Congruences and Exponential Sums with the Euler Function

Transcription

1 Fields Institute Communications Volume 00, 0000 Congruences and Exonential Sums with the Euler Function William D. Banks Deartment of Mathematics, University of Missouri Columbia, MO 652 USA Igor E. Sharlinski Deartment of Comuting, Macquarie University Sydney, NSW 209, Australia Dedicated to Hugh Williams on his sixtieth birthday Abstract. We give uer bounds for the number of solutions to congruences with the Euler function ϕn) and with the Carmichael function λn). We also give nontrivial bounds for certain exonential sums involving ϕn). Analogous results can also be obtained for the sum of divisors function and similar arithmetic functions. Let ϕn) denote the Euler function: Introduction ϕn) = #{ a n gcda, n) = }. Let be a rime number, fixed throughout, and ut e x) = ex2πix/) for all x R. In this aer, we give uer bounds for exonential sums of the form N S a, N) = e aϕn)), n= where gcda, ) =, and N is sufficiently large. Our bounds are nontrivial for a wide range of values of, starting with log 9 N. We remark that although it might be ossible to imrove on this ower of log N, for very small values of relative to N, it is simly not ossible to obtain nontrivial bounds. In fact, it has been shown in Theorem 3.5 of [5] that for any rime number of size = olog log N),.) 2000 Mathematics Subject Classification. Primary L07, N69; Secondary N37, L20. c 0000 American Mathematical Society

2 2 William D. Banks and Igor E. Sharlinski the congruence ϕn) 0 mod ) holds for all ositive integers n N with at most on) excetions; see 2.3) below for a more recise formulation of this statement. Thus, for rimes of this size, one has S a, N) = N + on). We also estimate more general sums of the form N S f, N) = e fϕn))), n= where fx) is a olynomial with integer coefficients and degree d 2 that is not constant modulo. In fact, our methods can be used without any further modifications to estimate the similar sums when fx) is a rational function one only needs to deal aroriately with the oles). We also exect that our methods can be alied to exonential sums with relaced by an arbitrary ositive integer m, although certain arguments would be more comlicated. One should also be able to work with various other arithmetic functions, including the sum of divisors function σn). We also give bounds for the number T a, N) of ositive integers n N such that ϕn) a mod ) and for the number L a, N) of ositive integers n N such that λn) a mod ), where λn) denotes the Carmichael function. We recall that λn) is defined to be the largest ossible order of any element in the unit grou of the residue ring modulo n. More recisely, for a rime ower q k, one has λ q k) { q = k q ) if q 3 or k 2, 2 k 2 if q = 2 and k 3, and for arbitrary n 2, λn) = lcm λ q k ),...,λ q k ν ) ) ν, where n = q k... qkν ν is the rime factorization of n. Of course, λ) =. Throughout the aer, the imlied constants in the symbols O, and may occasionally, where obvious, deend on a real arameter ε > 0 and an integer d but are absolute otherwise. We recall that the notations U V and V U are equivalent to the statement that U = OV ) for ositive functions U and V. We also use the symbol o with its usual meaning: the statement U = ov ) is equivalent to U/V 0. Acknowledgements. Most of this work was done during a visit by W. B. to Macquarie University, whose hositality and suort are gratefully acknowledged. Work also suorted in art, for W. B. by NSF grant DMS , and for I. S. by ARC grant DP Prearations Here we collect some known number-theoretic estimates which are used in the sequel. For any integer n 2, let Pn) denote the largest rime divisor of n, and ut P) =. As usual, we say that an integer n is Y -smooth if and only if Pn) Y. Let ψx, Y ) = #{ n X n is Y -smooth}. The following estimate is a substantially relaxed and simlified version of Corollary.3 of [3]; see also [3].

