Density of non-residues in Burgess-type intervals and applications

Transcription

1 Bull. London Math. Soc ) C 2008 London Mathematical Society doi:0.2/blms/bdm Density of non-residues in Burgess-type intervals and applications W. D. Banks, M. Z. Garaev, D. R. Heath-Brown and I. E. Shparlinski Abstract We show that for any fixed ε>0, there are numbers δ>0 and p 0 2 with the following property: for every prime p p 0 and every integer N such that p /4 e )+ε N p, the sequence, 2,...,N contains at least δn quadratic non-residues modulo p. We use this result to obtain strong upper bounds on the sizes of the least quadratic non-residues in Beatty and Piatetski-Shapiro sequences.. Introduction In 994 Heath-Brown conjectured the existence of an absolute constant c>0 such that, for all positive integers N and all prime numbers p, the interval [,N] contains at least cn quadratic residues modulo p. This conjecture has been established by Hall [2]. In the seminal work of Granville and Soundararajan [] it has been shown that if N is sufficiently large, then for every prime p more than 7.5% of the integers in [,N] are quadratic residues modulo p. On the other hand, for any fixed positive integer N there exist infinitely many primes p such that the interval [,N] is free of quadratic non-residues modulo p; see [0] for a more precise statement. In particular, complete analogues of the results of Hall [2] and of Granville and Soundararajan [] are not possible in the case of quadratic non-residues. In the present paper we show that for any given ε>0 there exists a constant cε) > 0 with the following property: for every sufficiently large prime p and every integer N in the range p /4 e )+ε N p, the interval [,N] contains at least cε)n quadratic non-residues modulo p. This is the partial analog of Hall s result for quadratic non-residues in Burgesstype intervals. We recall that the celebrated result of Burgess [6] states that the least positive quadratic non-residue modulo p is of size Op /4 e )+ε ) for any given ε>0, and the constant /4 e ) has never been improved. We apply our result on the density of non-residues to obtain strong upper bounds on the sizes of the least quadratic non-residues in Beatty and Piatetski-Shapiro sequences, which substantially improve all previously known results for these questions. 2. Statement of results For an odd prime p, we use p) to denote the Legendre symbol modulo p, and we put S p x) = n xn p) x ). Theorem 2.. p, the bound For every ε>0 there exists δ>0 such that, for all sufficiently large primes S p N) δ)n Received 8 January 2007; revised 3 September 2007; published online March Mathematics Subject Classification A5, L40, N37.

2 NON-RESIDUES IN BURGESS-TYPE INTERVALS AND APPLICATIONS 89 holds for all integers N in the range p /4 e )+ε N p. For two fixed real numbers α and β, the corresponding non-homogeneous Beatty sequence is the sequence of integers defined by B α,β = αn + β ) n=. Beatty sequences appear in a variety of apparently unrelated mathematical settings, and because of their versatility, the arithmetic properties of these sequences have been extensively explored in the literature; see, for example, [, 5, 7, 8, 2, 28] and the references contained therein. For each prime p, let N α,β p) denote the least positive integer n such that αn + β is a quadratic non-residue modulo p we formally put N α,β p) = if no such integer exists). Below, we show that Theorem 2. can be applied to establish the following Burgess-type bound, which substantially improves earlier results in [3, 4, 7, 22 24]. Theorem 2.2. Let α, β be fixed real numbers with α irrational. Then, for every ε>0 the bound N α,β p) p /4 e )+ε holds for all sufficiently large primes p. We remark that the irrationality of α is essential to our argument. Even in the simple case α =3,β =, we have not been able to improve upon the inequality N 3, p) p /4+o) which follows from the Burgess bound on the relevant character sum. Next, let N c p) be the least positive integer n such that n c is a quadratic non-residue modulo p. It is easy to show that N c p) exists for any non-integer c>. For values of c close to, good upper bounds for N c p) have been obtained in [7, 20]. Here, we establish a much stronger bound by appealing to Theorem 2.. It is formulated in terms of exponent pairs; we refer to [9, 5, 6, 25 27] for their exact definition and properties. Theorem 2.3. Let κ, λ) be an exponent pair, and suppose that <c<+ λ 2κ λ +3. Then, for every ε>0 the bound holds for all sufficiently large primes p. N c p) p /42 c) e )+ε The classical exponent pair κ, λ) =/2, /2) implies that Theorem 2.3 is valid for c in the range <c<8/7. Graham s optimization algorithm see [8, 9]) extends this range to <c<+ R = , 2 R where R = is Rankin s constant. Note that as c + our upper bound for N c p) tends to the Burgess bound, which illustrates the strength of our estimate.

