EXPONENTIAL SUMS WITH MULTIPLICATIVE COEFFICIENTS

Abstract. We show that if an exponential sum with multiplicative coefficients is large then the associated multiplicative function is “pretentious”. This leads to applications in the circle method.

1. Introduction 1a. Exponential sums with multiplicative coefficients. Throughout f is a totally multiplicative function with |f(n)| ≤ 1 for all n. Diverse investigations in analytic number theory involve sums like X S(α, N) := f(n)e(nα) n≤N where e(t) = e2iπt. Typically one wants non-trivial upper bounds on |S(α, N)|, which is usually less tractable on the major arcs (when α is close to a rational with small denomi- nator) than on the minor arcs (that is, all other α ∈ [0, 1]). Moreover one also often wants asymptotic estimates for S(α, N) on the major arcs, though hitherto this has only been achieved in certain special cases. Our main result is to give such an estimate for all such f (and we will explore several applications). First though we need to introduce a distance function D between two multiplicative functions f and g with |f(n)|, |g(n)| ≤ 1:1

X 1 − Re f(p)g(p) (f(n), g(n); x)2 := . Dm p p≤x p-m

Suppose that x ≥ R and T ≥ 1 are given. Select a primitive character ψ (mod r) with r ≤ R, and a real number t with |t| ≤ T for which D(f(n), ψ(n)nit; x) is minimized (where D = D1). Define κ(m) = 0 unless m is an integer, in which case

Y κ(m) := ((f(p)/pit)b − ψ(p)(f(p)/pit)b−1). pbkm

1 This distance function is particularly useful since it satisfies the triangle inequality D(f1(n), g1(n); x)+ D(f2(n), g2(n); x) ≥ D((f1f2)(n), (g1g2)(n); x), (see Lemma 3.1 of [7], or the general discussion in [8]).

Typeset by AMS-TEX 1 2 EXPONENTIAL SUMS WITH MULTIPLICATIVE COEFFICIENTS √ Theorem 1. Let T = log N and R = log N, and select ψ and t, as above, to minimize D(f(n), ψ(n)nit; N). If (a, q) = 1, and α = a/q + β where |β| ≤ 1/qQ then (4.6) √ κ(q/r) 1 X f(n)ψ(n)  qN  S(α, N) = ψ(a)g(ψ) I(N, β, t) · + O (1 + |βN|) , φ(q) N nit (log N)1/4 n≤N

R N it where g(ψ) is the Gauss sum and I(N, β, t) := 0 e(βv)v dv. It is important to note that, once one knows ψ and t, the only part of the formula on 1/4 1 P f(n)ψ(n) the right side of (4.6) involving the values of f(p) for p > (log N) is N n≤N nit , which is independent of α. This will be a key observation in what follows. This observation should compared to the “structure theorem” [5], which is best inter- preted as follows: Define

 f(p) if p ≤ y  1 if p ≤ y fS(p) = and fL(p) = 1 if p > y f(p) if p > y, so that f(p) = fS(p)fL(p) for all primes p. If the values of f(p) are restricted to a proper, closed subset of the unit disk then the mean value of f(n) more-or-less equals the mean value of fS(n) times the mean value of fL(n). Here we take y a little smaller than a fixed power of x. The consequence pursued in [5] (and several subsequent papers) is that the mean value of fS(n) is easily shown to be an appropriate Euler product (which is easy to handle), and the mean value of fL(n) is given by the solution to a certain integral delay equation, so that the analytic theory of integral delay equations becomes our focus of study. Theorem 1, and particularly the role of f(p) on the large primes, has caused us to reconsider the “structure theorem” of [5] and we now understand that our earlier restriction of f to a proper, closed subset of the unit disk was artificial and a more natural structure theorem in easily stated. √ Theorem 2. Let T = log N and select t, as above, to minimize D(f(n), nit; N). Select any y ≤ xlog log log x/4 log log x, and define

 f(p) if p ≤ y  1 if p ≤ y fs(p) = and fl(p) = pit if p > y f(p)p−it if p > y, so that f(p) = fs(p)fl(p) for all primes p. Then

1 X 1 X 1 X  1  f(n) = f (n) · f (n) + O , x x s x l (log x)β n≤x n≤x n≤x for some constant β > 0. 1 P xit 1 P 0 0 Note also that one can replace x n≤x fs(n) here by 1+it · x n≤x fs(n) where fs(p) = −it 0 f(p)p if p ≤ y and fs(p) = 1 if p > y. EXPONENTIAL SUMS WITH MULTIPLICATIVE COEFFICIENTS 3

1b. Upper bounds on exponential sums. Montgomery and Vaughan, improving on a breakthrough of Daboussi, showed that x x (1.1) S(α, x)  + √ (log R)3/2 log x R uniformly in f, if |α − a/q| ≤ q−2 with (a, q) = 1 where R := min{q, x/q}. In the form stated this bound cannot be improved other than perhaps improving the exponent 3/2. However the upper bound in (1.1) does not take into account any information concerning f, and doing so might allow some considerable improvement. Indeed, to this end, Bachmann conjectured [B2] that

x 1 X (1.2) S(α, x)  + √ |f(n)|, log x R n≤x uniformly in f, and showed that   x (log R)1/2+o(1) X 1 − |f(p)| (1.3) S(α, x)  + x √ exp − . log x  p  R p≤x

The most interesting α in most applications are those with q small. Both (1.2) and (1.3) only take account the size of f and not of its arithmetic nature. Theorem 1 implies better bounds for small q that do take into account f’s arithmetic nature:

