ON THE LEAST PRIMITIVE ROOT MODULO p 2

Transcription

1 ON THE LEAST PRIMITIVE ROOT MODULO p 2 S. D. COHEN, R. W. K. ODONI, AND W. W. STOTHERS Let h(p) be the least positive primitive root modulo p 2. Burgess [1] indicated that his work on character sums yields the estimate h(p) = In 1, we show that this can be improved to for sufficiently large p. h(p)<p i+b (1) In 2, we adapt the method of Burgess and Elliott [3] to obtain the estimate n{xy l I h{p) 4 (log X) 2 (log log X)* (2) where, as usual, Tr(A') denotes the number of primes up to X. This improves the result of Burgess [1; p. 263], who obtained {logxf (log log A") 6 on the right. Finally, in 3, we give estimates, valid except on a thin set of primes, for the number of small primes which are primitive roots modulo p 2. These follow from the analogous expressions for primitive roots modulo p, given by Elliott [4]. All our results depend on the following argument. Suppose that JC and y are positive integers less than p 2. If they are p-th powers modulo p 2, and are congruent modulo p, then x = x p = y" = y (mod. p 2 ). Since x,y < p 2, we must have x y. 1. Bad primitive roots A primitive root modulo p is bad if it is not primitive modulo p 2. Write d(k, n) for the number of ways of expressing the integer n as the product of k integers, order being important. LEMMA 1. For each positive e, and each positive integer k, there exists a constant M{k, B) such that, if N ^ M(k, e) and n^n, then d(k y n) < N*. Proof. Clearly, d(2, n) is the number of divisors of n, so that the existence of M(2, e) is well known. Received 15 June, 1973; revised 28 July, [BULL. LONDON MATH. SOC, 6 (1974), 42-46]

2 ON THE LEAST PRIMITIVE ROOT MODULO p 2 43 For k > 2, it is enough to observe that, for any n, Let T(p) be the set of integers less than p which are p-th power residues modulo p 2. THEOREM 1. For each positive e, there exists a constant P(e) such that, if p > P(e), then card(t(p)) < p* +e. Proof. If card (T(p)) 5* p* +, there are at least p 1+2e ordered pairs of elements of T(p). The products of these pairs are less than p 2, and at least p 2e of these products coincide modulo p. By the above argument, the products, being p-th power residues modulo p 2, must be equal when they coincide modulo p. Thus, we have an integer less than p 2 with d(2, n) ^ p 2e. If p is sufficiently large, this contradicts the lemma. Now we observe that a primitive root modulo p is bad if and only if it belongs to T(p). An application of the last result bounds the number of bad primitive roots, viz. p i+e COROLLARY. With P(e) as in the theorem, if p > P(e), then there are less than bad primitive roots. THEOREM 2. For each positive e, there exists a constant Q(e) such that, if p > Q(e), then h(p) < p i+e. Proof. Burgess [2] has shown that, for sufficiently large p, there are at least p i primitive roots less than p i+e. It suffices, therefore, to show that less than p* are bad. Without loss of generality, we assume that e < 3/20. If we can find at least p* bad primitive roots in the range [2,p i+e ], then we can form at least p* ordered 5-tuples of them. By our restriction on e, the product of any 5 is less than p 2. Noting that at least p* coincide modulo p, we see that there must be an integer n less than p 2 with Again, an application of Lemma 1 gives a contradiction. 2. The average of h(p) To obtain (2), we use the method of Burgess and Elliott [3], with suitable modifications. The essential differences consist in replacing the estimate g(p) = O(p i+e ), used on p. 45 of [3], by (1) (or even the weaker estimate h(p) = O(p i+b )), and also in replacing their Lemma 4 by the following result.

