwhen xis enormous, larger than exponential of a (small) power ofq, whereas we believe it holds whenxis just a little bigger thanq, that isx > q¹⁺. We can prove a far better range forxif we ask whether (1.1), or a weakened version of it, holds for mostq: Fix >0. There existsAsuch that

π(x;q, a)−π(x) φ(q)

≤π(x) (1.2) φ(q)

for all(a, q) = 1and for allq ≤ x^1/A, except possiblythoseq that are multiples of someexceptionalmodulusr(which depends only onandx). We do not believe that any suchr exists, but if it does thenr ≥ logx, so the result (1.2) applies to most q.

Moreover, ifq is divisible byrwe can still get an understanding of the distribution of

Reçu le 13 janvier 2010 et, sous forme définitive, le 30 octobre 2010.

primes in the arithmetic progressions(modq), albeit rather different from what we had expected: There exists a real characterψ (modr)such that

π(x;q, a)− π(x) φ(q)

=ψ(a)

π(x;q,1)−π(x) φ(q)

+O

π(x)

φ(q)

, (1.3)

whenever(a, q) = 1andrdividesq, withq≤x^1/A.

In the literature (e.g.[3, 15]) exceptional moduli are derived and discussed in terms ofL-functions and egregious counterexamples to the Generalized Riemann Hypothesis known as Siegel-Landau zeros. Our goal here is to develop this same theory without L-functions and in a way that generalizes to a wide class of functions. The first step of our approach comes in recognizing that results about the distribution of prime numbers can usually be rephrased as results about mean values of the Möbius function µ(n) or the Liouville functionλ(n), both multiplicative functions (see,e.g.[23]), as we run through certain sequences of integersn. To see this, write the von Mangoldt function as

Λ(n) =X

d|n

µ(d) log(n/d)

(whereΛ(p^e) = logpfor primepand integere≥1, andΛ(n) = 0otherwise), which is deduced fromlog = 1∗Λby Möbius inversion. Nowlogis a very “smooth” function, so that estimates for the sum ofΛ(n)fornin an arithmetic progression, follow from estimates for the mean value ofµin certain related arithmetic progressions. Thus (1.1) is equivalent to the statement that the mean value of µin any arithmetic progression (mod q)tends to 0 as we go further and further out in the arithmetic progression [23];

and indeed (1.2) is tantamount to the fact that

X

n≤x n≡a(modq)

µ(n)

≤x (1.2b) q

for all(a, q) = 1for allq≤x^1/A, except thoseqthat are multiples of some exceptional modulusr.

Ifq isa multiple ofr then the corresponding result forµ(n)is more enlightening than (1.3). It is well known that if (1.3) holds thenL(1, ψ)is surprisingly small, which can hold only ifψ(p) =−1for many of the “small” primesp. In other words we have µ(p) =ψ(p)for many small primesp, which implies thatµ(n)displays a bias towards looking likeψ(n). But thenµ(n)tends to look likeψ(n) =ψ(a)forn≡a (modq), so its mean value over such integersntends to be likeψ(a). All of these vagaries can be made more precise, but for now we content ourselves with the appropriate analogy to (1.3):

X

n≤x n≡a(modq)

µ(n) = ψ(a) X

n≤x n≡1 (modq)

µ(n) +O

x q

(1.3b)

whenever(a, q) = 1andr dividesq, withq ≤ x^1/A. The key point is to see thatµ pretendsto beψ, at least at small values of the argument in this exceptional case. This notion generalizes very well.

This paper is concerned with determining, for multiplicative functionsf such that

|f(n)| ≤1for alln, estimates for the mean values 1

x/q

X

n≤x n≡a(modq)

f(n) (1.4)

when(a, q) = 1. We will show that for any fixed >0there existsAsuch that

X

n≤x n≡a(modq)

f(n)

≤x (1.2c) q

for all(a, q) = 1for allq ≤x^1/A, except possibly thoseq that are multiples of some exceptional modulusr. Moreover, if such a modulusr exists, there is a character ψ (mod r)such that ifqis a multiple ofrwithq≤x^1/A, then

X

n≤x n≡a(modq)

f(n) = ψ(a) X

n≤x n≡1 (modq)

f(n) +O

x q

(1.3c)

whenever(a, q) = 1.

There are examples off for which exceptional charactersdoexist: indeed we sim- ply letf(n) =ψ(n)or evenf(n) =ψ(n)n^itfor some|t| 1/. Thus, in this theory, exceptional characters take on a different role in that they exist for anyfwhich pretends to be a function of the formψ(n)n^it. This pretentiousness can be made more precise in terms of the followingdistance functionD=D1 between two multiplicative functions f andgwith|f(n)|,|g(n)| ≤1:

Dr(f(n), g(n);x)²:=X

p≤x p-r

1−Ref(p)g(p)

p ·

This distance function is particularly useful since it satisfies the triangle inequality Dq(f₁(n), g₁(n);x) +Dr(f₂(n), g₂(n);x)≥Dqr((f₁f₂)(n),(g₁g₂)(n);x), which may be proved by squaring both sides and using the Cauchy-Schwarz inequality (see Lemma 3.1 of [15], or the general discussion in [16]).

The main thrust of this paper has been anticipated by Elliott [7]; indeed he obtains stronger results than we do here (we will give details below). However our method of proof is somewhat different, arguably easier, so it seems to be worth recording. In a further article, the second two authors [17] re-develop Elliott’s original argument and are able to get best possible results in several of the results mentioned below.

1.2. More precise results

Throughout f is a multiplicative function with|f(n)| ≤ 1for alln, andx ≥ Q, A ≥1are given. Of all primitive characters with conductor belowQ, let ψ (modr) be that character for which

min

|t|≤A Dr(f(n), ψ(n)n^it;x)² (1.5)

is a minimum.¹Lett=t(x, Q, A)denote a value oftthat gives the minimum value in (1.5)

Theorem 1.1. With the notation as above we have, whena (modq)is an arith- metic progression with(a, q) = 1andq≤Q, that

X

n≤x n≡a(modq)

f(n)Error(x, q, A) (1.2d)

ifrdoes not divideq, and X

n≤x n≡a(modq)

f(n) = ψ(a) φ(q)

X

n≤x (n,q)=1

f(n)ψ(n) +O Error(x, q, A) (1.3d)

ifrdoes divideq. Here we may take eitherQ=x^1/Awithlogx≥A≥20and Error(x, q, A) = x

q · 1 plogA

,

orQ= logxandA= log²xwith

Error(x, q,log²x) = x

q(logx)^1/3+o(1) + x (logx)^1−o(1)·

Throughout letχ0 be the primitive character modq; and here letχ=χ0 ifr does not divideq, andχ=ψχ₀ifr does divideq. Then (1.2d) and (1.3d) can be expressed in one equation as

X

n≤x n≡a(modq)

f(n) − χ(a) X

n≤x n≡1 (modq)

f(n)Error(x, q, A).