√ qx 1 r X −it (1.4) S(α, x)  + √ · f(n)ψ(n)n . (log x)1/4 q φ(r) n≤x

In fact if (a, r) = 1 then

√  √  r 1 X −it rx (1.5) |S(a/r, x)| = · √ · f(n)ψ(n)n + O , φ(r) 1 + t2 (log x)1/4 n≤x and so one can easily come up with many examples in which (1.4) cannot be improved. Two immediate consequences of (1.4) are √ qx x   it 2 (1.6) S(α, x)  + √ 1 + (f(n), ψ(n)nit; x)2 e−Dr (f(n),ψ(n)n ;x) . (log x)1/4 q D and the following improvement to (1.3): √ qx 1 q X (1.7) S(α, x)  + √ |f(n)|. (log x)1/4 q φ(q) n≤x 4 EXPONENTIAL SUMS WITH MULTIPLICATIVE COEFFICIENTS

1c. Applications involving the circle method. Our first application is consider the mean value of f(n) on the elements of A + B, counted with multiplicity. We will see again that the mean value can be separated into the effect of small and large primes, as in Theorem 2. Moreover the effect of large primes is pretty much independent of the particular choice of sets A and B:

√Theorem 3. Let f be a totally multiplicative function with each |f(n)| ≤ 1. Let T = log N and R = log N, and select ψ and t, as above, to minimize D(f(n), ψ(n)nit; N). Define

 f(p) if p ≤ y  1 if p ≤ y fs(p) = and fl(p) = ψ(p)pit if p > y f(p)ψ(p)p−it if p > y, so that f(p) = fs(p)fl(p). If A, B ⊂ {1, 2,...,N} with |A|, |B|  N then

1 X 1 X 1 X log y  f(a + b) = f (n) · f (a + b) + O , |A||B| N l |A||B| s y1/2 a∈A, b∈B n≤N a∈A, b∈B where y = (log N)1/6.

This is a classic local-global result. The mean value of fl(n) is the global contribution, while the mean value of fs(n) considers the effect of the small primes, and thus is the “local” contribution. It is not hard to be more explicit than Theorem 3: Corollary 1. With the hypothesis as in Theorem 2 we have

1 X Y (1 − f(p)ψ(p)/p1+it) 1 X f(n)ψ(n) f(a + b) = · × |A||B| (1 − 1/p) N nit a∈A, b∈B p≤y n≤N

X 1 X a + b log y  × κ(m) · ψ (a + b)it + O . |A||B| m y1/2 m≥1 a∈A, b∈B p|m =⇒ p≤y m|a+b

A particular case of interest is where f(p) = 1 for all p ≤ z := N 1/u and f(p) = 0 for p > z. Here ψ = 1 and t = 0. Hence κ(m) = 1 unless m = 1, so we get that a proportion ρ(u) of the sums are z-smooth (this result was already given in [], with some uniformity). Theorem 4. Let f, g, h be totally multiplicative functions with each |f(`)|, |g(m)|, |h(n)| ≤ 1. Select ψf and tf , as in Theorem 3, and similarly ψg, tg, ψh and th. For fixed non-zero integers a, b, c 1 X f(`)g(m)h(n) = N 2/2 `,m,n≤N a`+bm+cn=0 1 X 1 X 1 X 1 X log y  f (n)· g (n)· h (n)· f (`)g (m)h (n)+O , N l N l N l N 2/2 s s s y1/2 n≤N n≤N n≤N `,m,n≤N a`+bm+cn=0 EXPONENTIAL SUMS WITH MULTIPLICATIVE COEFFICIENTS 5

1/6 where y = (log N) . The main term can be taken to be 0 unless ψf ψgψh is principal.

We indicate in section 5.1 how this mean value of fs(`)gs(m)hs(n) over solutions to a` + bm + cn = 0, can be estimated by elementary methods. With an acceptably small error term, this works out to be CN i(tf +tg +th), the mean value of `itf mitg nith , where

1 Z itf itg ith C = 0≤u,v,w≤1 u v w d(au)d(bv), |abc|/2 au+bv+cw=0 Q times p≤y E(p), where E(p) are appropriate local factors for each prime p ≤ y, which can be obtained by applying the Chinese Remainder theorem to evaluate

1 X fs(`) gs(m) hs(n) , B2 `itf mitg nith `,m,n (mod B) a`+bm+cn≡0 (mod B) where B is the product of arbitrarily high powers of the primes ≤ y. (This is discussed in some detail in section 5.1.) Typically the factors are complicated (though elementary) to evaluate exactly; the simplest example being when p - c1c2c3r1r2r3 for which E(p) equals

 −1  3 3  −1 3   3 ! 1 Y f`(p)  1 Y f`(p) 1 Y f`(p)  1 − 1 − 1+it · 1 + 2 it − 1 1 − 2 it . p ψ`(p)p ` (p − 1) ψ`(p)p ` p p ` `=1  `=1 `=1 

−1  1   f(p)  1 P The factors 1 − p 1 − ψ(p)p1+it can be multiplied with the mean values N n≤N fl(n), 1 P 0 0 −it to obtain (with acceptable error), mean values N n≤N fl (n) where fl (p) = f(p)ψ(p)p for all primes p - c1c2c3r1r2r3. Analogous to Theorem 4 we have the following result. Theorem 5. Let f, g, h, . . . , y etc, be as in Theorem 4.