3 44 S. D. COHEN, R. W. K. ODONI, AND W. W. STOTHERS LEMMA 2. Suppose that the hypotheses of [3; Lemma 4] are satisfied, and that, in addition, (20P/(f>(P)) 2 (\ogp) 2 <H<p* (3) Then, if p and H are sufficiently large, h'(p) ^ H, where h'(p) is the least prime primitive root modulo p 2. Proof Let S = S(H, p) be the set of all prime primitive roots modulo p lying in [l,h]. Then, following exactly the proof of Lemma 4 of [3], we have s = card (S) > ($n(h) -1) 4>{F)IP > 0(P) H/5P log H (4) for sufficiently large H (absolute). Assume that Lemma 2 is false, so that all members of S are bad. r = [2 \ogpf\ogh], and consider Let Since all members of S are primes, there are at least a = s r /r\ distinct integers in S r. Since H r ^ p 2, these are distinct modulo p 2 ; indeed, since T(p) contains S r, the basic argument shows that the integers in S r are distinct modulo p. Thus, a ^ p. However, a ^ (s/r) r so that, by (3), (4) and the definition of r, a> p for large enough p and H. The contradiction proves the lemma. LEMMA 3. // S{2) = {p ^ X :d{2,p-l)h'(p) < {logxf}, then, for B as in [3; Lemma 5], h{p) 4 X/(log X) 2 S(2) Proof. In following the proof of [3; Lemma 5], we need only show that, for all but 0{X*), say, of values of p < X, the condition (3) is satisfied. For sufficiently large X, (log*) 54 ^ H = max{r?(logx) 3 } 18 ^ X*, i = l, 2 and P/</>(P) <^ log log X. Hence, for large X, and p > X*, ((20P/<KP))) 2 (logp) 2 < (log log X) 2. (log X) 2 < H ^ p* In place of [3; Lemma 6], we use: LEMMA 4. In the notation of [3; Lemma 6], let V - (log log X) 2 S(5) = S(2)n{p < X : h'(p) < D^logX) 2 [{p (1) (p) + p (2) (p)(p-l)/0(p-l)} 4 where D y is an absolute constant, to be chosen later. Then, h(p)<x/(log X) 2. S(S) and + (log log log X) 2 ]

4 ON THE LEAST PRIMITIVE ROOT MODULO p 2 45 Proof. In this case, P/</>(P) <^ log log log X, so that (20PAKP)) 2 (log/?) 2 ^ ^(log*) 2 Oog log logx) 2, say, Instead of the H used in [3; Lemma 6], we put H = E 2 (\ogx) 2.max{X 1 4 R l \X 2 "R 2 \ (log log log X) 2 } where E 2 = max {E, x }, and E is defined on p. 47 of [3]. Thus, (3) will again be satisfied for all but O(X*) values of p. Since our S(7) is a subset of the S 7 used by Burgess and Elliott, we obtain, in place of [3; eqn. (28)], h'(p) < X/(\ogX) 2, while, if p S(2)\S(7), then h'(p) ^H, and we take D x = 2 16 E 2 to complete the proof of the lemma. To prove (2), it remains to estimate Z OogX)V 1) (p) 4 + Z pes(s) pes(5) + Z (log X) 2 (log log log X) 2 pes(5) The estimates of Burgess and Elliott for the sums corresponding to the first two sums remain valid. Clearly, the third sum is <^ X log X. (log log log-y) 2. This completes the proof of (2). 3. Elliott's results For our extension of Elliott's results, we can replace the first lemma by the following simple observation: an integer n can be written as a product of k primes in at most k\ ways. As a trivial corollary, we see that the number of prime p-th. power residues modulo p 2 in [l,p 1] is less than (2/?)*. The case of prime primitive roots modulo p 2 is more interesting. Let N(H,p) denote the number of primes less than H which are primitive modulo p 2. Using the results of Elliott [4], together with the above technique to bound the number of bad primes, we can obtain a useful estimate. THEOREM 3. For positive constants e, B, there exist constants F = F(e, B) and G = G(e, B) such that N(H,p) = (i) when H $s exp(f(loglogp)(log log logp)), except on a set of primes, E, with E(x) = (ii) when H ^ p e, except on a set of primes E', with E'(x) = O((logx) G ). (A(x) denotes the number of integers in A which are less than x.)

5 46 ON THE LEAST PRIMITIVE ROOT MODULO p We have replaced Elliott's 0(j>-l)/(p-l) with the expression appropriate to the situation; the effect is negligible. The proof is like that of Theorem 2, but with the idea of the second paragraph of Lemma 2 in place of Lemma 1. References 1. D. A. Burgess, " The average of the least primitive root modulo/? 2 ", Acta Arithmetica, 18 (1971), , " On character sums and primitive roots ", Proc. London Math. Soc, 12 (1962), and P. D. T. A. Elliott," The average of the least primitive root modulo/? ", Mathematika, 15 (1968), P. D. T. A. Elliott, " The distribution of primitive roots ", Canadian Math. /., 21 (1969), Department of Mathematics, University Gardens, Glasgow, G12 8QW.