(1.3e)

We can deduce the following bound from Corollary 2.2 below, which is useful in bound- ing (1.3d) of Theorem 1.1:

(1.6) 1 φ(q)

X

n≤x (n,q)=1

f(n)ψ(n)

x q

(1 +Dq(f(n), ψ(n)n^it;x)²)e^{−Dq(f(n),ψ(n)nit;x)2}+ 1 (logx)^1/4

. We see that (1.2c) and (1.3c) follow immediately from this theorem. The theorem gives more than (1.2c) and (1.3c) as we get results here with some uniformity, and somewhat stronger results whenxis larger thane^q.

The constant “1/3” in Theorem 1.1 cannot be improved, as we will show in [17].

To explain the idea we need to construct a multiplicative functionf for which there are two different charactersχ_j, j = 1,2such that

D(f(n), χj(n);x)²∼ 1

3log logx .

1If there are several possibilities forψ, simply pick one of those choices.

(In [17] we show that no other characters contribute much in Theorem 1.1, so that this construction indeed works.) To do this letχ be any character of order 3, and define f(p) = 1whenχ(p) = 1, andf(p) =−1otherwise. Then

D(f(n), χ(n);x)²=D(f(n), χ²(n);x)² =X

p≤x

2−χ(p)−χ(p)

p ∼ 1

3log logx

as required.

Elliott [7] gave an exponentially stronger version of our Theorem 1.1 when x is small, obtaining

Error(x, q, A) = x q · 1

A^1/4−

uniformly forA1. (Actually he proved something a little more complicated but our claim can easily be deduced from [7] and the ideas herein.) In [17] we will improve the constant “1/4” and further develop Elliott’s ideas. It is worth noting that the same circle of ideas allowed Elliott to give a new proof of Linnik’s Theorem [8] without explicit mention of zeros of DirichletL-functions.

Otherwise earlier work in this area focused on the equidistribution off(n)in arith- metic progressions; that is, on obtaining bounds for

(1.7) X

n≤x n≡a(modq)

f(n)− 1 φ(q)

X

n≤x (n,q)=1

f(n)

whenever(a, q) = 1. Following Elliott [7] we now know that it is necessary to change focus, to key in on how “pretentious” the given characters are, since Theorem 1.1 im- plies that (1.7) will not necessarily be small for moduliq divisible by an exceptional modulus r (in other words, the “expected” asymptotics can be incorrect). This issue can be seen in two of the older key articles in this subject. Elliott [4, 5] showed that (1.7) is x(log logx/logx)¹⁸ for allq except possibly for multiples of a certain ex- ceptional modulusr: note that this is non-trivial only in the rangeq ≤(logx)^18+o(1). Hildebrand [18] showed that (1.7) isx/(qp

logA)for allq≤x^1/Aexcept possibly for the multiples of at most two exceptional modulirandr⁰. Our result improves both of these, giving the same bound on (1.7) in the same range as Hildebrand, but at worst in terms of one exceptional moduli, and also understanding what happens whenq is divisible by the (putative) exceptional modulus. Our proof derives from that of Hilde- brand – our improvements here stem more from moving our focus away from bounding (1.7), to classifying when (1.4) can be large, rather than from any particular technical innovations.

One can see many of the ideas in this paper, and particularly the dichotomy between the two cases, appearing in work on elementary proofs of the prime number theorem. In particular Selberg [22] essentially shows that there is no exceptional modulus by giving (what is equivalent to) an elementary proof that the minimum in (1.5) cannot be too small. This is developed in [11] more along the lines given here.

Daboussi and Delange [3] determined when

x→∞lim 1 x/q

X

n≤x n≡a(modq)

f(n) (1.8)

exists, for a given progressiona (modq) with (a, q) = 1. If f(n) = n^it for some t6= 0, orψ(n)n^itfor some characterψ (modq)then (1.8) evidently does not exist; or even whenf(n)pretends to beψ(n)n^it. Otherwise it does:

Corollary 1.2. If f is unpretentious, that is D(f(n), ψ(n)n^it;∞) = ∞ for all charactersψand real numberst, then

X

n≤x n≡a(modq)

f(n) =o(x) (1.9)

asx→ ∞, for anya (mod q)with(a, q) = 1. If there does exist a primitive character ψ (modr)and a real numbertfor whichD(f(n), ψ(n)n^it;∞)<∞, thenψandtare unique. In this case(1.9)still holds ifrdoes not divideq. However, ifrdividesqthen

X

n≤x n≡a(modq)

f(n)

= 1 +o(1)ψ(a)x^1+it q(1 +it)

Y

p-q

1−1

p

1 +f(p)ψ(p)

p^1+it +f(p²)ψ(p²)

p^2+2it +. . .

asx → ∞. In particular if f is real-valued andχ is that real character(modq)for which

Y

p

1−1

p

1 +f(p)χ(p)

p +f(p²)χ(p²) p² +. . .

is maximized,²then

x→∞lim 1 x/q

X

n≤x n≡a(modq)

f(n) = χ(a)Y

p-q

1−1

p

1 +f(p)χ(p)

p +f(p²)χ(p²)

p² +. . . .

1.3. Mean values of multiplicative functions, twisted by Dirichlet characters

Traditionally one estimates mean values of functions in an arithmetic progression by using characters, as in the identity

X

n≤x n≡a(modq)

f(n) = 1 φ(q)

X

χ(modq)

χ(a)X

n≤x

f(n)χ(n). (1.10)

We therefore give estimates on such characters sums.

2Note that|Q

p≤x(1−1/p)(1 +g(p)/p+g(p²)/p²+. . .)| exp(−D(1, g(n);x)²).

Theorem 1.3. Defineψas above. For fixedQ≤√

xandA= log²xthe estimate

X

n≤x

f(n)χ(n)xlogQ logx

₂₀¹

holds for all charactersχof conductorq ≤ Q, except perhaps those induced fromψ.

Moreover the estimate

X

n≤x

f(n)χ(n) x

(logx)^1/3+o(1)

holds for all charactersχof conductorq ≤logx, except perhaps those induced fromψ.

Whenq is small compared tox(that isq ≤logx) we may obtain good estimates on (1.4) by using Theorem 1.3 in (1.10). Moreover Theorem 1.3 for large moduli q suggests that Theorem 1.1 should hold for these moduli, but deriving this from (1.10) is not straightforward since many characters are now involved.