1 X f(`)g(m)h(n) = N 2/2 `,m,n≥1 `+m+n=N

1 X 1 X 1 X 1 X log y  f (n) · g (n) · h (n) · f (`)g (m)h (n) + O . N l N l N l N 2/2 s s s y1/2 n≤N n≤N n≤N `,m,n≥1 `+m+n=N

This mean value of fs(`)gs(m)hs(n) over solutions to ` + m + n = N, can also be estimated by elementary methods. With an acceptably small error term, this works out to be C0N i(tf +tg +th), the mean value of `itf mitg nith , where Z 0 itf itg ith C = 2 0≤u,v,w≤1 u v w dudv, u+v+w=1 6 EXPONENTIAL SUMS WITH MULTIPLICATIVE COEFFICIENTS

Q 0 0 times p≤y E (p), where E (p) are again appropriate local factors. We now present an application of Theorem 4: In [10] we showed that if f is a totally multiplicative function which can only take the values 1 and −1 then there are at least (1 − δ0)N/2 solutions to f(n) = 1 with n ≤ N, and that this is best possible, where √ √ Z e log t δ0 = −1 + 2 log(1 + e) − 4 dt = 0.656999 .... 1 t + 1 If these solutions are “randomly-distributed with respect to addition” then we would expect 3 2 at least (1 − δ0) N /16 solutions to a + b = c with 1 ≤ a, b, c ≤ N and f(a) = f(b) = f(c) = 1. Corollary 2. If f, g, h are totally multiplicative functions taking on only the values 1 and −1 then there are at most κN 2/2 solutions to a + b = c with 1 ≤ a, b, c ≤ N and 3 f(a) = g(b) = h(c) = −1, where κ = (1 + δ0) /8 = .56869 ... . Moreover there are at least κ0N 2/2 solutions to a + b = c with 1 ≤ a, b, c ≤ N and f(a) = f(b) = f(c) = 1, where 0 3 κ = (1 − δ0) /8 = .005044 ... . One amusing application of this result is to bound the number of Pythagorean triples mod p up to any given point: The number of triples x, y, z (mod p) for which x2 + y2 ≡ z2 (mod p) and the least residues of x2, y2 and z2 (mod p) all lie between 1 and N (with N < p/2), is at least 4κN 2.

2. Old and new results; setting the scene 2a. Collecting technical results. Theorem 2.1 (Hal´asz). For any T ≥ 1 we have

1 X  it 2 − (f(n),nit;x)2 1 (2.1) f(n)  max 1 + D(f(n), n ; x) e D + √ . x |t|≤T n≤x T √ From the proof of Corollary 2.2 in [3] we have, for d, r ≤ x and |t| ≤ T (≤ (log x)1/2),

X 1 Y  f(p)  X  r x log log x  (2.2) f(n) = 1 − f(n) + O · √ . d1+it p1+it φ(r) d (log x)2− 3 n≤x/d p|r n≤x (n,r)=1 √ (Note that 2 − 3 > 1/4. Remark: It would be nice to improve (log x)1/2 to (log x)2/3 for the application below.) In the second-to-last displayed equation in the proof of Corollary 3 in [6] we showed that it   X x X −it x log log x (2.3) f(n) = f(n)n + O √ . 1 + it 2− 3 n≤x n≤x (log x) In [3] we proved that f cannot pretend to be two different functions of the form ψ(n)nit: EXPONENTIAL SUMS WITH MULTIPLICATIVE COEFFICIENTS 7

Corollary to Lemmas 3.1 and 3.3 of [3]. Fix T ≤ log2 x. For each primitive character it ψ with conductor below log x select tψ, |tψ| ≤ T for which D(f(n), ψ(n)n ψ ; x) is minimal, and then label these pairs so that (ψj, tj) is that pair which gives the j-th smallest distance itj D(f(n), ψj(n)n ; x). We have

1  (f(n), ψ (n)nit2 ; x)2 ≥ + o(1) log log x; D 2 3 and itj 2 p D(f(n), ψj(n)n ; x) ≥ log log x + O( log log x); for all j ≥ log log x. 2b. Bounds on exponential sums deduced from Theorem 1. Herein we prove the results of section 1b: (1.4) is trivial if q > (log x)1/2. Otherwise take N = x and Q = x/q R N it in Theorem 1, with the facts that |I(N, β, t)| ≤ 0 |e(βv)v |dv = N and √ |g(ψ)| r r 1 pq/r √ r 1 · |κ(q/r)| ≤ 2ω(q/r) ≤ √ · 2ω(q/r) ≤ 30 √ , φ(q) φ(q) φ(r) q φ(q/r) φ(r) q to obtain (1.4). Now (1.7) follows from (1.4) by noting that r/φ(r) ≤ q/φ(q) as r|q, and each |ψ(n)n−it| ≤ 1; and (1.6) is an immediate consequence of Theorem 2.1. R N it 1+it √ Now I(N, 0, t) := 0 v dv = N /(1 + it) and |κ(r)g(r)| = r, and so Theorem 1 implies (1.5). If we select f(n) = ψ(n)nit then the main term in (1.5) is x/(r(1 + t2))1/2 and, hence, (1.4) cannot be improved when x is sufficiently large. 2c. A new structure theorem; proof of Theorem 2. Let g(p) = 1 for all p > y. Then, letting g = 1 ∗ h, we have

X X X X X X X F (n)g(n) = F (n) h(d) = h(d) F (n) = h(d)F (d) F (m), n≤x n≤x d|n d≤x n≤x d≤x m≤x/d d|n √ writing n = dm. Now by (2.2) for d ≤ x and trivial bounds for larger d, for a function F (n) where t = 0, this equals

    X h(d)F (d) X X x log log x X x · F (n) + O  |h(d)| √  + O  |h(d)|  . √ d √ d (log x)2− 3 √ d d≤ x n≤x d≤ x x