3 90 W. D. BANKS ET AL. 3. Proofs 3.. Proof of Theorem 2. We can assume that 0 <ε 0.0. In view of the identities #{n x : n p) =±} = n x 2 ± n p)) = 2 x ±S px)) x ), and taking into account the result of Hall [2] mentioned earlier, it suffices to establish only the lower bound #{n N : n p) = } 2 δn with N in the stated range. By the character sum estimate of Hildebrand [4] which extends the range of validity of the Burgess bound [6]) it follows that S p p /4 )=op /4 )asp ; therefore, #{n p /4 : n p) = } =0.5 + o))p /4. Since every non-residue n is divisible by a prime non-residue q, wehave o))p /4 p /4 q, and thus p /4 e )+0.5ε <q p / o) n p /4 q n q p)= s j= q j + q p /4 q p)= p /4 e )+0.5ε <q p /4 where q <...<q s are the prime quadratic non-residues modulo p that do not exceed p /4 e )+0.5ε. Using Mertens formula see [3, Theorem 427]), we bound the latter sum by q = log log p /4 ) ) +O ε, log p /4 e )+0.5ε log p where the inequality holds for all sufficiently large p. Consequently, s ε q j j= if the prime p is large enough. For each j =,...,k, let N j denote the set of positive quadratic residues modulo p which do not exceed N/q j. From the result of Granville and Soundararajan [] wehave #N j 0.N j =,...,s). q j In particular, if q ε, then the numbers {q n : n N } are all positive non-residues of size at most N, and the theorem follows from the lower bound #N 0.εN. Now suppose that q >ε. In this case, we can choose k such that ε k l= q l 2 ε. q,

4 NON-RESIDUES IN BURGESS-TYPE INTERVALS AND APPLICATIONS 9 For each j =,...,s, let M j be the set of numbers in N j that are not divisible by any of the primes q,...,q k ; then k N 0. 2 ε)n #M j #N j 0.08N, q j q l q j q j l= where we have used the fact that ε 0.0 for the last inequality. It is easy to see that the numbers of the form q j n with j {,...,k} and n M j are distinct non-residues of size at most N, and the number of such integers is k k 0.08N #M j 0.08εN. q j= j= j This completes the proof of Theorem Proof of Theorem 2.2 Using Theorem 2., we immediately obtain the following result, which is needed in our proof of Theorem 2.2 below. Lemma 3.. Let σ {±} be fixed. For every ε>0 there exists a constant η>0 such that, for all sufficiently large primes p, the lower bound # {n, m) : n N, m M, nm p) =σ} ηnm holds with N = p /4 e )+ε and an arbitrary positive integer M. The next elementary result characterizes the set of values taken by the Beatty sequence B α,β in the case that α>. Lemma 3.2. Let α>. A positive integer m>β belongs to the Beatty sequence B α,β if and only if 0 < {α m β +)} α, and in this case m = αn + β if and only if n = α m β). The following estimate is a particular case of a series of similar estimates dating back to the early works of Vinogradov see, for example, [29]). Lemma 3.3. Let λ be a real number and suppose that the inequality λ r q q 2 holds for some integers r and q with gcdr, q) =. Then, for any complex numbers a n,b m such that max { a n } and max { b m }, n N m M the following bound holds: n N m M where ez) = exp2πiz) for all z R. a n b m eλnm) NM N + M + q + q NM,

5 92 W. D. BANKS ET AL. Considering for every integer h the sequence of convergents in the continued fraction expansion of λh, from Lemma 3.3 we derive the following statement. Corollary 3.4. For every irrational λ, there are functions H λ K) and ρ λ K) 0 as K such that for any complex numbers a n,b m such that max n N { a n } and max m M { b m }, the bound a n b m eλhnm) n N m M ρ λk)nm holds for all integers h in the range h H λ K), where K = min{n,m}. In particular, if λ is irrational and h 0 is fixed, then a n b m eλhnm) =onm) n N m M whenever min{n,m}. We now turn to the proof of Theorem 2.2. Case : α>. Put λ = α, and let σ {±} be fixed. For all integers N,M and primes p, we consider the set of ordered pairs W σ p N,M) ={n, m) : n N, m M, nm p) =σ}. For every ε>0, Lemma 3. shows that there is a constant η>0 such that, for all sufficiently large primes p, the inequality #Wp σ N,M) ηnm holds with N = p /4 e )+ε/2 and an arbitrary positive integer M. For every large prime p, let N be such an integer, and put M = p ε/2. To prove Theorem 2.2 when α>, by Lemma 3.2 it suffices to show that the set Vp σ N,M) = { n, m) Wp σ N,M) : 0 < {λnm λβ + λ} λ } is non-empty for σ = when p is sufficiently large. In fact, we shall prove this result for either choice of σ {±}. To simplify the notation, write W σ = Wp σ N,M) and V σ = Vp σ N,M). To estimate #V σ, we use the well-known Erdős Turán inequality between the discrepancy of a sequence and its associated exponential sums; for example, see [9, Theorem 2.5, Chapter 2]. For any integer H>, we have #V σ λ #W σ #Wσ H H + h eλhnm) h= n,m) W. σ Applying Corollary 3.4 with the choice { H = min H λ K), exp ρ λ K) /2)}, where K = min{n,m} as before, we see that #V σ λ #W σ #Wσ H + ρ λk) NM log H NM log H.

