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Exponential Sums with Multiplicative Coefficients

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Exponential Sums with Multiplicative Coefficients 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 jf(n)j ≤ 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 jS(α; N)j, 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 2 [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 jf(n)j; jg(n)j ≤ 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 (mod r) with r ≤ R, and a real number t with jtj ≤ T for which D(f(n); (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 − (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 p Theorem 1. Let T = log N and R = log N, and select and t, as above, to minimize D(f(n); (n)nit; N). If (a; q) = 1, and α = a=q + β where jβj ≤ 1=qQ then (4.6) p κ(q=r) 1 X f(n) (n) qN S(α; N) = (a)g( ) I(N; β; t) · + O (1 + jβNj) ; φ(q) N nit (log N)1=4 n≤N R N it where g( ) is the Gauss sum and I(N; β; t) := 0 e(βv)v dv. It is important to note that, once one knows and t, the only part of the formula on 1=4 1 P f(n) (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. p 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) + p (log R)3=2 log x R uniformly in f, if jα − a=qj ≤ q−2 with (a; q) = 1 where R := minfq; x=qg. 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) + p jf(n)j; log x R n≤x uniformly in f, and showed that 0 1 x (log R)1=2+o(1) X 1 − jf(p)j (1.3) S(α; x) + x p exp − : log x @ p A 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: p qx 1 r X −it (1.4) S(α; x) + p · f(n) (n)n : (log x)1=4 q φ(r) n≤x In fact if (a; r) = 1 then p p r 1 X −it rx (1.5) jS(a=r; x)j = · p · f(n) (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 p qx x it 2 (1.6) S(α; x) + p 1 + (f(n); (n)nit; x)2 e−Dr (f(n); (n)n ;x) : (log x)1=4 q D and the following improvement to (1.3): p qx 1 q X (1.7) S(α; x) + p jf(n)j: (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: pTheorem 3. Let f be a totally multiplicative function with each jf(n)j ≤ 1. Let T = log N and R = log N, and select and t, as above, to minimize D(f(n); (n)nit; N). Define f(p) if p ≤ y 1 if p ≤ y fs(p) = it and fl(p) = −it (p)p if p > y f(p) (p)p if p > y; so that f(p) = fs(p)fl(p). If A; B ⊂ f1; 2;:::;Ng with jAj; jBj N then 1 X 1 X 1 X log y f(a + b) = f (n) · f (a + b) + O ; jAjjBj N l jAjjBj s y1=2 a2A; b2B n≤N a2A; b2B 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) (p)=p1+it) 1 X f(n) (n) f(a + b) = · × jAjjBj (1 − 1=p) N nit a2A; b2B p≤y n≤N X 1 X a + b log y × κ(m) · (a + b)it + O : jAjjBj m y1=2 m≥1 a2A; b2B pjm =) p≤y mja+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 = 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 jf(`)j; jg(m)j; jh(n)j ≤ 1. Select f and tf , as in Theorem 3, and similarly g; tg; h and th.
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