Congruences and exponential sums with the sum of aliquot divisors function

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1 Congruences and exonential sums with the sum of aliquot divisors function Sanka Balasuriya Deartment of Comuting Macquarie University Sydney, SW 209, Australia William D. Banks Deartment of Mathematics University of Missouri Columbia, MO 652 USA Igor E. Sharlinski Deartment of Comuting Macquarie University Sydney, SW 209, Australia

2 Abstract We give bounds on the number of integers n such that s(n), where is a rime and s(n) is the sum of aliquot divisors function given by s(n) = σ(n) n, where σ(n) is the sum of divisors function. Using this result we obtain nontrivial bounds in certain ranges for rational exonential sums of the form S (a,) = n ex(2πias(n)/), gcd(a, ) =. Keywords: divisors, congruences, exonential sums 2000 Mathematics Subject Classification: A07, L07, 60 Introduction For every ositive integer n, let s(n) be the sum of the aliquot divisors of n: s(n) = d n d n d = σ(n) n, where σ(n) is the sum of divisors function. In this aer we consider arithmetic roerties of the aliquot sequence (s(n)) n. In articular, for a fixed rime we obtain nontrivial uer bounds in certain ranges for exonential sums of the form: S (a, ) = e (as(n)) (a Z, ), where n= e (x) = ex(2πix/) (x R). Our results for the sums S (a, ) rely on uer bounds for the cardinalities #T () of the sets T () = { n s(n) 0 (mod )} ( ). We remark that analogous results for the Euler function ϕ(n) have been obtained in [, 2, 3], and we aly similar methods in the resent aer. Various modifications are needed, however, since s(n) is not a multilicative function. 2

3 Theorem. For v = (log )/(log ), the following bound holds: Using this result we show: #T () v v/2+o(v) + v. Theorem 2. The following bound holds: max gcd(a,)= S (a, ) ( log 4 /2 + log ) log log. log log log log In the statements above and throughout the aer, any imlied constants in the symbols, and O are absolute unless indicated otherwise. We recall that for ositive functions F and G the notations F = O(G), F G and G F are all equivalent to the assertion that the inequality F c G holds for some constant c > 0. Throughout the aer, the letters, q are used to denote rime numbers, and m, n are ositive integers. 2 Preliminaries Let P(n) be the largest rime factor of an integer n 2, and ut P() =. An integer n is said to be y-smooth if P(n) y. As usual, we define ψ(x, y) = #{n x : n is y-smooth} (x y > ). The following bound is a relaxed and simlified version of [7, Corollary.3] (see also [4]): Lemma 3. For u = (log x)/(log y) with u y /2, we have ψ(x, y) xu u+o(u). The next statement is a simlified form of the Brun-Titchmarsh theorem; see, for examle, [5, Section 2.3., Theorem ] or [6, Chater 3, Theorem 3.7]. Lemma 4. Let π(x; k, a) be the number of rimes x such that a (mod k). Then, for any x > k we have π(x; k, a) x ϕ(k) log(2x/k). 3

4 Finally, our rincial tool is the following bound for exonential sums with rime numbers, which follows immediately from Theorem 2 of [8]. Lemma 5. For any rime and real number x 2, the following bound holds: max e (aq) gcd(a,)= ( /2 + x /4 /8 + x /2 /2) x log 3 x. q x 3 Proof of Theorem We can assume that v since the result is trivial otherwise. Thus, taking we see that u = v 2 = log 2 log, 2u log u v log = log. Defining the smoothness bound K = /u = 2, it follows that u K /2. In articular, if E is the set of integers n such that n is K-smooth, then we can aly Lemma 3 to derive the bound #E = ψ(, K) u u+o(u) = v v/2+o(v). ext, let E 2 be the set of integers n such that q 2 n for some rime q > K. Then, #E 2 /q 2 /K / 2. q>k q>k n q 2 n Finally, let E 3 be the set of integers n which are multiles of. Then, #E 3 = / /. ow let = {,..., }\ (E E 2 E 3 ). Using the bounds established above, it follows that #T () v v/2+o(v) + / + # ( T () ). () 4