3 Congruences and Exonential Sums with the Euler Function 3 Lemma 2. Let u = log X/ logy. For any u with u Y /2, we have ψx, Y ) Xu u+ou). Throughout the sequel, we denote by P the set of all rime numbers, P[Y, X] the set of l P with Y < l X, and P[X] = P[, X]. We also need the following simlified form of the Brun-Titchmarsh theorem; see Theorem in Section 2.3. of [9] or Theorem 3.7 in Chater 3 of [0]. Lemma 2.2 For any X > k, let πx; k, a) be the number of rimes l P[X] such that l a mod k). Then πx; k, a) X ϕk)log2x/k). Finally, our rincial tool is the following bound for exonential sums with rime numbers, which follows immediately from Theorem 2 of [6]. Lemma 2.3 For any X 2, the following bound holds: max gcdc,)= e cl) /2 + X /4 /8 + /2 X /2 )X log 3 X. l P[X] Proof Let Λn) denote the von Mangoldt function: { log l if n is a ositive ower of some rime l, Λn) = 0 otherwise. According to Theorem 2 of [6], we have for any integer m 2: m max Λn)e cn) gcdc,)= /2 + m /4 /8 + /2 m /2 )m log 4 m. Taking M = X, we aly artial summation: M e cl) = log n Λn)e cn) + OM /2 ) l P[X] = log M M Λn)e cn) M ) + log m m Λn)e cn) + OX /2 ) logm + ) m=2 M M Λn)e cn) log M + m m log 2 Λn)e cn) m + M/2. The result follows. m=2 Let F d, denote the set of olynomials with integer coefficients of degree d whose leading coefficient is relatively rime to ; that is, F d, = { fx) = a d x d a x + a 0 Z[x] ad 0 mod ) }. 2.) For exonential sums over rimes with olynomials from F d,, d 2, we use the following bound of [7]. We remark that the condition d 2 is imortant; thus Lemma 2.3 and Lemma 2.4 do not overla.

4 4 William D. Banks and Igor E. Sharlinski Lemma 2.4 For any ε > 0, d 2 and X 2, the following bound holds: max g F d, e gl)) 3/6+ε X 25/32. For any integer a, let l P[X] T a, N) = { n N ϕn) a mod )}, 2.2) and ut T a, N) = #T a, N). In the secial case where a 0 mod ), we have the following bound, which is a artial case of Theorem 3.5 of [5]. Lemma 2.5 For any N 2, we have T 0, N) N log log N. We remark that the bound of Lemma 2.5 becomes trivial when = Olog log N), which is very close to the threshold.) below which it is not ossible to obtain nontrivial uer bounds. Indeed, for satisfying.), we have by inequality 4.2) of [5]: T 0, N) = N + ON ex c log log N)), 2.3) for some absolute constant c > 0. To study congruences with the Carmichael function, we need the following statement, which is Theorem 5 of [8]. Lemma 2.6 For all sufficiently large numbers N and any log log N) 3, the number of ositive integers n N with λn) n ex ) is at most N ex 0.69 log ) /3). 3 Congruences with ϕn) As in Section 2, let T a, N) be defined by 2.2), and let T a, N) = #T a, N). In this section, we consider the roblem of estimating T a, N) in the case where a 0 mod ). Theorem 3. The following bound holds: max T a, N) Nw w/2+ow) + Nw gcda,)=, where w = log N/ log. Moreover, if ex 0.5 logn log log N ), then max T a, N) N +o). gcda,)= Proof Without loss of generality, we can assume that as N, since the bounds are trivial otherwise. Throughout the roof, let a be fixed with gcda, ) =. We define log N if ex 0.5 logn log log N ), 2 log u = log if < ex 0.5 logn log log N ). log log

5 Congruences and Exonential Sums with the Euler Function 5 We also define a smoothness bound K = N /u. Note that K 2/3 and u K /2 for sufficiently large values of N, and that u as N. Let E be the set of integers n [, N] such that n is K-smooth. Since all of the conditions of Lemma 2. hold, we have l P[K,N /2 ] K<n N l 2 n #E Nu u+ou). Next, let E 2 be the set of integers n [, N] such that Pn) > K and Pn) 2 n. Then #E 2 k 2 N/K. l P[K,N /2 ] N/l 2 N k>k Now define N = {,..., N}\ E E 2 ). Using the results above, we obtain that T a, N) Nu u+ou) + N/K +. ϕn) a mod ) Since every n N can be exressed in the form n = lm with l P[K, N/m] and gcdl, m) =, and since gcda, ) =, it follows that, l P[K,N/m] l a m mod ) ϕn) a mod ) <m N/K l P[K,N/m] ϕm) 0 mod ) l a m mod ) where a m + aϕm) mod ). By Lemma 2.2, N m log2n/m) since K 2/3. Hence, Therefore ϕn) a mod ) N logk <m N/K N m log2k/) N m logk, m N logn/k) log K < Nu. T a, N) Nu u+ou) + N/K + Nu/. Because K 2/3, we see that the second term never dominates, and the result follows. The bound 2.3) suggests that when is very small relative to N, one might exect that T a, N) = on/). By refining the arguments in Theorem 3. in order to make use of 2.3), we show that this is indeed the case in the following quantitative form: Theorem 3.2 There exists an absolute constant C > 0, such that for the following bound holds: log log N C log log log N, max T a, N) N gcda,)= ex C log log N).