We now record one more consequence of this circle of ideas. Letfbe a real-valued completely multiplicative function with |f(n)| ≤ 1for alln. Proving a conjecture of Hall, Heath-Brown and Montgomery, Granville and Soundararajan showed in [13] that

X

n≤x

f(n)≥(δ1+o(1))x,

uniformly for allf, whereo(1)→0asx→ ∞, and δ1 = 1−2 log(1 +√

e) + 4 Z

√e

1

logt

t+ 1dt=−0.656999. . .

Further this lower bound is best possible, and is attained whenf(`) = 1for all primes

`≤x^1/(1+

√e), and f(`) =−1forx^1/(1+

√e) < `≤ x. For any totally multiplicative functiongwith eachg(`) = 1or−1and anyx, there exists infinitely many primespfor which _p^`

=g(`)for all primes`≤ x, as may be proved using quadratic reciprocity and Dirichlet’s theorem for primes in arithmetic progression. Thus takingg = f we see that there exist primes p such that at least 17.15% of the integers below x are quadratic residues (modp). Here17.15%is an approximation toδ₀, and this constant δ₀ = (1 +δ₁)/2 = 0.1715. . . is best possible. We now give a generalization of this result to arithmetic progressions.

Corollary 1.4. Leta (modq)be a progression with(a, q) = 1. Then

lim inf

x→∞

infp

p-q

1 x/q

X

n≤x n≡a(modq)

n

p =

(δ1 ifa≡ (modq),

−1 ifa6≡ (modq).

Colloquially, ifa ≡ (modq)then at least17.15%of the integersn ≤xsuch that n≡a (modq)are quadratic residues (modp)for any primep.

2. Hallowed Halász

Our main results rest on the large sieve and development of Halász’s pioneering results on mean values of multiplicative functions, given here after incorporating sig- nificant refinements due to Montgomery and also Tenenbaum [24] (and see [14] for an explicit version).

Theorem 2.1 (Halász). For anyT ≥1we have 1

x X

n≤x

f(n) max

|t|≤T

1 +D(f(n), n^it;x)²

e^{−D(f(n),nit;x)2} + 1

√ T ·

Halász’s result shows that if the mean value off is large, thenf(p)pretends to bep^it for some small value oft, the quantityDgiving an appropriate measure of the distance betweenf(p) and the valuesp^it. Note that iff(p) = p^it then the mean value off is indeed large sinceP

n≤xn^it∼x^1+it/(1 +it)asx→ ∞.

We need the following consequence of Theorem 2.1.

Corollary 2.2. For1≤T ≤(logx)¹² selecttas in Theorem 2.1. Ifris an integer, smaller than or equal to√

x, then 1

φ(r) r x

! X

n≤x (n,r)=1

f(n)

1 +Dr(f(n), n^it;x)²

e^{−Dr(f(n),nit;x)2} + 1

√ T ·

Halász’s theorem allows us to estimateP

n≤xf(n)χ(n), and deduce that this mean value is small unless f pretends to be χ(p)p^it for some small t. We will show in Lemma 3.1 below thatfcan pretend to beχ(p)p^itfor at most one characterχ (modq) providedqis not too large, and hence Halász’s theorem implies Theorem 1.3 for large moduliq.

Proving an old conjecture of Erd˝os and Wintner, Wirsing [25] showed that every real-valued multiplicative functionfwith|f(n)| ≤1has a mean value; and it is natural to ask whether this holds asnranges over an arithmetic progression: that is, whether (1.8) exists? We determined this mean value in Corollary 1.2 above, and then gave the analogy to Wirsing’s result for arithmetic progressions. These results rest on a further beautiful result of Halász, which gives a qualitative version of Theorem 2.1.³

Theorem 2.3 (Halász). If D(f(n), n^it;∞)<∞for some real numbert, then X

n≤x

f(n) = 1 +o(1) x^1+it 1 +it

Y

p

1− 1

p

1 +f(p)p^−it

p +f(p²)p^−2it

p² +. . . ,

asx→ ∞, where theo(1)→0in a manner that depends onf. If D(f(n), n^it;∞) =∞ for alltthenP

n≤xf(n) =o(x)asx→ ∞

3Although unrelated to the main topic of this paper, we take this opportunity to mention a beautiful, but not widely known consequence of these results of Halász and Wirsing. LetGbe a finite abelian group, and letf :N→Gbe multiplicative. Then for anyg∈Gthe densitylimx→∞1

x#{n≤x:f(n) =g}

exists! This is stated as a Conjecture “of the sixteen year old Hungarian mathematician I. Ruzsa” by Erd˝os [9].

LetSbe a subset of the unit disc such that|arg(1−z)| ≤θ < π/2for allz ∈S (here we take the argument to always be between−πandπ). We can also deduce from Theorem 2.3 that iff is such that f(p) ∈ S for all p, then the limit in (1.8) exists (extending Theorem 1.3).

Proof of Corollary 2.2. By Theorem 4 of [14] and the last two displayed equa- tions in the proof of Corollary 3 of [14],⁴we have, fortselected withDr(f(n), n^it;x) minimal andd≤√

x,

X

n≤x/d

f(n) = 1 d^1+it

X

n≤x

f(n) +O x d

log logx (logx)²⁻

√3

! .

We deduce, from the combinatorial sieve that, forr ≤√ x,

r φ(r)

X

n≤x (n,r)=1

f(n)

=Y

p|r

1− f(p)

p^1+it 1−1 p

−1

X

n≤x

f(n) +O r

φ(r) 2

x log logx (logx)²⁻

√ 3

! .

There is no loss of generality in settingf(p) =p^itfor each primepdividingr, and so we deduce the result since2−√

3>1/4.

3. Pretentious characters are repulsive

We show thatf cannot pretend to be two different functions of the formψ(n)n^it: Lemma 3.1. For each primitive character ψ with conductor below logx select

|t| ≤ log²x for whichD(f(n), ψ(n)n^it;x) is minimal, and then label these pairs so that(ψj, tj) is that pair which gives thej-th smallest distance D(f(n), ψj(n)n^itj;x).

Letr_jbe the conductor ofψ_j. For eachj ≥1we have

Drj(f(n), ψ_j(n)n^itj;x)² ≥ 1− 1

√j

log logx+O p

log logx .

Proof. This is essentially Lemma 3.4 of [16], except that there we dealt with the case wheref itself was a character, and took allt_j = 0. The same proof applies, and

4We take this opportunity to correct an error in the statement of those two equations: in each, one should divide the sum on the right side of the equation by the number of terms in the sum.

for completeness we give the details. Note that D(f(n), ψ_j(n)n^itj;x)²≥ 1

j

j

X

k=1

D(f(n), ψ_k(n)n^itk;x)²

≥ 1 j

X

p≤x

1 p

j

X

k=1

1−Ref(p)ψ_k(p)p^itk

≥ 1 j

X

p≤x

1 p

j−

j

X

k=1

ψ_k(p)p^itk

.

