PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 11, November 2012, Pages 3693–3701 S 0002-9939(2012)11257-2 Article electronically published on March 2, 2012

SHORT SUMS OF MULTIPLICATIVE FUNCTIONS

VISHAAL KAPOOR

(Communicated by Matthew A. Papanikolas)

Abstract. We show that on a short interval, x

The literature is rich with asymptotic formulae for the sum of a multiplicative function over long intervals n ≤ x. In contrast, little is known about multiplicative functions summed over short intervals x0. In this article, we extend Bordell`es’s theorem to a larger class of functions, lifting the requirement that f(n) be real-valued and weakening the hypothesis on the values of f(n)on the primes. More specifically, we allow any complex-valued multiplicative function that is uniformly bounded on the prime powers and “close” to 1 on the primes.

Theorem 1. Let A and θ be positive constants with θ ≥ 4/5. Define FA,θ to be the class of complex-valued multiplicative functions f(n) which are bounded by A on prime powers and satisfy the inequality |f(p) − 1|≤Ap−θ, for all primes p. For any f ∈F if θ ≥ 4/5,then A,θ (1) f(n)=wCf + O(Δ(x, w, ε)) xx3/5. Here ε>0 is sufficiently small, and the implicit constant in the Vinogradov notation depends on A, θ and ε.

Received by the editors October 6, 2010 and, in revised form, April 21, 2011. 2010 Mathematics Subject Classification. Primary 11N37.

c 2012 American Mathematical Society Reverts to public domain 28 years from publication 3693

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When w ≤ x3/5, the three expressions x1/5+ε, x1/15+εw2/3,andx−1/10+εw dom- inate the error term in the ranges w ≤ x1/5, x1/5 x1/2 respectively. In particular this theorem is nontrivial when w ≥ x1/5+ε. We note that the range of θ can be enlarged to θ>1/2. However, in the range 1/2 <θ<4/5 the error term depends heavily on the choice of θ,soweomitthe details and refer the reader to the comments following the proof of Theorem 1. For an application of Theorem 1, we find asymptotic formulae for short sums of the functions n/ϕ(n)andσ(n)/n over squarefree positive integers n.Bothof these sums involve functions that are not identically 1 on the set of primes and that have values outside [0, 1] and thus cannot be handled using Bordell`es’s theorem. Applying Theorem 1 with θ = 1 gives the short interval asymptotic formulae: Corollary 2. For 1 ≤ w ≤ x we have both n = w + O(Δ(x, w, ε)) ϕ(n) x

where ε>0. Both of these formulae are nontrivial when w>x1/5+ε. In what follows, we will not state the implicit dependence of A, θ, or ε in the Vinogradov and big-Oh notation. We use the notation a to denote the largest integer ≤ a. The Dirichlet convolution of two arithmetic functions f and g is defined as (f ∗ g)(n)= f(a)g(b), ab=n where a, b range over positive integers. Recall that a squarefree number is a natural number not divisible by the square of any prime, and a squarefull number is a natural number with every prime factor appearing with multiplicity at least 2. We outline our strategy to prove Theorem 1. We begin by establishing two simple lemmata and stating the more technical Lemma 5; this is followed by the proof of Theorem 1. We will require two results to prove Lemma 5. We then state these results and prove the lemma. Lemma 3. Let ε>0. For z ≥ 1 we have (2) mε  z1/2+ε m≤z m squarefull and we have 1 1 (3)  m1−ε z1/2−ε m>z m squarefull if ε<1/2.

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Proof. We prove the second result; the proof of the first is analogous. Let S(t)be the number of squarefull numbers ≤ t. Using integration by parts on the Riemann- Steltjies integral, ∞ ∞ ∞ 1 1 S(t) S(t) (4) = dS(t)  + dt. m1−ε t1−ε t1−ε t2−ε m>z z z z m squarefull √ Golomb has shown that S(t)  t (see [4]), so (4) is  z−1/2+ε, as required. 

Lemma 4. Suppose f ∈FA,θ for some A>0,andθ ≥ 1/2+ε,whereε>0.Let g = f ∗ μ. Then for y ≥ 1 we have |g(d)|y1/2+ε d≤y

and we have g(d)  yε−1/2 d d>y when ε<1/2.

Proof. On primes p, |g(p)|≤A/pθ andonhigherprimepowers|g(pk)|≤2A.Thus for squarefree n, one has

(5) g(n)  nε−θ

ω(n) ε following from the well-known upper bound A A,ε n . For arbitrary n,we have

(6) g(n)  nε. | | We use (5) and (6) to find upper bounds on the sums d≤y g(d) and d>y g(d)/d. One can uniquely parameterize d ≤ y as the product of relatively prime m and n, with m squarefull and n squarefree. Thus, |g(d)| = |g(n)| |g(m)| d≤y n≤y nm≤y n squarefree (m,n)=1 m squarefull 1  mε. nθ−ε n≤y m≤y/n n squarefree m squarefull

By (2) this is 1 y 1/2+ε 1  = y1/2+ε  y1/2+ε nθ−ε n nθ+1/2 n≤y n n squarefree

since θ>1/2. This establishes the first estimate.

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To deal with the second estimate, we write g(d) g(n) g(m) = d n m d>y n squarefree nm>y (m,n)=1 m squarefull 1 1  . nθ+1−ε m1−ε n squarefree m≥y/n m squarefull By(3)thisis 1/2−ε  1 n nθ+1−ε y n squarefree 1  yε−1/2 nθ+1/2 n ε−1/2 (7) θ y , when ε<1/2. The inner sum converges because θ>1/2. 