Completing the first sum this becomes

    X h(d)F (d) X X |h(d)| x log log x X x · F (n) + O  √  + O  |h(d)|  . d d (log x)2− 3 √ d d n≤x d≥1 d> x 8 EXPONENTIAL SUMS WITH MULTIPLICATIVE COEFFICIENTS so that −1 X Y  F (p)  g(p)F (p) X (2.4) F (n)g(n) = 1 − 1 − · F (n) + O (Err(h, x)) , p p n≤x p≤y n≤x where X |h(d)| x log log x X x Err(h, x) := √ + |h(d)| . d (log x)2− 3 √ d d≥1 d> x We shall apply (2.4) twice with g(p) = f(p)/pit for p ≤ y, and = 1 for p > y. First −it let F (n) = fl(n) so that f(n)n = fl(n)g(n) so, by (2.3), and then (2.4) (with t = 0) we have it   it X x X −it x log log x Rx X f(n) = f(n)n + O √ = fl(n) + O (Err(h, x)) 1 + it 2− 3 1 + it n≤x n≤x (log x) n≤x −1 Q  1   g(p)  it where R := p≤y 1 − p 1 − p . Similarly with F (n) = fs(n) = g(n)n we have X xit X  x log log x  Rx1+it fs(n) = g(n) + O √ = + O (Err(h, x)) 1 + it 2− 3 1 + it n≤x n≤x (log x) Substituting the value for R from the second equation here into the first, we obtain 1 X 1 X 1 X (2.5) f(n) = f (n) · f (n) + O ( Err(h, x)/x) x x s x l n≤x n≤x n≤x Suppose that D(f(n), nit; x)2 = α log log x. Now,   X |h(d)| X |1 − f(p)/pit|  exp , d  p  d≥1 p≤y and, using Cauchy-Schwarz,  2 X |1 − f(p)/pit| X 1 − Re(f(p)/pit)  {2+o(1)} log log x = {2α+o(1)}(log log x)2.  p  p p≤y p≤y Also, by Cauchy-Schwarz,  2   X |h(d)| X 1 X |h(d)|2 X 1 − Re(f(p)/pit) log x ≤  exp 2 ·     u+o(u) √ d √ d √ d p u d> x d> x d> x p≤y P (d)≤y it u P 1−Re(f(p)/p ) where x = y , using estimates on smooth numbers. Now p≤y p ≤ {2 + o(1)} log log x so, if u ≥ {4 + o(1)} log log x/ log log log x then this contribution to Err(h, x) √ 1/2 is  x/(log x)1/2. Overall we have Err(h, x)  x/(log x)2− 3−(2α) +o(1) in (2.5). Using Halasz’s theorem (2.1), we obtain 1 X 1 X 1 X  1  f(n) = f (n) · f (n) + O . x x s x l (log x)α+o(1) n≤x n≤x n≤x √ The result follows by combining these two results, selecting β so that β = 2− 3−(2β)1/2. EXPONENTIAL SUMS WITH MULTIPLICATIVE COEFFICIENTS 9

3. Evaluating S(a/q, N) for q ≤ log N We begin with the identity, if (b, d) = 1 then

d−1  b  X  j  1 X e = e · χ(b)χ(j). d d φ(d) j=0 χ (mod d)

Therefore if (a, q) = 1 and if f is totally multiplicative then, writing n = mq/d when (n, q) = q/d,

X X am S(a/q, N) = f(mq/d)e d d|q m≤N/(q/d) (m,d)=1 d−1 X f(q/d) X  j  X X = e f(m) χ(am)χ(j) φ(d) d d|q j=0 m≤dN/q χ (mod d) (m,d)=1 X f(q/d) X X = χ(a)g(χ) f(m)χ(m) φ(d) d|q χ (mod d) m≤dN/q (m,d)=1

Pd−1 j
which g(χ) = j=0 χ(j)e d is the Gauss sum. Now suppose χ (mod d) was induced from ψ (mod r) (where r|d) so that g(χ) = µ(d/r)ψ(d/r)g(ψ) (see Lemma 4.1 of [7]), so that the last line equals

X f(q/d) X X X = ψ(a)µ(d/r)ψ(d/r)g(ψ) f(m)ψ(m) φ(d) d|q r|d ψ (mod r) m≤dN/q ψ primitive (m,d)=1 X 1 X X f((q/r)/k) X = ψ(a)g(ψ) µ(k)ψ(k) f(m)ψ(m) φ(r) φ(k) r|q ψ (mod r) k|(q/r) m≤krN/q ψ primitive (k,r)=1 (m,kr)=1 writing d = kr and noting that if ψ(k) 6= 0 then (k, r) = 1. We now apply (2.2), where 1/2 each |tψ| ≤ T (≤ (log N) ), in the form

X 1 Y  (fψ)(p) X  k krN log log N  (fψ)(n) = 1 − (fψ)(n)+O · √ , (q/kr)1+itψ p1+itψ φ(k) q (log N)2− 3 n≤krN/q p|k n≤N (n,k)=1 to obtain a main term of

X 1 X X f((q/r)/k) µ(k)ψ(k) Y  (fψ)(p) X ψ(a)g(ψ) 1 − (fψ)(n) φ(r) (q/kr)1+itψ φ(k) p1+itψ r|q ψ (mod r) k|(q/r) p|k n≤N ψ primitive (k,r)=1 10 EXPONENTIAL SUMS WITH MULTIPLICATIVE COEFFICIENTS and an error term which is √ X 1 X √ X 1 µ(k)2k N log log N q log log N  r √  N · √ . φ(r) q/kr φ(k)2 (log N)2− 3 (log N)2− 3 r|q ψ (mod r) k|(q/r) ψ primitive (k,r)=1 We need to bound the contribution of the part of the main term that comes from the 1/2 primitive characters χj, j ≥ 2, so we take T = (log N) in the Corollary above and then in (2.1), to get that X N (fψ)(n)  √ . n≤N T Thus, in total, these characters contribute √ X 1 X √ X 1 µ(k)2k N q N  r √  √ . φ(r) (q/kr) φ(k)2 T T r|q ψ (mod r) k|(q/r) ψ primitive (k,r)=1