By Cauchy-Schwarz we have that X

p≤x

1 p

j

X

k=1

ψ_k(p)p^itk

2

≤ X

p≤x

1 p

X

p≤x

1 p

j

X

k=1

ψ_k(p)p^itk

2 .

The first factor above is equal tolog logx+O(1), while the second term is bounded above by

X

p≤x

1 p

j+ X

1≤k,`≤j k6=`

ψ_k(p)ψ<sub>`</sub>(p)p<sup>i(tk−t`)</sup>

=jlog logx+O(j²),

upon using the prime number theorem in arithmetic progressions. For the transition fromDtoDrsimply note thatP

p|r1/plog log log logx. The Lemma follows.

Corollary 3.2. Ifmis a positive integer for whichf^mis real-valued and

D(f(n), ψ(n)n^it;x)² ≤ 1

16m² log logx with|t| ≤ _m¹ log²x, thenψ^mis real and|t| ¹

m√ logx· Proof. Letm= 1. Taking complex conjugates we have

D(f(n), ψ(n)n^it;x) =D(f(n), ψ(n)n^−it;x).

By Lemma 3.1 this is impossible unlessψ =ψ, that isψis real. Now, by the triangle inequality we have

D(f(n)², n^2it;x)≤2D(f(n), ψ(n)n^it;x).

Asf²≥0we have

D(f(n)², n^2it;x)² ≥X

p

1/p

where the sum is over those primesp ≤ x withcos(2tlogp) < 0. Since we get the

“expected” number of primes in intervals[y, y+y^1−δ]for some fixedδ > 0, we can deduce that

X

p

1 p ≥ 1

2

log logx−log(max{1/|t|,log|t|}) +O(1)

for|t|logx≥1, which implies that|t| 1/p

logx, as required. For largermwe use the triangle inequality to note that

D(f(n)^m, ψ^m(n)n^imt;x)≤mD(f(n), ψ(n)n^it;x)

and the result then follows from them= 1case.

Lemma 3.3. With the hypothesis of Lemma 3.1, we have Dr2(f(n), ψ2(n)n^it2;x)² ≥

1

3 +o(1)

log logx .

Proof. The proof of Lemma 3.1 gives D(f(n), ψ2(n)n^it2;x)² ≥X

p≤x

1 p

1−1 2

2

X

k=1

ψk(p)p^itk

=X

p≤x

1 p

1−1 2

1+χ(p)p^it

,

taking χ = ψ₂ψ₁ and t = t₂ −t₁. By the prime number theorem for arithmetic progressions, this equals

1 φ(q)

X

a(modq) (a,q)=1

Z _logx

1

1− 1

2

1 +χ(a)e^itv

dv

v +o(log logx).

Ifχhas order m > 1then there are exactlyφ(q)/mvalues ofj (modm) for which χ(a) =e^2iπj/m, and so our integral equals, takingv= 2u,

1 m

m−1

X

j=0

Z logx 1

1−

cos(tu+πj/m)

du

u +o(log logx). (3.1)

Now considerRlogx

1 |cos(tu+β)|du/ufor arbitraryβ. Ifu≤/tthen cos(tu+β) = cos(β) +O().

The contribution of u in the range /t < u < 1/(t) to the integral is O(1). For u ≥ 1/(t) we cut the range up into intervals [U, U + 2π/t), and use the fact that 1/u= 1/U +O(1/U²t), to obtain

Z U+2π/t U

|cos(tu+β)| du u =

Z U+2π/t U

1 2π

Z 2π

w=0

|cosw|dw+O(1/tu) du

u · The total error that arises here is R

u≥1/(t)du/(tu²). We deduce that (3.1) equalslog logxtimes

α· 1 2π

Z 2π

w=0

(1− |cosw|)dw+ (1−α)· 1 m

m−1

X

j=0

(1− |cos(πj/m)|) +o(1)

for someαin the range0≤α≤1. Now 1

2π Z 2π

w=0

(1− |cosw|)dw= 1− Z 1/2

−1/2

cos(πθ)dθ= 1−2/π .

Moreover, 1− 1

m

X

−m/2<j≤m/2

cos(πj/m) =

(1−1/(msin(π/2m) ifmis odd, 1−1/(mtan(π/2m) ifmis even.

The minimum occurs form= 3, and the result follows.

Our next lemma (which is essentially Proposition 7 of [16]) formulates a similar repulsion principle involving characters of substantially larger conductors, at the cost of obtaining a smaller separation.

Lemma 3.4. Letχ (modq)be a non-principal character (withq≥3), andt∈R. There is an absolute constantc >0such that for allx≥qwe have

Dq(1, χ(n)n^it;x)² ≥ 1 2log

clogx log(q(1 +|t|))

·

Consequently, iffis a multiplicative function, andχandψare any two characters with conductorsq, r ≤Q, respectively, such thatχψis non-principal, then forx ≥Qwe have

Dq(f(n), χ(n)n^it;x)²+Dr(f(n), ψ(n)n^iu;x)²≥ 1 8log

clogx

2 log(Q(1 +|t−u|))

·

Proof. For completeness we sketch the proof. We consider dχ,t(n) = X

ab=n

χ(a)a^itχ(b)b^−it.

The proof of the Pólya-Vinogradov inequality is easily modified to show that X

n≤N

χ(n)(φ(q)/q)√ qlogq .

Using this and partial summation, we see that X

n≤x

χ(n)n^it=x^itX

n≤x

χ(n)−it Z x

1

u^it−1X

n≤u

χ(n)du

φ(q) q

√q(logq)(1 +|t|logx).

Therefore, grouping the termsab=naccording to whethera≤√

xorb≤√ x < a, we obtain

X

n≤x

dχ,t(n) φ(q)

q

2 √

qx(logq)(1 +|t|logx). (3.2)

We write

d(n) =X

`|n

dχ,t(n/`)h(`)

whereh is a multiplicative function withh(p) = 2−2 Re(χ(p)p^it), and notice that

|h(n)| ≤ d₄(n)for all n, whered₄(n) is the divisor function counting the number of factorizations ofninto four factors. Then

xlogx+O(x) =X

n≤x

d(n) =X

`≤x

h(`) X

n≤x/`

d_χ,t(n).

When`≤x/(q<sup>2</sup>(1 +|t|)<sup>2</sup>)we use (3.2) to bound the sum overn. For larger`we use the fact that |d_χ,t(n)| ≤ χ0(n)d(n) whereχ0(n) is the principal character(modq).