6 6 William D. Banks and Igor E. Sharlinski Proof Without loss of generality, we can assume that as N since otherwise the result follows trivially from 2.3). Let a be fixed with gcda, ) =. Put u = ex0.5c log log N), where c > 0 is the constant from 2.3); by our hyothesis on the size of, we see that u as N. Finally, define the smoothness bound K = N /u. Since u = log N) o), we have that K = exlog N) +o) ), hence u K /2 and K /2 if N is sufficiently large. Proceeding as in Theorem 3. with these choices for u and K, we obtain the estimate T a, N) Nu u+ou) + N/K + N log K <m N/K ϕm) 0 mod ) Using artial summation together with the estimate 2.3), we also have with M = N/K ): Since <m N/K ϕm) 0 mod ) m = M M T 0, M)) + M + we derive the estimate m=3 M m=2 ex c log log m) m + log M) c log N) c. m. m ) m T 0, m)) m + M = + m=3 mlog m) c N log N) c = N log K ex 0.5c log log N), T a, N) Nu u+ou) + NK + N ex 0.5c log log N). Now let C = c/3, and suose that satisfies the hyothesis of the theorem. It is easily seen that Also, K = exlog N) +o) ) ex0.5c log log N). u = ex0.5c log log N) log log N).5 log C + log log N log log log N log, thus and therefore u logu + o)) u/2)logu log u log, The result follows. u u+ou) u = ex 0.5c log log N).

7 Congruences and Exonential Sums with the Euler Function 7 Let us define 4 Congruences with λn) L a, N) = { n N λn) a mod )}, and ut L a, N) = #L a, N). Since the integers λn) and ϕn) always have the same set of rime divisors, it follows that L 0, N) = T 0, N); thus the estimate for T 0, N) in Lemma 2.5 alies to L 0, N) as well. In this section, we combine Theorem 3. with Lemma 2.6 to estimate L a, N) in certain ranges when a 0 mod ). Theorem 4. For ex 3log log N) 3) log N log log log N ex 5log log N) 3 the following bound holds: max L a, N) gcda,)= N ex 0.4w /3 log w) 2/3) + ex 0.5 log loglog ) /3)), where w = log N/ log. Proof Denote hn) = ϕn)/λn). For any η, we have L a, N) + T ah, N) + h<η n L a,n) hn)=h n L a,n) hn) η h<η ), n L a,n) hn) η. Then, rovided that η ex log log N) 3), 4.) Theorem 3. and Lemma 2.6 together imly L a, N) η Nw w/2+ow) + Nw Now ut ) + N ex { η = min w w/4, /2} log η log log η) /3). It follows from the conditions of the theorem that 4.) is satisfied and also that w = o) ; consequently, L a, N) Nη /2+o) + N ex 0.69 log η log log η) /3) N ex 0.69 log η log log η) /3), and the result follows. It is easy to see that one can slightly imrove the constants that occur in the statement of Theorem 4., both in the range secified for and in the bound, but we have not done so in order to rovide a cleaner statement.

8 8 William D. Banks and Igor E. Sharlinski 5 Exonential Sums with ϕn) We now show that the same arguments used in the roof of Theorem 3., combined with the bound of Lemma 2.5, can be used to estimate exonential sums with the Euler function. Theorem 5. The following bound holds: log 4 max S N a, N) N gcda,)= where w = log N/ log. /2 ) + w 2w/5+ow), Proof Without loss of generality, we can assume that log 8 N since the bound is trivial otherwise. In articular, we can assume that is sufficiently large for our uroses. Throughout the roof, fix a with gcda, ) =. We define K = 2.5 and denote by E the set of n [, N] which are K-smooth. Let u = log N log K = 2w/5. It is easy to see that if w /3, then log 3 N and the bound is trivial; thus we can assume that u /2 K /2. By Lemma 2., we have that #E Nu u+ou). Denote by E 2 the set of n [, N] for which Pn) > K and Pn) 2 n. Then #E 2 N/l 2 N/K. l P[K,N] Denote by E 3 the set of n [, N] such that Pn) 2 n and ϕm), where m = n/pn). Since ϕn) for every n E 3, Lemma 2.5 yields the estimate #E 3 N log log N N log. Finally, let N = {,..., N}\ E E 2 E 3 ). From the receding bounds, it follows that S a, N) = )) log e aϕn)) + O N + u u+ou) + K = )) log e aϕn)) + O N + w 2w/5+ow). Now, every integer n N has a unique reresentation of the form n = ml, where l P[K, N], l > Pm), and ϕm). Conversely, if L m = max{k, Pm)}, then for any m N/K such that ϕm) and any l P[L m, N/m], we have n = ml N. Observing that ϕn) = ϕm)l ), we obtain e aϕn)) = e aϕml)) = m N/K ϕm) m N/K ϕm) e aϕm)) e aϕm)l).

