On the counting function for the generalized. Niven numbers

Transcription

1 On the counting function for the generalized Niven numbers R. C. Daileda, Jessica Jou, Robert Lemke-Oliver, Elizabeth Rossolimo, Enrique Treviño Introduction Let q 2 be a fied integer and let f be an arbitrary comple-valued function defined on the set of nonnegative integers. We say that f is completely q-additive if faq j + b fa + fb for all nonnegative integers a, b, j satisfying b < q j. Given a nonnegative integer n, there eists a unique sequence a 0 n, a n, a 2 n,..., 2000 Mathematics Subject Classification: A25, A63, K65. The second, third, fourth and fifth authors received support from the National Science Foundation under grant DMS

2 2 Ryan C. Daileda et al. {0,,..., q } so that n a j nq j. j0 The right hand side of epression will be called the base q epansion of n. Using the base q epansion, we find that the function f is completely q-additive if and only if f0 0 and fn fa j n. j It follows that a completely q-additive function is completely determined by its values on the set {0,,..., q }. The prototypical eample of a completely q-additive function is the base q sum of digits function s q, which is defined by s q n a j n. j0 A q-niven number is a nonnegative integer n that is divisible by s q n. The question of the distribution of the q-niven numbers can be answered by studying the counting function N q #{0 n < : s q n n}, a task which has been undertaken by several authors see, for eample, the papers of Cooper and Kennedy [, 2, 3, 4] or De Koninck and Doyon [5]. The best known result is the asymptotic formula N q c q + o log, where c q 2 log q q 2 q j, q, j

3 The counting function for the generalized Niven numbers 3 which was proven only recently by De Koninck, Doyon and Kátai [6], and independently by Mauduit, Pomerance and Sárközy [8]. Given an arbitrary non-zero, integer-valued, completely q additive function f, we define an f-niven number to be a nonnegative integer n that is divisible by fn. In the final section of [6] it is suggested that the techniques used therein could be applied to derive an asymptotic epression for the counting function of the f-niven numbers, N f # {0 n < fn n}. It is the goal of this paper to show that, under an additional mild restriction on f, this is indeed the case. Our main result is the following. Theorem. Let f be an arbitrary non-zero, integer-valued, completely q- additive function and set m q fj, σ 2 q fj 2 m 2, q q j0 F f, f2,..., fq and j0 d gcd {rfs sfr r, s {, 2,..., q }}. Assume F, q. i If m 0 then for any ɛ 0, /2 N f c f log + O log 3 2 ɛ

4 4 Ryan C. Daileda et al. where c f log q m q j, q, d. q j The implied constant depends only on f and ɛ. ii If m 0 then N f c f log log log 2 + O log 2 where log q c f 2πσ 2 2 q q j, q, d. j The implied constant depends only on f. Most of the proof of this theorem is a straightforward generalization of the methods used in [6]. It is the intent of this paper, then, to indicate where significant and perhaps non-obvious modifications to the original work must be made. The first notable change is the introduction and use of the quantities d and F. Because d 0 and F when f s q these quantities play no role in the earlier work. Finally, in order to shorten the presentation, we have attempted to unify the m 0 and m 0 cases for as long as possible. The result is a slightly different approach in the m 0 case than appears in [6]. The final steps when m 0 are entirely new.

5 The counting function for the generalized Niven numbers 5 2 Notation We denote by Z, Z +, Z + 0, R and C the sets of integers, positive integers, nonnegative integers, real numbers and comple numbers, respectively. For the remainder of this paper, we fi an integer q 2 and a non-zero completely q additive function f : Z + 0 Z. The variable will always be assumed real and positive and n will always be an integer. Following [6] we set A k, l, t #{0 n < n l mod k and fn t}, a t #{0 n < fn t}. As usual, we use the notation F OG or F G to mean that there is a constant C so that for all sufficiently large, F CG. The constant C and the size of are allowed to depend on f and hence on q, but on no other quantity unless specified. 3 Preliminary Lemmas For real y we define y to be the distance from y to the nearest integer. Lemma. Let M ma j q fj. Let s, k Z + with s, k and suppose that there is a pair j, j 2 {0,,..., q } so that k j fj 2 j 2 fj. Then for any ξ R ma 0 j q js fjξ + k 2Mk. 2