LetB = (logq)^q/φ(q). Thus ifx/B≤`≤xthen X

n≤x/`

|d_χ,t(n)| ≤ X

n≤x/`

d(n) x

` log(ex/`)

x

` logB x

` q

φ(q) log logq x

` logq

φ(q) q

2

,

sinceq/φ(q)log logq. Now, ifN > Bthen X

n≤N

χ0(n)d(n) = X

ab≤N (ab,q)=1

1

X

a≤√ N (a,q)=1

X

b≤N/a (b,q)=1

1 X

a≤√ N (a,q)=1

φ(q) q

N

a

φ(q) q

2

NlogN

by the sieve which we use for`in the rangex/(q(1 +|t|))<sup>2</sup> ≤`≤x/BwithN =x/`.

Combining these estimates we obtain

xlogx X

`≤x/(q(1+|t|))²

|h(`)|

φ(q) q

2

pqx/`(logq)(1 +|t|log(x/`))

+ X

x/(q(1+|t|))²<`≤x

|h(`)|

φ(q) q

2

x

` log(q(1 +|t|))

φ(q)

q 2

xlog(q(1 +|t|))X

`≤x

|h(`)|

` ·

Since

X

`≤x

|h(`)|/`exp X

p≤x

|h(p)|/p

= exp

2D(1, χ(n)n^it;x)²

we obtain the first statement of the Lemma.

To obtain the second estimate note that the triangle inequality gives (Dq(f(n), χ(n)n^it;x) +Dr(f(n), ψ(n)n^iu;x))²

≥(Dqr(f(n), χ(n)n^it;x) +Dqr(f(n), ψ(n)n^iu;x))²

≥X

p≤x p-qr

1−Re|f(p)|²χ(p)ψ(p)p^i(t−u) p

≥ 1 2

X

p≤x p-qr

1−Reχ(p)ψ(p)p^i(t−u) p

,

and now we appeal to the first part of the Lemma.

4. The results for small moduli

Proof of Theorem 1.3 for small moduli. Select ψ as at the start of section 1.2 with Q = logx and A = log²x. Let χ⁰ be the primitive character that inducesχ (mod q). By assumptionχ⁰ 6=ψ. By Lemma 3.3 we know that for all|t| ≤log²x,

D(f(n), χ⁰(n)n^it;x)²≥1

3 +o(1)

log logx . Plainly,

D(f(n), χ(n)n^it;x)² =D(f(n), χ⁰(n)n^it;x)²+O X

p|q

1 p

≥1

3+o(1)

log logx .

Theorem 1.3 for small moduli now follows from Halász’s Theorem 2.1.

Proof of Theorem 1.1 for small moduli. From the discussion above, we know that for anyχ(modq)not induced byψwe have (for all|t| ≤log²x) the inequality

D(f(n), χ(n)n^it;x)² ≥1

3 +o(1)

log logx .

By a similar argument, Lemma 3.1 implies that for all but at mostp

log logxcharacters (modq)we have

D(f(n), χ(n)n^it;x)² ≥log logx+O p

log logx ,

for all|t| ≤log²x. Therefore, from (1.10), Halász’s Theorem 2.1 and these estimates, it follows that, forr -q, we have

X

n≤x n≡a(modq)

f(n) x

q(logx)^1/3+o(1) + x (logx)^1−o(1)·

Whenr

qthere is an extra contribution to (1.10) from the characterψ˜ (modq)which is induced byψ (mod r). Thus in this case we obtain

X

n≤x n≡a(modq)

f(n) = ψ(a) φ(q)

X

n≤x

f(n)ψ(n) +e O x

q(logx)^1/3+o(1) + x (logx)^1−o(1)

,

which gives the result.

Proof of Corollary 1.2. If f is unpretentious then the result follows from Theo- rem 1.1 for small moduli and (1.6); and similarly if f is pretentious but r does not divide q. Iff is pretentious andr dividesq then the result follows from (1.3d) and Theorem 2.3. To see thatψ andtare unique, suppose we have another pairψ⁰ andt⁰ so thatD(ψ⁰(p)p^it0, ψ(p)p^it;∞) < ∞by the triangle inequality, and thusL(s, ψ⁰ψ) has a pole at1 +i(t⁰−t), which is false. Finally, iff is real-valued then, by takingx arbitrary large in Corollary 3.2, we discover thatψis real, andtis zero, and the result

follows.

Proof of Corollary 1.4. If|P

n≤x,n≡a(modq) n p

| x/q, then

X

n≤x,n≡b(modq)

n p

x/q

whenever(b, q) = 1by two applications of (1.3e). Selectc (modq) so thatψ(c) is as close as possible toiand letb≡ a/c (modq); we then deduce from (1.3e) thatψ must be real since both sums are real (see also Corollary 3.2). Therefore (1.3d) tells us that our sum is

(ψ(a)/φ(q))X

n≤x

n p

ψ(n) +o(x/q). This is

(ψ(a)/q)X

n≤x

g(n) +o(x/q)

wheregis the multiplicative function with g(p) =

( n p

ψ(p) ifp-q, 1 ifp|q, by Proposition 4.4 of [13] (withφ=π/2and= logq/logx).

Now suppose thata6≡ (modq), so that there is a real characterψ (modq)for whichψ(a) =−1. There exist infinitely many primespfor which _p^`

=ψ(`)for any prime`-qwith`≤x, by quadratic reciprocity and Dirichlet’s theorem. In this case

X

n≤x n≡a(modq)

n p

=ψ(a)x

q +O(1)

=−x

q +O(1),

and so the result follows.

Now suppose that a ≡ (modq). Then ψ(a) = 1 for all real characters ψ (modq)and so our sum is(1/q)P

n≤xg(n) +o(x/q). The result then follows from

Corollary 1 of [13], the minimal value being obtained by selecting prime p so that

` p

= 1for all primes ` ≤ x^1/(1+

√e) and _p^`

= −1 for x^1/(1+

√e) < ` ≤ x as described in section 1.3, and withψbeing the trivial character (mod 1), so that

X

n≤x n≡a(modq)

n p

= 1

φ(q) X

n≤x (n,q)=1

n p

+o(x) =

δ₁+o(1)x q ·

(The reader can either directly verify the last equality, or consult [13].)

5. The case of large moduli

Proof of Theorem 1.3 for large moduli. Let us takeT = (logx/logQ)¹⁰¹ in The- orem 2.1. Suppose that there exists a characterψwith conductor≤Qand

X

n≤x

f(n)ψ(n)

≥xlogQ logx

₂₀¹ .

From Theorem 2.1 we conclude that for some real number|t| ≤T we have D(f(n), ψ(n)n^−it;x)² ≤ 1

20+o(1)

log logx logQ·

From Lemma 3.4 it follows that for any character χwith conductor belowQandχψ non-principal, we have

D(f(n), χ(n)n^−iu;x)² ≥ 1

14log logx logQ

,

for all|u| ≤ T. Appealing to Theorem 2.1, we obtain Theorem 1.3 for large moduli.