9 Congruences and Exonential Sums with the Euler Function 9 Write e aϕm)l) = l P[N/m] e aϕm)l) l P[L m] e aϕm)l), and observe that the right hand side of the bound in Lemma 2.3 is a monotonically increasing function of X. Then it follows that e aϕml)) N /2 + N /4 m /4 /8 + /2 m /2 N /2) log 3 N. m Recalling that m N/K = N 5/2, we see that the first term always dominates the other two. Hence, Therefore, and we obtain the stated result. e aϕn)) N log3 N /2 e aϕml)) N log3 N m /2, N m= m N log4 N, /2 It is easy to see that the bound of Theorem 5. is nontrivial when the conditions log N = o /8 ) and = N o) both hold. We now turn our attention to sums with olynomials f from the class F d, given by 2.), with d 2. As before, we remark that the condition d 2 is imortant, thus Theorem 5. and Theorem 5.2 do not overla. Theorem 5.2 For any ε > 0 and d 2, the following bound holds: max S f, N) N /4+ε + w w/2+ow)), f F d, for log N, where w = log N/ log. Proof We use the same notation as in the roof of Theorem 5. excet that we ut K = 2 ; thus u = log N/ logk = w/2 = K /2. As in the roof of Theorem 5. we obtain S f, N) = = m N/K ϕm) e fϕn))) + O )) log N + u u+ou) + K e fϕm)l ))) + O )) log N + w w/2+ow). It is clear that for each m in the sum above, the olynomial fϕm)x )) F d,. Therefore using Lemma 2.4 instead of Lemma 2.3, as in the roof of Theorem 5., we derive that e fϕm)l ))) 3/6+ε N/m) 25/32 m N/K ϕm) m N/K 3/6+ε NK 7/32 = N /4+ε,

10 0 William D. Banks and Igor E. Sharlinski and we obtain the stated result. Clearly, there are many ossible admissible choices for K and thus a trade-off between the exonent of and w w and the required bottom range of ). 6 Remarks As we have already remarked, our aroach should work in rincile for exonential sums with fϕn)), where f is a olynomial with integer coefficients. Aroriate analogues of Lemma 2.3 can be found in [7,, 2]. It would also be interesting to consider such sums when f is a olynomial with irrational coefficients, however our methods do not seem to extend to this case. One can also fix an integer g of multilicative order t modulo m 2 and consider the rather exotic sums N V m a, g, N) = e m ag ϕn)). n= At least when t is rime, our method, combined with the uer bounds from [] on sums with ag l taken over rimes l, should work in rincile to estimate such sums. It would also be interesting to estimate exonential sums with the Euler function over integers taken from various secial sets S, such as shifted rimes. It would also be interesting to extend our results in another direction, namely to sums with arbitrary integer denominator m. In this case, as well as in the estimate of sums V m a, g, N) for comosite multilicative orders t, one would need an analogue of Lemma 2.5 for arbitrary m. Clearly, roving such an analogue seems ossible, but it requires some additional arguments. The roblem of estimating the number of solutions to the congruence ϕn) a mod m), n N, is an interesting question in its own right, and it certainly deserves more attention. For moduli that are roducts of a fixed number of rimes not necessarily distinct), a generalization of Lemma 2.5 is given in [2]; in fact, it also alies to iterations of the Euler function. In the case where the modulus m is fixed and N is growing, rather detailed information about the distribution of ϕn), n N, in residue classes modulo m can be found in [4, 6, 5]. However, this question remains oen in the general case. We also remark that by Lemma 2 of [4], almost all values of ϕn), n N, are divisible by all rime owers α log log N log log log N. Therefore, for some constant c > 0 and some integer c/log log log N m log N) one has T m 0, N) = N + on). Sums with multilicative characters might also be considered; in rincile, our methods should rovide nontrivial bounds in certain ranges, similar to those of Theorem 5.. Finally, we mention that our methods can be alied to the sum of divisors function σn). However, it is still not clear how to estimate exonential sums with the Carmichael function λn), even given its close relationshi to the Euler function.

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