Finally we must consider the terms with ψ = ψ1. We have X ψ(a)g(ψ) (fψ)(n) n≤N times 1 X f((q/r)/k) µ(k)ψ(k) Y  (fψ)(p) 1 − . φ(r) (q/kr)1+itψ φ(k) p1+itψ k|(q/r) p|k (k,r)=1 Write q = RS where (R,S) = 1 and R has exactly the same prime factors as r. Then this Q becomes, with t = tψ and s = p|S p, 1 r f(R/r) X f(S/k) µ(k)ψ(k) Y  (fψ)(p) · 1 − r φ(r) (R/r)1+it (S/k)1+it φ(k) p1+it k|S p|k 1 R f(R/r) S 1 Y = · f(p)b−1(f(p) − pitψ(p)) r φ(R) (R/r)1+it φ(S) S1+it pbkS b b−1! 1 f(q/rs) Y f(p)  1 Y f(p) f(p) = − ψ(p) = − ψ(p) . φ(q) (q/rs)it pit φ(q) pit pit p|s pbk(q/r) Collecting all of this together we have proved: If ψ, t are chosen as in (1.6) then √ g(ψ) X  qN  (3.1) S(a/q, N) = ψ(a) κ(q/r) (fψ)(n) + O . φ(q) (log N)1/4 n≤N By (2.3) we can rewrite (3.1) as √ g(ψ) N it X f(n)  qN  (3.2) S(a/q, N) = ψ(a) κ(q/r) + O . φ(q) 1 + it ψ(n)nit (log N)1/4 n≤N EXPONENTIAL SUMS WITH MULTIPLICATIVE COEFFICIENTS 11

4. Evaluating S(α, N) for α irrational 4.1. The proof of Theorem 1. Theorem 1 of [9] states that if q ≤ N and (a, q) = 1 then N N S(a/q, N)  + + (qN)1/2(log(2N/q))3/2. log N φ(q)1/2 Fix Q. For any α there exists a/q with q ≤ Q and |α − a/q| ≤ 1/qQ. From the identity

Z N (4.1) S(α, N) = e((α − a/q)N)S(a/q, N) − 2πi(α − a/q) e((α − a/q)v)S(a/q, v)dv 1 we immediately deduce that

 N N  (4.2) S(α, N)  (1 + |α − a/q|N) + + (qN)1/2(log(2N/q))3/2 . log N φ(q)1/2 R q since the contribution to the integral from values of v ≤ q is  |α − a/q| 1 vdv  2 (1/qQ)q = q/Q ≤ √1. Assume Q > 2 N. If q > 2N/Q then |α − a/q|N < N/qQ < 1/2, so (4.2) yields

N (log(N/Q))3/2 (4.3) S(α, N)  + N log N (N/Q)1/2 √ If q ≤ 2N/Q < N. Select r < 2N/q so that |α − b/r| ≤ q/(2rN). Note that either r = q or 1/qr ≤ 1/qQ + q/(2rN) ≤ 1/qQ + 1/(2qr), so that r ≥ Q/2 ≥ q/2. Hence |α − b/r| ≤ 1/N so that by (4.2) (with q = r) we get

N N (log q)3/2 N (log q)3/2 (4.4) S(α, N)  + + N  + N . log N φ(r)1/2 q1/2 log N q1/2

With (4.3) and (4.4) we recover Montgomery and Vaughan’s (1.1). g(ψ) Now consider (3.1). By (4.1), S(α, N) equals ψ(a) φ(q) κ(q/r) times

X Z N X (4.5) e((α − a/q)N) (fψ)(n) − 2πi(α − a/q) e((α − a/q)v) (fψ)(n)dv n≤N 1 n≤v plus an error term which is √ √ √ qN Z N qv qN  + |α − a/q| dv  (1 + N|α − a/q|). 1/4 1/4 1/4 (log N) 1 (log v) (log N) √ Using the bound |(fψ)(n)| ≤ 1 for v ≤ N, and (2.2) otherwise, we find that (4.5) equals ( ) Z N e((α − a/q)v) X e((α − a/q)N) − 2πi(α − a/q) dv (fψ)(n) √ (N/v)1+it N n≤N 12 EXPONENTIAL SUMS WITH MULTIPLICATIVE COEFFICIENTS √ !! Z N Z N log log N +O |α − a/q| vdv + v √ dv . √ 2− 3 1 N (log N) This error term is certainly smaller than the previous one. Extending the remaining integral down to 0 gives more acceptable error term, and then integrating by parts and changing variable we get, with β = α − a/q,

Z N e(βv) (1 + it) e(βN) − 2πiβ 1+it dv = 1+it I(N, β, t). 0 (N/v) N Combining this with (2.3) we deduce Theorem 1 4.2. Exponential sum estimates. Select Q = N/((log N)1/6(log log N)). Let S(α, N) = M(α, N) + E(α, N) where

g(ψ) 1 X f(n)ψ(n) M(α) = M(α, N) := ψ(a) κ(q/r)I(N, β, t) · φ(q) N nit n≤N when q ≥ N/Q, and M(α) = 0 otherwise. Theorem 1 for q ≤ N/Q yields that √ qN  N  N 2 N(N/Q) (4.6) E(α, N)  1 +  √  ; (log N)1/4 qQ qQ(log N)1/4 (log N)1/4 combining this with (4.3) for q ≥ N/Q, we have log log N (4.7) |E(α)|  N (log N)1/12 uniformly for all α. R 1 2 P 2 Now 0 |S(α)| = n≤N |f(n)| ≤ N. Moreover, by (4.6),

a + 1 Z X X Z q qQ N 2(N/Q)2 |E(α)|2dα  dα 1/2 α: M(α)6=0 a − 1 q(log N) q≤N/Q (a,q)=1 q qQ X X N(N/Q)3 N(N/Q)3   log(N/Q) q2(log N)1/2 (log N)1/2 q≤N/Q (a,q)=1 (4.8)  N(log log N)4.