6 NON-RESIDUES IN BURGESS-TYPE INTERVALS AND APPLICATIONS 93 Since H as p, and the lower bound holds by Lemma 3., it follows that #W σ ηnm #V σ λη+ o)) NM p ). In particular, V σ for either choice of σ {±} once p is sufficiently large. Case 2: 0 <α<. In this case, Theorem 2.2 follows easily from the classical Burgess bound for the least quadratic non-residue modulo p since the sequence B α,β contains all integers exceeding α + β. Case 3: α<0. We note that the identity αn + β = αn β + holds for all n with at most O) exceptions since α is irrational), and hence the sequences B α,β and B α, β+ are essentially the same. If α<, then we argue as in Case with α replaced by α > and β replaced by β +. Choosing σ = p), Theorem 2.2 then follows from the fact that V σ once p is sufficiently large. Finally, if <α<0, then we note that the sequence B α,β contains all integers up to α + β. Hence, the result follows from the Burgess bound in the case that p) =+and from the ubiquity of quadratic residues modulo p in the case that p) = Proof of Theorem 2.3 The following statement is a variant of [9, Lemma 4.3] we omit the proof, which follows the same lines). Lemma 3.5. Let L and M be large positive parameters, and let κ, λ) be an exponent pair. Then for any complex numbers a l,b m such that the bound L/2<l L M/2<m M max { a l } and max { b m }, L/2<l L M/2<m M a l b m ehl /c m /c ) h κ0 L κ0/c+λ0 M κ0+κ0/c + h /2 LM) /2c) + LM /2) log L holds for any h, where κ 0 = κ 2κ +2 and λ 0 = κ + λ + 2κ +2. Turning to the proof of Theorem 2.3, let us fix c in the range <c<+ λ 2κ λ +3. If L and M are sufficiently large, and l L/2,L], m M/2,M] are integers such that 2LM) /c {l/c m /c },

7 94 W. D. BANKS ET AL. then n c = lm for some integer n. Indeed, with n = l /c m /c + we see that n c lm, and also ) c n c = l /c m /c + {l /c m /c } ) c lm + 2l /c m /c LM) /c lm + ) c <lm+, 2lm where the last inequality holds if L and M are large enough. Below, we work with integers L, M that tend to infinity with the prime p. Let log2/δ) δ J = and δ = log 2 2J +), where δ is as in Theorem 2.. Since 2 J <δ/2, by considering the intervals 2 j p /4 e+ε, 2 j p /4 e+ε ] for j =0,...,J we see that there is an integer L with 2 J p /4 e+ε <L p /4 e+ε such that the interval L/2,L] contains a set L with #L δ L quadratic non-residues modulo p. Let A be a large positive constant. From the aforementioned result of Hall [2] we see that there exists an integer M with L 2c )/2 c) log L) A M L 2c )/2 c) log L) A such that the interval M/2,M] contains a set M with #M δ 2 M quadratic residues modulo p, where δ 2 > 0 is an absolute constant. It suffices to show that for some integers l L, m M the inequality 2LM) /c {l/c m /c } holds. As in the proof of Theorem 2.2, from the Erdős Turán inequality we see that for any H the number of solutions T of this inequality is T = #L#M 2LM) /c +O LM H + H h= 0.5δ δ 2 LM) /c c 0LM H c 0 h H h= L/2<l L M/2<m M h ehl /c m /c ) ehl /c m /c ), L/2<l L M/2<m M where c 0 is an absolute constant. Take H = 4c 0 LM) /c /δ δ 2 ). With this choice it suffices to prove that H c 0 h ehl /c m /c ) h= L/2<l L M/2<m M < 0.δ δ 2 LM) /c. If A is large enough, then this inequality follows from Lemma 3.5, which in turn implies that T>0and concludes the proof. 4. Remarks We are grateful to the referee who has pointed that some recent work of Granville and Soundararajan unpublished) contains the following result, which yields a stronger form of our Theorem 2..