We now embark on the proof of Theorem 1.1 for large moduli. It is convenient to define

F(x;q, a) = X

n≤x n≡a(modq)

f(n).

We will studyF(x;q, a) using the large sieve. First we show that for many modulir, the distribution off(n)for n ≤ x,n ≡ a (modq), is uniform in sub-progressions (modr).

Proposition 5.1. Let1≤a≤q. Forη >1/p

logx, the setB(a, η)of bad moduli 1< r≤p

x/qfor which X∗

ψ (modr)

X

n≤x/q

f(nq+a)ψ(n) ≥ηx

q, (where X∗

denotes, as usual, a sum over primitive characters) satisfies X

r∈B(a,η)

1 φ(r) ≤ 2

η² · (5.1)

LetG(a, η)denote the set of good modulir ≤p

x/q, that is thoserwhich are square- free and not divisible by any element inB(a, η). Ifr ∈ G(a, η)with(r, q) = 1then

F(x;q, a) =rf(r)F(x/r;q, a/r) +Ox q

ηd(r) +

1−φ(r) r

.

Herea/rdenotes the residue classar⁻¹ (modq).

Proof. By Cauchy’s inequality and then the large sieve (as in Theorem 7.13 of [19]) we have

X

r≤√

x/q

1 φ(r)

X∗

ψ (modr)

X

n≤x/q

f(nq+a)ψ(n)

!2

≤ X

r≤√

x/q

X∗

ψ (modr)

X

n≤x/q

f(nq+a)ψ(n)

2

≤2 x q

X

n≤x/q

|f(nq+a)|² ≤2x q

2

·

The estimate (5.1) follows at once.

Now suppose thatr is a good modulus, and consider any progression b (modr) with(b, r) = 1. Sinceris square-free we check easily that

X

`|r

X∗

ψ (mod`)

ψ(b)ψ(n)

=

(φ(r/d) if(n, r) =d, andn≡b(modr/d)for somed r,

0 otherwise.

Therefore X

n≤x/q n≡b (modr)

f(nq+a) = 1 φ(r)

X

`|r

X∗

ψ (mod`)

ψ(b) X

n≤x/q

f(nq+a)ψ(n)

+O X

d|r d>1

φ(r/d) φ(r)

X

n≤x/q d|n n≡b (modr/d)

1

.

The error term above is plainly

X

d|r d>1

φ(r/d) φ(r)

x

qr = (r−φ(r)) φ(r)

x qr·

Further, sinceris good, if` >1and`dividesrthen`is good so that X∗

ψ(mod`)

ψ(b) X

n≤x/q

f(nq+a)ψ(n) X∗ ψ (mod`)

X

n≤x/q

f(nq+a)ψ(n) ≤ηx

q ·

We conclude that X

n≤x/q n≡b(modr)

f(nq+a) = 1 φ(r)

X

n≤x/q

f(nq+a)+O d(r)

φ(r)ηx q

+O

x qφ(r)

1−φ(r) r

.

If(r, q) = 1then we may choosebso that bq+a ≡ 0 (mod r), and so the left hand side of the last displayed equation equals

X

n≤x n≡a(modq)

r|n

f(n) =f(r)F(x/r;q, a/r) +O x

qr

1−φ(r) r

.

Hence, for good modulirthat are coprime toqwe have F(x;q, a) =f(r)φ(r)F(x/r;q, a/r) +O x

q

ηd(r) +

1−φ(r) r

!

=rf(r)F(x/r;q, a/r) +O x q

ηd(r) +

1− φ(r) r

! ,

proving our Proposition.

Proposition 5.2. Letx ≥ q²⁰and setA = logx/logq. Letaandbbe any two given reduced residues(modq). Then we have

F(x;q, ab)F(x;q,1) =F(x;q, a)F(x;q, b) +O x

q 1

plogA max

c(modq) (c,q)=1

|F(x;q, c)|

.

Proof. We may assume that A is large, and setη = C/p

logA for some suffi- ciently large constantC >0. We will produce two numbersrandswith the following properties:

(i) bothrandshave at most two prime factors and these prime factors are≥1/η, (ii) randsare both inG(1, η),G(a, η),G(b, η)andG(ab, η),

(iii) the ratior/slies between1−ηand1 +η, and (iv) r≡bs (mod q).

Assuming that this can be done, let us now prove Proposition 5.2. Four applications of Proposition 5.1 give:

F(x;q, ab) =rf(r)F(x/r;q, ab/r) +O(ηx/q), (5.2)

F(x;q,1) =sf(s)F(x/s;q,1/s) +O(ηx/q), (5.3)

F(x;q, a) =sf(s)F(x/s;q, a/s) +O(ηx/q), (5.4)

F(x;q, b) =rf(r)F(x/r;q, b/r) +O(ηx/q). (5.5)

By assumption we havea/s≡ab/r (modq), and moreover|r/s−1| ≤η. Hence F(x/s;q, a/s) =F(x/s;q, ab/r)

=F(x/r;q, ab/r) +O(|x/r−x/s|/q)

=F(x/r;q, ab/r) +O(ηx/(qr)), and, takinga= 1,

F(x/r;q, b/r) =F(x/s;q,1/s) +O(ηx/(qr)).

Therefore multiplying (5.2) and (5.3), and (5.4) and (5.5) we obtain that F(x;q, ab)F(x;q,1) =rsf(r)f(s)F(x/r;q, ab/r)F(x/s;q,1/s)

+O

ηx q max

c(modq) (c,q)=1

|F(x;q, c)|

=F(x;q, a)F(x;q, b) +O

ηx q max

c(modq) (c,q)=1

|F(x;q, c)|

,

which proves the Proposition.

It remains to show the existence ofrands. If there is no Siegel-Landau zero for a character(modq)then we can takerandsto be primes. To see this note that if there is no Siegel-Landau zero then minor modifications to the proof of Proposition 18.5 of [19] imply that there exist constantsc, B >0such that

X

z<p≤(1+η)z p≡a(modq)

logp= ηz φ(q)

1 +O

1

z^c/logq + 1 logz

(5.6)

for allz≥q^Band(a, q) = 1. We may assume thatA ≥B². LetP₁ denote the set of primes in the interval[q^B,p

x/q]which are in the set

G(1, η)∩ G(a, η)∩ G(b, η)∩ G(ab, η). By Proposition 5.1,

X

p∈P₁

1

p = X

q^B≤p≤√

x/q

1

p+O1 η²

· (5.7)

We divide the interval[q^B,p

x/q]into intervals of the form(z,(1 +η)z]. Any two primesrandsinP₁∩(z,(1+η)z]meet criteria (i), (ii) and (iii) above, so it only remains to satisfy criterion (iv). If criterion (iv) does not hold then the primes inP₁∩(z,(1+η)z]

lie in at mostφ(q)/2reduced residue classes(modq). Therefore, by (5.6) we must have X

z<p≤(1+η)z p∈P₁

logp≤ ηz 2

1 +O

1

z^c/logq + 1 logz

,

so that

X

p∈P1

1 p ≤ 1

2

X

q^B≤p≤√

x/q

1

p +O(1), which contradicts (5.7), sinceP

q^B≤p≤√

x/q 1/plogA.