From this we deduce Z Z (4.9) |M(α)|2dα ≤ 2 (|S(α)|2 + |E(α)|2)dα  N(log log N)4, α α: M(α)6=0 and Z Z (4.10) |E(α)|2dα ≤ 2 (|S(α)|2 + |M(α)|2)dα  N(log log N)4. α α EXPONENTIAL SUMS WITH MULTIPLICATIVE COEFFICIENTS 13

5. Mean value of a multiplicative function on a weighted set of integers Let w , n ≤ N be a set of weights, and W (t) = P w e(−nt). For a given f, let √ n n n T = log N and R = log N, and select ψ and t, as above, to minimize D(f(n), ψ(n)nit; N). Now X Z 1 wnf(n) = S(α)W (α)dα. n 0 R 1 By (4.7) this is 0 M(α)W (α)dα plus an error term which is

log log N Z 1 log log N  N |W (α)|dα  N 2 , 1/12 1/12 (log N) 0 (log N)

R 1 P 2 under the assumption that 0 |W (α)|dα  N. We will also assume that n |wn|  N . Moreover

Z 1 X X g(ψ) Z 1/qQ M(α)W (α)dα = ξ · ψ(a) κ(q/r) I(N, β, t)W (a/q + β)dβ, φ(q) 0 q≤N/Q (a,q)=1 −1/qQ r|q

1 P f(n)ψ(n) where ξ := N n≤N nit . The values of f(p) with p > y := N/Q have no effect on this sum (their values appear only in ξ). Therefore we obtain the same sum in estimating P n wnfs(n) where  f(p) if p ≤ y  1 if p ≤ y fs(p) = and fl(p) = ψ(p)pit if p > y f(p)ψ(p)p−it if p > y.

Observing that   1 X fs(n)ψ(n) Y (1 − 1/p) 1 ξ0 := = + O , N nit (1 − g(p)/p) (log N)1/2 n≤N p≤y where g(p) = f(p)ψ(p)/pit, we deduce that

X Y (1 − g(p)/p) X  log log N  (5.1) w f(n) = · ξ · w f (n) + O N 2 . n (1 − 1/p) n s (log N)1/12 n p≤y n P We now consider n wnfs(n) without the circle method, using only elementary meth- P it ods. It is only different at the small primes from n wnψ(n)n which itself depends only it on how wn is distributed modulo r, and in size (for the contribution of the n ). Hence we it can proceed best by writing fs(n) = ψ(n)n g(n), when (n, r) = 1, and then considering how far g is from 1, by considering the multiplicative function f given by g = 1 ∗ h:

X X it X it X X X it wnfs(n) = wnψ(n)n g(n) = wnψ(n)n h(d) = h(d) wnψ(n)n . n n n d|n d:(d,r)=1 n, d|n (n,r)=1 14 EXPONENTIAL SUMS WITH MULTIPLICATIVE COEFFICIENTS

If p|R =⇒ p|r then the same reasoning yields

X X X it wnfs(n) = f(R) h(d) wnψ(n/R)(n/R) , n: R|n d:(d,r)=1 n: dR|n (n/R,r)=1 (n/R,r)=1

We sum this up over all such R to obtain

X X X it (5.2) wnfs(n) = κ(m) wnn ψ(n/m). n m≥1 n: m|n p|m =⇒ p≤y

Proof of Theorem 3. Let wn = #{a ∈ A, b ∈ B : n = a + b}, and proceed as above by noting that W (α) = A(α)B(α) so that

Z 1 Z 1 Z 1 Z 1 1/2 |W (α)|dα = |A(α)B(α)|dα ≤ |A(α)|2dα |B(α)|2dα = (|A||B|)1/2 ≤ N. 0 0 0 0

Combining this with (5.2) and (5.1) gives the result.

6. Circle method

Proof of Theorem 4. It is notationally convenient here to replace f, g and h by f1, f2 and f3. Now X f1(n1)f2(n2)f3(n3)

n1,n2,n3≤N c1n1+c2n2+c3n3=0 X Z 1 = f1(n1)f2(n2)f3(n3) e(t(c1n1 + c2n2 + c3n3))dt 0 n1,n2,n3≤N 3 Z 1 Y X = f`(n`)e(c`n`t)dt 0 `=1 n`≤N Z 1 (6.1) = S1(c1t)S2(c2t)S3(c3t)dt 0

Now, by (4.7),

Z 1 Z 1

S1(c1t)S2(c2t)E3(c3t)dt ≤ max |E3(c3t)| |S1(c1t)S2(c2t)|dt 0 t 0 1/2 log log N Z 1 Z 1   N |S (c t)|2dt |S (c t)|2dt 1/12 1 1 2 2 (log N) 0 0 log log N  N 2 . (log N)1/12 EXPONENTIAL SUMS WITH MULTIPLICATIVE COEFFICIENTS 15

Similarly by (4.9)

Z 1 Z 1 Z 1 1/2 2 2 S1(c1t)E2(c2t)M3(c3t)dt ≤ max |E2(c2t)| |S1(c1t)| dt |M3(c3t)| dt 0 t 0 0 (log log N)3  N 2 , (log N)1/12 and

Z 1 Z 1 Z 1 1/2 2 2 E1(c1t)M2(c2t)M3(c3t)dt ≤ max |E1(c1t)| |M2(c1t)| dt |M3(c3t)| dt 0 t 0 0 (log log N)5  N 2 . (log N)1/12

Therefore (6.1) equals

Z 1  (log log N)5  (6.2) M (c t)M (c t)M (c t)dt + O N 2 . 1 1 2 2 3 3 1/12 0 (log N)

The integrand here, divided by

1 X f1(n)ψ (n) 1 X f2(n)ψ (n) 1 X f3(n)ψ (n) (6.3) 1 · 2 · 3 , N nit1 N nit2 N nit3 n≤N n≤N n≤N retains the same value if we replace f by fs since it depends only the values of f(p) with p ≤ y (as well as ψ and t). Hence we may proceed analogously to as in the proof of Theorem 3, to obtain the estimate in Theorem 4.