6 6 Ryan C. Daileda et al. Proof. We first claim that the maimum in question is at least positive. To see this we argue by contradiction and assume that fjξ + js/k 0 for 0 j q. Then fjξ + js/k n j Z for each j. Solving for ξ and equating the resulting epressions, we find that if fj, fj 2 0 then kn j fj 2 n j2 fj sj fj 2 j 2 fj. 3 Since s, k, this implies that k j fj 2 j 2 fj, which is impossible. Since this same condition is trivially verified if fj 0 or fj 2 0, we have a contradiction in any case. To prove the lemma we again argue by contradiction. Assume the contrary. Then k j fj 2 j 2 fj s k fj fj 2 ξ + j fj 2 s fj fj 2 ξ + j 2fj s k k fj 2 fj ξ + j s + k fj fj 2 ξ + j 2s. k By our assumption, fj M < 2k ma 0 j q fjξ + js/k 2 fj 2 ξ + j 2 s/k and similarly fj 2 < 2 fj ξ + j s/k.

7 The counting function for the generalized Niven numbers 7 Therefore, by properties of, k fj 2 fj ξ + j s k + fj fj 2ξ + j 2s k 2M ma js 0 j q fjξ + k < 2M 2Mk k which is a contradiction. It is proven in [7] that if z, z 2,..., z q C satisfy z j for j, 2,..., q. then q + z j ma q 2q Re z j. 4 j q j Since for real y we have y 2 cos2πy and + y e y, the net lemma is an immediate consequence. Lemma 2. Let ey e 2πiy. There is a positive constant c c f so that for any ξ, r R q e fjξ + rj q ep c j0 ma j q fjξ + rj The distribution of the values of f 4. Argument restricted to a congruence class Let k Z +, l Z + 0 and t Z. Our first goal in this section is to relate A k, l, t to the function a t through a series of reductions on the mod-

8 8 Ryan C. Daileda et al. ulus k. The first three reductions are proven eactly as in [6], substituting our Lemma 2 for their Lemma 3. We state them for the convenience of the reader. Reduction. Write k k k 2 where k is the largest divisor of k so that k, q. Then the primes dividing k 2 also divide q and we let h be the smallest positive integer so that k 2 divides q h. Since k divides k q h, the congruence class l mod k is the union of classes l j mod k q h, j,..., q h /k 2. For each j write l j l j + q h l j 2 where 0 l j < q h. Then A k, l, t q h /k 2 j A l j q h k, l j 2, t fl j. Since k, q, we are led to the net reduction. Reduction 2. Suppose that k, q. Let k k k 2 where k is the largest divisor of k so that k, q. Then there is a positive constant c 3 c 3 f so that A k, l, t A k 2, l, t + O c 3 log 2k. k Before moving to the net reduction, we note that the primes dividing k 2 must also divide q.

9 The counting function for the generalized Niven numbers 9 Reduction 3. Suppose that the prime divisors of k also divide q. Then there is a positive constant c 4 c 4 f so that A k, l, t k, q A k, q, l, t + O c 4 log 2k. k We have therefore reduced to the case in which the modulus is a divisor of q. Reduction 4. We now come to the first new reduction. The proof closely follows that of Reductions 2 and 3 of [6], substituting our Lemma for their Lemma and our Lemma 2 for their Lemma 3. We therefore choose to omit it. Suppose that k q and let d denote the greatest common divisor of j fj 2 j 2 fj for j, j 2 {0,,..., q }. Then there is a positive constant c 5 c 5 f so that A k, l, t k, d k A k, d, l, t + O c 5. We note that in the case f s q we have d 0, making this reduction unnecessary. As such, it has no analogue in [6]. Reduction 5. Suppose that k divides q, d. Then q j mod k for all j and j fj 2 j 2 fj mod k for all 0 j, j 2 q. From this and the complete q-additivity of f it follows that mfn nfm mod k for all m, n Z + 0. Let F f, f2,..., fq and suppose further that F, q. Since the values of f are linear combinations of f, f2,..., fq with nonnegative integer coefficients, the condition