Now assume that there is a real characterχ (modq)with a Siegel-Landau zeroβ.

Then Proposition 18.5 of [19] yields that X

p≤z p≡a(modq)

logp= 1 φ(q)

z−χ(a)z^β β +O

z log²z

(5.8)

for allz ≥ q^B and(a, q) = 1. Notice that the primes are concentrated in the residue classesc(modq)withχ(c) =−1, and it is therefore difficult to solver ≡bs(modq) in primesr andsifχ(b) = −1. However products of two primes, will now be con- centrated on the residue classesc (modq) for whichχ(c) = 1, as one may deduce directly from (5.8):

X

p1p2≤z p1p2≡a(modq)

p1>p2≥q^B

logp₁logp₂= 1

φ(q) log(√ z/q^B)

z+χ(a)z^β β +O

z logz

, (5.9)

for allz≥q^3Band(a, q) = 1.

So now let P₂ denote the set of products of two primesp1 > p2 ≥ q^B with the property thatp₁p₂ ≤p

x/q, which are in the set

G(1, η)∩ G(a, η)∩ G(b, η)∩ G(ab, η),

and imitate as best we can the proof above. Proposition 5.1 tells us, again, thatP₂ contains lots of elements, and that we can find an interval(z,(1 +η)z]containing many elements fromP₁ andP₂. Then criterion (iv) is met by choosing either two elements fromP₁(whenχ(b) = 1), or an element each fromP₁ andP₂(whenχ(b) =−1), and the proof of the Proposition is complete. Although we will not go into the details, the easiest way to formulate this proof is to combine (5.8) and (5.9) into the equation

(5.10) X

p≤z p≡a(modq)

logp+ 1 log(√

z/q^B)

X

p1p2≤z p1p2≡a(modq)

p1>p2≥q^B

logp1logp2

= 2z φ(q)

1 +O

1 logz

, for allz ≥q^3B and(a, q) = 1, removing the effect of the Siegel-Landau zero, so that we do not need to split this proof into cases depending on the value ofχ(b).

Remark. It is amusing to note that we cannot prove (5.10) under the assumption that there is no Siegel-Landau zero. In this case there is an additional1/z^c/logqin the error term, as in (5.6). It would be interesting to know whether this can be removed, so as to prove (5.10) uniformly. Note that (5.10) is a complicated formulation of Selberg’s formula [22] forzsufficiently large.

The proof of Proposition 5.2 is the only proof in this paper that requires non-trivial information about the zeros ofL-functions and it is pleasing to give an alternate proof that is “elementary”. Thus, in the spirit of Selberg’s paper, one can ask whether one can give an elementary proof of (5.10) in this range? The proof of Proposition 5.2

goes through easily with∼ 2z/φ(q) on the right side of (5.10), a formula that was proved, whether or not there is a Siegel-Landau zero, by Friedlander [10] using only sieve methods. Our proof of Proposition 5.2 can easily be modified to work provided the quantity on the right side of (5.10) is betweenz/φ(q) and3z/φ(q), forz > q^B for some fixedB >0; in fact our (non-elementary) proof gives such a result (as in the discussion of the previous paragraph) but, better, Friedlander’s proof is easily modified to give this result, so that all of the results in this paper can be proved avoiding use of zeros ofL-functions.

Lastly, we need a result which characterizes periodic functions that are almost mul- tiplicative. We will show that such functions are close to being characters. L. Babai, K. Friedl and A. Lukács [1] have explored such questions in greater generality recently, but for the sake of completeness we provide a proof.

Proposition 5.3. Let0 ≤ < 1/2. Letg : (Z/qZ)^∗ → Cbe a function with g(1) = 1, and|g(ab)−g(a)g(b)| ≤for allaandbcoprime toq. Then there exists a characterχ (modq)such that|χ(a)−g(a)| ≤/(1−2), for all(a, q) = 1.

Proof. For any characterχ (modq), define ˆ

g(χ) = X

a(modq)

g(a)χ(a).

We claim that there exists a characterχ (modq)with|ˆg(χ)| ≥(1−2)φ(q). Granting this claim, we see that for any(a, q) = 1

|(g(a)−χ(a))ˆg(χ)|=

X

b(modq)

g(a)g(b)χ(b)−χ(a) X

ab(modq)

g(ab)χ(ab)

=

X

b(modq)

(g(a)g(b)−g(ab))χ(b)

≤φ(q), which proves the Proposition.

Note that

X

χ(modq)

|ˆg(χ)|² =φ(q) X∗ a(modq)

|g(a)|² (5.11)

(where, as usual, X∗

denotes a sum over integers coprime withq). Further, X

χ(modq)

ˆ

g(χ)²g(χ) =ˆ φ(q)X∗ a,b

g(a)g(b)g(ab) =φ(q)X∗ a,b

|g(a)g(b)|²+E

where

|E| ≤φ(q)X∗ a,b

|g(ab)−g(a)g(b)||g(a)g(b)|

≤φ(q)X∗ a,b

|g(a)g(b)| ≤φ(q)²X∗ a

|g(a)|²,

so that, by (5.11), max

χ(modq)

|ˆg(χ)| ·φ(q)X∗ a

|g(a)|² ≥

X

χ(modq)

ˆ

g(χ)²ˆg(χ)

≥φ(q) X∗ a

|g(a)|²2

−φ(q)²X∗ a

|g(a)|². We deduce that there exists aχ (modq)with |g(χ)| ≥ X∗

a|g(a)|²−φ(q). Now, using Cauchy’s inequality, we have

X^∗

a

|g(a)|² ≥

X^∗

a

g(a)g(a⁻¹)

≥(1−)φ(q),

since|g(1)−g(a)g(a⁻¹)| ≤, so that we may conclude the existence of aχ (modq)

with|ˆg(χ)| ≥(1−2)φ(q), as claimed.