We now indicate how to evaluate the mean value of fs(`)gs(m)hs(n) over solutions to a` + bm + cn = 0, by elementary methods. Let B = lcmn≤yn where x is sufficiently large that y is large in comparison to |a|, |b|, |c|, rf , rg and rh. Let us define a multiplicative b b it b b b b function F such that F (p ) = fs(p )/(ψ(p)p ) if p kB, and F (p ) = F ((p ,B)) otherwise, b b ibt when p - r, and F (p ) = fs(p )/p when p|r. For a given integer n define nf to be the largest divisor of n which is coprime to rf . We make analogous definitions for g and h. it There are very few integers ` for which f(`) differs from F (`)` ψf (`f ) (and similarly g(m) and h(n)), and indeed one can show that

1 X f (`)g (m)h (n) = N 2/2 s s s `,m,n≤N a`+bm+cn=0

1 X log y  F (`)G(m)H(n)ψ (` )ψ (m )ψ (n )`itf mitg nith + O . N 2/2 f f g g h h y1/2 `,m,n≤N a`+bm+cn=0 16 EXPONENTIAL SUMS WITH MULTIPLICATIVE COEFFICIENTS

The summand F (`)G(m)H(n)ψf (`f )ψg(mg)ψh(nh) depends only the values of `, m and n (mod B). Thus by cutting the range into boxes of side length B, we can show that

1 X log y  F (`)G(m)H(n)ψ (` )ψ (m )ψ (n )`itf mitg nith = O N 2/2 f f g g h h y1/2 `,m,n≤N a`+bm+cn=0

1 X + F (`)G(m)H(n)ψ (` )ψ (m )ψ (n )× B2 f f g g h h `,m,n (mod B) a`+bm+cn≡0 (mod B) 1 X `itf mitg nith . N 2/2 `,m,n≤N a`+bm+cn=0 The last mean value is easily estimated as the integral

1 Z itf itg ith i(tf +tg +th) 2 0≤U,V,W ≤N U V W dUdV = CN . |c|N /2 aU+bV +cW =0

For any u coprime to B, we can replace `, m, n (mod B) by u`, um, un (mod B) in the sum over B above. This does not change the range of the sum, nor the values of F,G and H, since these are all 1 on primes that do not divide B, so each term in the sum is multiplied through by (ψf ψgψh)(u). Hence the sum equals 0 unless (ψf ψgψh)(u) = 1 for each such u, that is, unless ψf ψgψh is principal. Since all the functions in the first sum are multiplicative, the sum is multiplicative. Hence it is the product, over the prime powers pbkB of the mean values mod pb. One can show that these sums can be extended to arbitrary powers of p, so as to obtain E(p). In the special case that p - abcrf rgrh one defines fp to be the multiplicative function such that fp(p) = F (p) and fp(q) = 1 for all other primes q, and similarly gp and hp, and then

1 X E(p) = lim fp(`)gp(m)hp(n) ν→∞ p2ν `,m,n (mod pν ) a`+bm+cn≡0 (mod pν )

To evaluate this we write fp = 1 ∗ Fp (and similarly gp and hp) and so the above equals

i j k 1 X i j k X X Fp(p )Gp(p )Hp(p ) lim Fp(p )Gp(p )Hp(p ) 1 = . ν→∞ p2ν pi+j+k−min{i,j,k} i,j,k≥0 `,m,n (mod pν ) i,j,k≥0 a`+bm+cn≡0 (mod pν ) pi`, pj |m, pk|n

 F (p)   G(p)   H(p)  Multiplying this through by 1 − p 1 − p 1 − p all terms cancel except those with i = j = k ≥ 1, which get multiplied by a factor (1 − 1/p)3, and those with i, j, k ≤ 1. Keeping track of what happens we obtain the formula for E(p) given in the introduction. EXPONENTIAL SUMS WITH MULTIPLICATIVE COEFFICIENTS 17

6.2. Summing to N. Proof of Theorem 5.

X f1(n1)f2(n2)f3(n3)

n1,n2,n3≥1 n1+n2+n3=N X Z 1 = f1(n1)f2(n2)f3(n3) e(t(n1 + n2 + n3 − N))dt 0 n1,n2,n3≥1 3 Z 1 Y X = e(−Nt) f`(n`)e(n`t)dt 0 `=1 n`≤N Z 1 (6.4) = e(−Nt)S1(t)S2(t)S3(t)dt 0

As in section 6.1, (6.4) equals

Z 1  (log log N)5  (6.5) e(−Nt)M (t)M (t)M (t)dt + O N 2 . 1 2 3 1/12 0 (log N)

The integrand here, divided by (6.3) retains the same value if we replace f by fs since it depends only the values of f(p) with p ≤ y (as well as ψ and t). Hence we may proceed analogously to as in the proof of Theorems 3 and 4, to obtain the estimate in Theorem 6.