10 0 Ryan C. Daileda et al. F, q implies that we can find an m Z + so that fm, q. Therefore if n Z + 0 satisfies fn t we have mt mfn nfm mod k and it follows that a t if mt lfm mod k, A k, l, t 0 otherwise. This is the analogue of the final reduction in section 4.5 of [6]. 4.2 Unrestricted argument We now turn to the distribution of the values of f when its argument is free to take on any value. We let m and σ denote the mean and standard deviation of f on the set D {0,,..., q } of base q digits. That is, m q fj, σ 2 q fj 2 m 2. q q j0 j0 We also set N log log q where y is the greatest integer not eceeding y. Following [], we view D as a probability space in which each element is assigned a measure of /q. We use the digits of the base q epansion to identify the set of nonnegative integers n strictly less than q N N Z + with the product space D N. In this way f can be viewed as the random variable on D N obtained by summing together N independent copies of f

11 The counting function for the generalized Niven numbers acting on D alone. Applying Theorem 5, Chapter III of [9] in this setting, it is not difficult to deduce the net result. Proposition. Let ɛ 0, /2 and set I [mn N /2+ɛ, mn +N /2+ɛ ]. Then #{0 n < fn I} log 2 the implied constant depending only on f and ɛ. We remark that in the case f s q, this proposition follows from Lemma 4 of [8]. In fact, using the line of reasoning we have suggested, it is possible to obtain a more general result very similar to that lemma in which N /2+ɛ is replaced by N /2 λ, where λ on /2. As observed in [6] in the case f s q, Theorem 6, Chapter VII of [9] can be used to obtain the following result. Proposition 2. If f, f2,..., fq then uniformly for t Z we have a t t σ mn ϕ N σ N + O N where ϕy 2π /2 e y2 /2. The implied constant depends only on f. Corollary. If F f, f2,..., fq then a t F t σ mn ϕ N σ N when t 0 mod F and a t 0 otherwise. + O N Proof. Apply the proposition to the function g f/f.

12 2 Ryan C. Daileda et al. 5 Proof of Theorem It suffices to consider the case in which m 0, since N f N f. Fiing ɛ 0, /2, according to Proposition we have N f t 0 mod F A t, 0, t + O. 6 log 2 As in Section 5 of [6], one can show that Reductions through 5 together with the corollary to Proposition 2 yield A t, 0, t t, q, d a t + O t log N tn 7 uniformly for nonzero t I. Since t, q, d depends only on the residue of t mod q, substitution of 7 into 6 yields where q N f j, q, d j t 0 t 0 mod F t j mod q a t t N ɛ /2 if m > 0, E log N if m 0. log N + O E N 8 Lemma 3. For j, 2,..., q we have where t 0 t 0 mod F t j mod q a t t q E 2 t 0 t 0 mod F N ɛ /2 a t t if m > 0, N /2 if m 0. + O E 2 N

13 The counting function for the generalized Niven numbers 3 Proof. Since we have assumed F, q, for each j, 2,..., q there is a unique b j mod F q so that the two simultaneous congruences t 0 mod F and t j mod q are equivalent to the single congruence t b j mod F q. The corollary to Proposition 2 implies that for t, k 0 mod F we have a t + k t + k a t t + O t log + t 2 log /2 9 when t + k 0, the implied constant depending only on f and k. When m > 0 we therefore have t 0 t 0 mod F t j mod q a t t t b j q mod F q t b j +O q N 3/2 ɛ +O a t t q 2 k0 mod F q s I s 0 mod F N 3/2 ɛ a s s a t + kf t + kf + O s J s 0 mod F a s s where J is the set of integers with distance at most qf from the endpoints of I. Since a s /N /2, the sum over J contributes no more than the second error term, proving the lemma in this case. When m 0 we may carry through the same analysis, being careful in the second step to omit from the summation the single value of t for which