Proof of Theorem 1.1 for large moduli. Select c (modq) with (c, q) = 1 for which|F(x;q, c)|is maximized. We may suppose that

|F(x;q, c)| x q

1 plogA

,

else there is nothing to prove. With this assumption, taking a = b = c in Proposi- tion 5.2, and noting that|F(x;q, c²)| ≤ |F(x;q, c)|, we deduce that

|F(x;q,1)| ≥ 1

2|F(x;q, c)|. (5.12)

Let us definegby settingF(x;q, a) =g(a)F(x;q,1). Thengis periodic with periodq, g(1) = 1, and by Proposition 5.2 we have

|g(ab)−g(a)g(b)|=O 1 plogA

x q|F(x;q,1)|

·

By Proposition 5.3 it follows that there is a character χ (modq) such that for all (a, q) = 1

|g(a)−χ(a)|=O 1 plogA

x q|F(x;q,1)|

; (5.13)

that is

F(x;q, a) =χ(a)F(x;q,1) +Ox q

1 plogA

which is (1.3e); and we also deduce that X

n≤x

f(n)χ(n) = X

a(modq)

χ(a)F(x;q, a) =F(x;q,1) X

a(modq)

χ(a)g(a) (5.14)

=φ(q)F(x;q,1) +O 1

plogAxφ(q) q

.

Ifχψis non-principal then

Dq(f, χ(n)n^−it;x)² ≥ 1 17logA,

for all|t| ≤Aby Lemma 3.4. Therefore by Corollary 2.2 we have X

n≤x

f(n)χ(n) φ(q) q

x A^1/20

,

which implies the result, using (5.12) and (5.14), whenrdoes not divideq. Ifrdivides q, withχψnon-principal, then, proceeding as above,

X

n≤x (n,q)=1

f(n)ψ(n) =F(x;q,1) X

a(modq) (a,q)=1

ψ(a)g(a) =O xφ(q)

q 1 plogA

,

by (5.13) and the orthogonality of the charactersχandψ, which gives the result.

Ifr|qandχis induced byψthen from (5.14) we get F(x;q, a) =g(a)F(x;q,1) =χ(a)F(x;q,1) +Ox

q 1 plogA

= ψ(a) φ(q)

X

n≤x (n,q)=1

f(n)ψ(n) +O x

q 1 plogA

,

as desired.

Acknowledgements. We are grateful to Ben Green for some interesting conver- sations regarding Proposition 5.3, and for drawing our attention to [1], and to John Friedlander for drawing our attention to [10]. We are also grateful to the referee for his suggestions.

The first author would like to acknowledge the support of the Hungarian National Science Foundation, as well as of the Centre de recherches mathématiques in Montréal.

The second author was partially supported by a grant from Natural Sciences and En- gineering Research Council of Canada. The third author is partially supported by the National Science Foundation and the American Institute of Mathematics.

R

[1] L. Babai, K. Friedl and A. Lukács,Near representations of finite groups, preprint.

[2] E. Bombieri, Le grand crible dans la théorie analytique des nombres, Second Edition, Astérisque No. 18 (1987), 103 pp.

[3] H. Daboussi and H. Delange,Quelques propriétés des fonctions multiplicatives de module au plus égal à1, C. R. Acad. Sci. Paris Sér. A278(1974), 657–660.

[4] H. Davenport,Multiplicative number theory, Second edition. Revised by Hugh L. Mont- gomery. Graduate Texts in Mathematics, 74, Springer-Verlag, New York-Berlin, 1980, xiii+177 pp.

[5] P. D. T. A. Elliott,Multiplicative functions on arithmetic progressions, Mathematika34 (1987), no. 2, 199–206.

[6] P. D. T. A. Elliott,Multiplicative functions on arithmetic progressions II, Mathematika35 (1988), no. 1, 38–50.

[7] P. D. T. A. Elliott,Multiplicative functions on arithmetic progressions VII. Large moduli, J. London Math. Soc. (2)66(2002), no. 1, 14–28.

[8] P. D. T. A. Elliott,The least prime primitive root and Linnik’s theorem, in Number theory for the millennium, I (Urbana, IL, 2000), 393–418, A K Peters, Natick, MA, 2002.

[9] P. Erd˝os,Some problems in number theory, in Computers in Number Theory (eds. Atkin and Birch), 405–414, Academic Press, London-New York, 1971.

[10] J. B. Friedlander,Selberg’s formula and Siegel’s zero, in Recent progress in analytic num- ber theory, Vol. 1 (Durham, 1979), 15–23, Academic Press, London-New York, 1981.

[11] A. Granville,On elementary proofs of the prime number theorem for arithmetic progres- sions, without characters, in Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), 157–194, Univ. Salerno, Salerno, 1992.

[12] A. Granville and K. Soundararajan,Large character sums, J. Amer. Math. Soc.14(2001), 365–397.

[13] A. Granville and K. Soundararajan, The spectrum of multiplicative functions, Ann. of Math. (2)153(2001), no. 2, 407–470.

[14] A. Granville and K. Soundararajan, Decay of mean values of multiplicative functions, Canad. J. Math.55(2003), no. 6, 1191–1230.

[15] A. Granville and K. Soundararajan,Large character sums: pretentious characters and the Pólya-Vinogradov theorem, J. Amer. Math. Soc.20(2007), no. 2, 357–384.

[16] A. Granville and K. Soundararajan,Pretentious multiplicative functions and an inequality for the zeta-function, in Anatomy of integers, 191–197, CRM Proc. Lecture Notes, 46, Amer. Math. Soc., Providence, RI, 2008.

[17] A. Granville and K. Soundararajan,The pretentious large sieve, to appear.

[18] A. Hildebrand, Multiplicative functions on arithmetic progressions, Proc. Amer. Math.

Soc.108(1990), no. 2, 307–318.

[19] H. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004.

xii+615 pp.

[20] H. L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, 84, American Mathematical Society, Providence, RI, 1994. xiv+220 pp.

[21] H. L. Montgomery and R. C. Vaughan,Exponential sums with multiplicative coefficients, Invent. Math.43(1977), no. 1, 69–82.

[22] A. Selberg,An elementary proof of the prime number theorem for arithmetic progressions, Canadian J. Math.2(1950), 66–78.

[23] H. N. Shapiro,Some assertions equivalent to the prime number theorem for arithmetic progressions, Comm. Pure Appl. Math.2(1949), 293–308.

[24] G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge Studies in Advanced Mathematics, 46, Cambridge University Press, Cambridge, 1995.

xvi+448 pp.

[25] E. Wirsing,Das asymptotische Verhalten von Summen über multiplikative Funktionen II, Acta Math. Acad. Sci. Hungar.18(1967), 411–467.

A. BALOG, ALFRÉDRÉNYIINST.OFMATH., POB 127, 1364 BUDAPEST, HUNGARY

[email protected]

A. GRANVILLE, DÉP. DEMATH. ETSTAT., U.DEMONTRÉAL, CASE POSTALE6128,SUCCURSALE

CENTRE-VILLE, MONTRÉALQC H3C 3J7, CANADA

[email protected]

K. SOUNDARARAJAN, DEPT.OFMATH., 450 SERRAMALL, BLDG. 380, STANFORDU., PALOALTO

CA 94305, USA

[email protected]