7. Special case of Theorem 4

1 X f(`)g(m)h(n) = N 2/2 `,m,n≤N a`+bm+cn=0

1 X 1 X 1 X 1 X log y  f (n)· g (n)· h (n)· f (`)g (m)h (n)+O , N l N l N l N 2/2 s s s y1/2 n≤N n≤N n≤N `,m,n≤N a`+bm+cn=0

We take c1 = c2 = 1, c3 = −1 with each fj real-valued, from which we can deduce that each tj = 0 and each ψj is real-valued, if our sum is to have a significant size. We may assume that each ψj is primitive and so is the primitive quadratic character mod rj; moreover, since their product is principal, we may assume that r1 = uv, r2 = uw, r3 = vw where uvw is squarefree. Now let F (p) = f(p)ψf (p) if p - rf , and F (p) = f(p) if p|rf . By (2.2) we have

−1 1 X Y  F (p)  1 1 X f (n) = 1 − 1 − · F (n) N l p p N n≤N p≤y n≤N 18 EXPONENTIAL SUMS WITH MULTIPLICATIVE COEFFICIENTS plus an acceptable error term (and the same can be said for g and h). We saw above that if p - uvw then E(p) equals

 3 3  −1 ( Q3  −1) 1 Y F`(p) (F`(p) − 1) (F GH)(p) (7.1) 1 − 1 − · 1 + `=1 1 − . p p (p − 1)2 p2 `=1

Now suppose p|w. Then E(p) equals   1 X k i j 1 X k i j MN lim fs(p )gs(p )hs(p ) = F (p )G(p )H(p ) . ν→∞ 2ν 2ν p ν p p `,m,n (mod p ) Lpk,Mpi,Npj (mod pν ) ν `+m≡n (mod p ) Lpk+Mpi≡Npj (mod pν ) i j k p km, p kn, p k` (LMN,p)=1

There are no terms unless the minimum of i, j, k is not unique. If i = k < j then N runs equally over each residue class mod p equally often, so that subsum is 0; amnd a similar thing happens if j = k < i. Hence we may assume i = j ≤ k, and our sum becomes

ν ν X X X MN  F (pk)(GH)(pi) p i=0 k=i M,N (mod pν−i) pk−ikN−M

ν ν ν ν X X X X = − (F GH)(pi)φ(p2ν−2i−1) + F (pk)(GH)(pi)φ(pν−i)φ(pν−k) + O(1) i=0 k=i+1 i=0 k=i+1 Therefore  1  (F GH)(p)−1 F (p) − 1 E(p) = 1 − 1 − · . p p2 p − F (p) If we now consider only those functions where f(p) = ±1 then E(p) = 0 unless −1 −2(p−1)  (gh)(p)  F (p) = −1, that is f(p) = −ψf (p), whence E(p) = p(p+1) 1 + p2 .

Remark. If f(n) = 1 for all n, then ψf = 1, so that u = v = 1, and so F (n) = 1. If p|w then E(p) = 0. Hence if we want E(p) 6= 0 then there are no primes dividing w; that is ψf = ψg = ψh = 1. If p - w then we see that the final term in (7.1) is 1, and hence our mean value is the product of the mean values of G(n) and of H(n), up to N.

Proof of Corollary 2. We now only consider fj which take values 1 and −1. The number of solutions to f1(a) = f2(b) = f3(c) = −1 with a + b = c ≤ N is

1 X 1 (7.2) (1 − f (a))(1 − f (b))(1 − f (c)). N 2/2 8 1 2 3 a,b,c≤N a+b=c

We now prove that if (7.2) is > 1/2 then each ψj = 1 : If ψ1ψ2ψ3 is not principal Q then i∈I ψi is non-principal for at least four of the subsets I of {1, 2, 3}; each of the EXPONENTIAL SUMS WITH MULTIPLICATIVE COEFFICIENTS 19 corresponding mean values is o(1) by Theorem 4. When we multiply out (7.2) it is one- eighth the sum of eight averages, each of which lies between −1 and 1, and so totals ≤ 4 × 1/8 + o(1), as required.. Now suppose that ψ1ψ2ψ3 is principal, but that some ψj 6= 1. Then there are at least two ψj 6= 1 since ψ1ψ2ψ3 = 1. When we multiply out (7.2) it is the sum of eight averages, each of which lies between −1 and 1. At least four of these terms contain a ψj 6= 1 and so the mean value is o(1) by the remark before the the start of this proof. But then the value of (7.2) is ≤ 4 × 1/8 + o(1), completing the proof that each ψj = 1. Now, by the remarks at the start of this section, we know that (7.2) equals   ( −1) 1 X Y Y 8  1  (7.4)  δ + δ δ δ · 1 − 1 +  + o(1),  i 1 2 3 2 2  8 p (p − 1) p I({1,2,3} i∈I fj (p)=−1∀j

1 P where δj := − N n≤N fj(n) Evidently every δi > 0, else there are at most four positive terms in (7.4) and so the value is ≤ 4/8. Our goal now is to maximize the expression in (7.4). Now if, for a given prime p, we have fi(p) = −1 then let Fi(p) = 1, and Fi(q) = fi(q) for all primes q 6= p, so that

δFi = (1 + 2/(p − 1))δi. But then each term in (7.4) is increased, or at worst stays the same. So (7.4) is maximized by letting each fj(p) = 1 for all small primes p, so that (7.4) becomes (1 + δ1)(1 + δ2)(1 + δ3) + o(1). The maximum value that this can take is when each δi = δ0, as in [5]. To prove the second part of Corollary 2, we must minimize

(  −1) 1 2 3
Y 8 1 1 − 3δ + 3δ − Cf δ where Cf := 1 − 2 1 + 2 . 8 p (p − 1) p f(p)=−1

If δ ≤ 0 then this is ≥ 1/8 so we may assume δ > 0. The function 1 − 3δ + 3δ2 − Cδ3 is thus minimized with C maximal in its range, that is C = 1. But then (1−δ)3 is minimized when δ is maximized in its range, that is with δ = δ0. Remark. One might “guess” that the probability that f(a) = f(b) = f(c) = 1 amongst solutions to a + b = c is roughly the cube of the probability that f(a) = 1. We see in the proof of Corollary 2 that this guess is incorrect if f, g and h are all 1-pretentious, unless f(p) = 1 for all small primes p.

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