5 For any n T (), we write n = mq, where q = P(n) > P(m). Since s(n) = σ(n) n, and σ(n) is multilicative, the condition s(n) 0 (mod ) imlies mq σ(mq) σ(m)(q + ) (mod ). Then σ(m) 0 (mod ) since n, hence the same relation also imlies that σ(m) m (mod ); consequently, q a m (mod ) for any integer a m σ(m)(m σ(m)) (mod ). Since q > K we see that # ( T () ). m /K σ(m) m (mod ) For the inner sum, we have by Lemma 4: K<q /m q a m (mod ) m log(2/m) K<q /m q a m (mod ) m log(2k/) m log K, where we have used the inequality K /2 in the last ste. Therefore, # ( T () ) log K m /K m log(/k) log K Inserting this bound into (), we obtain the desired result. 4 Proof of Theorem 2 u v. We can assume that log 8 and that v = (log )/(log ) as since the result is trivial otherwise. Let u, K and the sets E, E 2, E 3 be defined as in the roof of Theorem. Then, #(E E 2 E 3 ) v v/2+o(v) + /. Also, ut M = /w, where w 2 is a arameter to be secified later, and let E 4 be the set of integers n for which P(n) n/m. Every integer n E 4 can factored as n = mq, where P(m) P(n) = q /m and m M. 5

6 Therefore, #E 4 m M q /m m M /m log(/m) log m M m log M log = w. ow let = {,..., }\ (E E 2 E 3 E 4 ). From the bounds above it follows that S (a, ) = n e (as(n)) + O ( ( v v/2+o(v) + + w )). (2) Every integer n can be uniquely reresented in the form n = mq, where M < m < /K and max{k, P(m)} < q /m. Conversely, if the numbers m, q satisfy these inequalities, then n = mq lies in. Observing that s(mq) = s(m)q + σ(m), we have e (as(n)) = e (as(mq)) = Σ + Σ 2, (3) n M<m</K where L m = max{k, P(m)}, and Σ = e (aσ(m)) Σ 2 = M<m</K s(m) M<m</K s(m) e (aσ(m)) e (as(m)q),. Write e (as(m)q) = e (as(m)q) q /m q L m e (as(m)q), and observe that the right side of the bound in Lemma 5 is a monotonically increasing function of x; thus, if s(m) we have e (as(m)q) ( /2 + (/m) /4 /8 + (/m) /2 /2) log 3 m 6.

7 For m < /K = / 2 the first term inside the arentheses dominates the other two; therefore, Σ log3 /2 M<m</K s(m) m log4 /2. (4) ext, we turn our attention to the roblem of bounding Σ 2. Writing we have trivially: I = log M + and J = log(/k) +, Σ 2 M<m</K s(m) J j=i e j J m m e j s(m) = j=i e j <m e j s(m) m J e j #T (e j ). j=i Define and note that v j = v w = log M log < v j = j log j log (I j J), log + log w (I j J). Thus if then Theorem imlies that v/w (5) e j #T (e j ) v v j/2+o(v j ) j + v j. Hence, Σ 2 J j=i ( v v j/2+o(v j ) j + v ) ( j (v/w) v/(2w)+o(v/w) + w ) log. 7

8 ow, combining the revious bound with (2), (3) and (4), and droing terms which are clearly dominated by other terms, it follows that S (a, ) log4 /2 + w + (v/w) v/(2w)+o(v/w) log + w log. (6) ote that the last term in this bound can also be droed. Indeed, we can assume that w v, for otherwise the bound is trivial, and thus w log v log = log2 log log4 /2. We now choose w = v log log log 6 log log to (essentially) balance the middle two terms in (6). We also note that the condition (5) is satisfied. With this choice of w, it is easily seen that (v/w) v/(2w)+o(v/w) log = (log ) 2+o() (log ) 3/2, whereas for log 8 we have w = 6 log log log log log log log (log ) 3/2. Therefore, the third term in (6) can be droed, and the result follows. References [] S. Balasuriya, I. E. Sharlinski and D. Sutantyo Multilicative character sums with the Euler function, Studia Sci. Math. Hungarica, (to aear). [2] W. Banks and I. E. Sharlinski, Congruences and exonential sums with the Euler function, High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, Amer. Math. Soc., 2004, [3] W. Banks and I. E. Sharlinski, Congruences and rational exonential sums with the Euler function, Rocky Mountain J. Math., 36 (2006),

9 [4] E. R. Canfield, P. Erdős and C. Pomerance, On a roblem of Oenheim concerning Factorisatio umerorum, J. umber Theory, 7 (983), 28. [5] G. Greaves, Sieves in number theory, Sringer-Verlag, Berlin, 200. [6] H. Halberstam and H. E. Richert, Sieve methods, Academic Press, London, 974. [7] A. Hildebrand and G. Tenenbaum, Integers without large rime factors, J. de Théorie des ombres de Bordeaux, 5 (993), [8] R. C. Vaughan, Mean value theorems in rime number theory, J. Lond. Math. Soc., 0 (975),