Lemma 5. There exists a sufficiently small positive constant c0 such that if Y ≤ 1/2 c0X and X ≥ 1/2,then X + Y X (8) −  XεE(X, Y ), m m Y0. Lemma 5 is based on lattice point estimates of Huxley and Sargos (see [5, 6]), and Filaseta and Trifonov [3, Theorem 7]; we summarize these results in Lemma 7. Lemma 5 is essentially proved by Bordell`es over the two papers [1, 2]. For the convenience of the reader, we will collect the details of the proof following the proof of Theorem 1. Now, we are ready to prove Theorem 1. Like Bordell`es, we express f(n)asthe Dirichlet convolution of the identity function and a second function, g(n), which is small in magnitude in lieu of the hypotheses imposed on f(n). However, because we have relaxed the conditions on f(n) at primes, we are left with more technical estimates on g(n). Further, we will consider short intervals of length ≤ x,where Bordell`es considers intervals of length  x1/2. In this wider consideration, we will use the estimate of Filaseta and Trifonov (18) in as large a range as possible and use the estimates of Huxley and Sargos (17) otherwise.

Proof of Theorem 1. Let g = f ∗ μ and 1 ≤ y ≤ x. Write f(n)= g(d)((x + y)/d−x/d) x

=Σ1 +Σ2.

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Then ∞ (9) Σ1 = y g(d)/d + O y g(d)/d + O |g(d)| . d=1 d>y d≤y Applying Lemma 4 to the error terms and expressing the infinite sum as an Euler product gives 1/2+ε (10) Σ1 = yCf + O(y ), C − 2 2 where f = p(1 1/p)(1 + f(p)/p + f(p )/p + ...). Now we consider Σ2: (x + y)/n x/n |Σ |≤ |g(n)| |g(m)| − 2 m m n squarefree y/n

1/2 3/5 1/2 Case c0x ≤ w

where z = w − yw/y.

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We note that z ≤ y,andifw ≤ x,thenbothx +(k − 1)y and x + yw/y are ≤ 2x.ThusbothofthetermsE(x +(k − 1)y, y), and E(x + yw/y,z)are 2/5 1/2 E(x, y)  x , plugging in y = c0x . Thus the error term in (12) is (13)

 wy−1/2+ε +y1/2+ε +(w/y +1)x2/5+2ε  x−1/10+2εw + x2/5+2ε  x−1/10+2εw,

1/2 with the last upper bound following from the fact that w ≥ c0x . Summing the errors obtained in the previous two cases and replacing ‘2ε’ with ‘ε’ gives the asymptotic formula ε 1/5 1/15 2/3 −1/10 f(n)=wCf + x O(x + x w + x w), wu Applying the estimates of Lemma 4 gives ∞ g(k) f(n)=u + O(u1/2+ε) k n≤u k=1 1/2+ε = uCf + O(u ). Applying this result for u = x + w and u = x and subtracting give 1/2+ε f(n)=wCf + O(x ), x≤n≤x+w as required. 

In Theorem 1, although one requires θ ≥ 4/5 for the bound on Σ2 in equation (11), we remark that some mileage may be obtained in the range 1/2 <θ<4/5. In this range of θ, one may sum the series involving n up to x to obtain the upper bound  3εE  2εE (14) Σ2 x θ(x, y)+x (x, y), where E  1/3−θ 2/3 11/15−θ 4/15 1−θ 5/6−θ 1/6 θ(X, Y )=X Y + X Y + X + X Y . a  1+ε−a Here we have used the bound n≤u 1/n 1+u valid for a>0 and including a uε to bound log u in the case a = 1. One may then proceed similarly as in the proof of Theorem 1, again using standard convolution arguments applied to the long sums when appropriate. We omit the details as they depend on the choice of θ.

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The proof of Lemma 5 requires the following: Lemma 6. For 1 ≤ y ≤ x,andr a positive integer, one has x + y x (15) T (x, y; r):= −  xε/r(x + y−x). cr cr x0. We define R(ϕ, N, δ)=|{n ∈ Z : N

Then for a function γ(n) > 0 with δ = δ(N)=maxN0.Ifthereexistsλk = λk(N) such that |ϕ (x)| λk when NY,oneofa and b is greater than Y 1/5. Thus X + Y − X m m Y

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The trivial estimates (X + Y )/a2−X/a2≤Y/a2 and (X + Y )/b3−X/b3≤Y/b3 applied for the ranges Y 1/5

References

1. Olivier Bordell`es, On short sums of certain multiplicative functions, JIPAM, Journal of In- equalities in Pure and Applied Mathematics 3 (2002), no. 5, Article 70, 6 pp. (electronic). MR1966505 (2004a:11097) 2. , Corrigendum to: “On short sums of certain multiplicative functions”, JIPAM, Journal of Inequalities in Pure and Applied Mathematics 5 (2004), no. 3, Article 81, 3 pp. (electronic). MR2084890 (2005d:11141) 3. Michael Filaseta and Ognian Trifonov, The distribution of fractional parts with applications to gap results in number theory, Proc. London Math. Soc. (3) 73 (1996), no. 2, 241–278. MR1397690 (2000i:11110)

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4. S. W. Golomb, Powerful numbers, Amer. Math. Monthly 77 (1970), 848–855. MR0266878 (42:1780) 5. M. N. Huxley and P. Sargos, Points entiers au voisinage d’une courbe plane de classe Cn, Acta Arith. 69 (1995), no. 4, 359–366. MR1318755 (96a:11065) 6. Martin N. Huxley and Patrick Sargos, Points entiers au voisinage d’une courbe plane de classe Cn.II, Funct. Approx. Comment. Math. 35 (2006), 91–115. MR2271609 (2007m:11092)

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