14 4 Ryan C. Daileda et al. t + kf 0. This introduces an error of size O/N /2, which is consistent with the statement of the lemma. Combining Lemma 3 with equation 8 we can now complete the proof. When m > 0 we have N f q j, q, d q j t 0 mod F q j, q, d mn q j log log +O log 3/2 ɛ log q m log q log log +O log 3/2 ɛ log q m q a t t t 0 mod F q j, q, d j q j, q, d j t 0 mod F log N + O a t a t N 3/2 ɛ log log log + O log 3/2 ɛ where in the final line we have used Proposition to replace the sum with + O/log 2. Since ɛ 0, /2 was arbitrary, we may discard the log log term in the error, giving the stated result. When m 0 we have q N f j, q, d q j t 0 t 0 mod F a t t + O N /2 0 Set ɛ /4 for convenience. Writing t F s and using the corollary to

15 The counting function for the generalized Niven numbers 5 Proposition 2 we obtain t 0 t 0 mod F a t t 2 σ N s N 3/4 /F sf s ϕ σ + O N log N N. Now s N 3/4 /F sf s ϕ σ N N 3/4 /F N 2π sf s ϕ σ ds + O N /4 /σ 2 F/σ 2N e u2 u du + O e u2 du + O 2π F/σ 2N u du + O 2π F/σ 2N u 2 2π log N + O 2 log log + O. 2π Returning to equations 0 and we find that N f q j, q, d 2πσ 2 /2 q j /2 q log q j, q, d 2πσ 2 q +O log /2 j log log N /2 log log log /2 + O N /2 which concludes the proof.

16 6 Ryan C. Daileda et al. References [] Cooper, C. N.; Kennedy, R. E., On the natural density of the Niven numbers, College Math. J , [2] Cooper, C. N.; Kennedy, R. E., On an asymptotic formula for the Niven numbers, Internat. J. Math. Sci , [3] Cooper, C. N.; Kennedy, R. E., A partial asymptotic formula for the Niven numbers, Fibonacci Quart , [4] Cooper, C. N.; Kennedy, R. E., Chebyshev s inequality and natural density, Amer. Math. Monthly , [5] De Koninck, J. M.; Doyon, N., On the number of Niven numbers up to, Fibonacci Quart , [6] De Koninck, J.-M.; Doyon, N.; Kátai, I., On the counting function for the Niven numbers, Acta Arithmetica , [7] Delange, H., Sur les fonctions q-additives ou q-multiplicatives, Acta Arithmetica 2 972, [8] Mauduit, C.; Pomerance, C.; Sárközy, A., On the distribution in residue classes of integers with a fied sum of digits, Ramanujan J ,

17 The counting function for the generalized Niven numbers 7 [9] Petrov, V. V., Sums of Independent Random Variables, Ergebnisse der Mathematik und ihrer Grenzgebiete 82, Springer, 975. Mathematics Department, Trinity University, One Trinity Place, San Antonio, TX rdaileda@trinity.edu Penn State University Mathematics Department, University Park, State College, PA jjj73@psu.edu Department of Mathematics, Rose-Hulman Institute of Technology, 5000 Wabash Avenue, Terre Haute, IN lemkeorj@rose-hulman.edu Department of Mathematics and Statistics, Lederle Graduate Research Tower, Bo 3455, University of Massachusetts Amherst, Amherst, MA erossoli@student.umass.edu Department of Mathematics, 688 Kemeny Hall, Dartmouth College, Hanover, NH enrique.trevino@dartmouth.edu