8 NON-RESIDUES IN BURGESS-TYPE INTERVALS AND APPLICATIONS 95 Theorem 4.. Let x be large, and let f be a completely multiplicative function with fn) for all n. Suppose that fn) =ox). Then for / e α we have fn) n x max{ ξ, /2 + 2log α)2 } + o))x α, α where ξ = 2log + e log t e )+4 dt = t + n x We note that ξ is the same constant that appears in [, Theorem ] where it is called δ, which has a different meaning in our paper). The referee has suggested that the following conjecture seems natural. Conjecture. Let x be large, let f be a completely multiplicative function with fn) for all n, and suppose that fn) =ox). n x Then for / e α wehave fn) n x 2 log α + o))xα. α Finally, the referee also observes that Theorem 2.2 holds also for rational α 0. The proof uses recent work of Balog, Granville and Soundararajan [2]. References. A. G. Abercrombie, Beatty sequences and multiplicative number theory, Acta Arith ) A. Balog, A. Granville and K. Soundararajan, Multiplicative functions in arithmetic progressions, Preprint, 2007, arxiv:math/ W. Banks and I. E. Shparlinski, Non-residues and primitive roots in Beatty sequences, Bull. Austral. Math. Soc ) W. Banks and I. E. Shparlinski, Short character sums with Beatty sequences, Math. Res. Lett ) A. V. Begunts, An analogue of the Dirichlet divisor problem, Moscow Univ. Math. Bull ) D. A. Burgess, The distribution of quadratic residues and non-residues, Mathematika 4 957) M. Z. Garaev, A note on the least quadratic non-residue of the integer-sequences, Bull. Austral. Math. Soc ). 8. S. W. Graham, An algorithm for computing optimal exponent pairs, J. London Math. Soc. 2) ) S. W. Graham and G. Kolesnik, Van der Corput s method of exponential sums Cambridge University Press, Cambridge, 99). 0. S. W. Graham and C. J. Ringrose, Lower bounds for least quadratic nonresidues, Analytic number theory, Allerton Park, IL, 989 Birkhäuser, Boston, MA, 990) A. Granville and K. Soundararajan, The spectrum of multiplicative functions, Ann. of Math ) R. R. Hall, Proof of a conjecture of Heath-Brown concerning quadratic residues, Proc. Edinb. Math. Soc. 2) ) G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th edn The Clarendon Press, Oxford University Press, New York, 979).

9 96 NON-RESIDUES IN BURGESS-TYPE INTERVALS AND APPLICATIONS 4. A. Hildebrand, A note on Burgess character sum estimate, C. R. Math. Rep. Acad. Sci. Canada 8 986) M. N. Huxley, Area, lattice points and exponential sums The Clarendon Press, Oxford University Press, New York, 996). 6. A. Ivić, The Riemann zeta-function John Wiley & Sons, Inc., New York, 985). 7. T. Komatsu, The fractional part of nϑ + ϕ and Beatty sequences, J. Théor. Nombres Bordeaux 7 995) T. Komatsu, A certain power series associated with a Beatty sequence, Acta Arith ) L. Kuipers and H. Niederreiter, Uniform distribution of sequences Wiley-Interscience, New York, 974). 20. Y. K. Lau and J. Wu, On the least quadratic non-residue, Preprint, 2006, 2. K. O Bryant, A generating function technique for Beatty sequences and other step sequences, J. Number Theory ) S. N. Preobrazhenskiĭ, On the least quadratic non-residue in an arithmetic sequence, Moscow Univ. Math. Bull ) S. N. Preobrazhenskiĭ, On power non-residues modulo a prime number in a special integer sequence, Moscow Univ. Math. Bull ) S. N. Preobrazhenskiĭ, On the least power non-residue in an integer sequence, Moscow Univ. Math. Bull ) O. Robert and P. Sargos, A fourth derivative test for exponential sums, Compositio Math ) O. Robert and P. Sargos, A third derivative test for mean values of exponential sums with application to lattice point problems, Acta Arith ) P. Sargos, An analog of van der Corput s A 4 -process for exponential sums, Acta Arith ) R. Tijdeman, Exact covers of balanced sequences and Fraenkel s conjecture, Algebraic number theory and Diophantine analysis, Graz, 998 de Gruyter, Berlin, 2000) I. M. Vinogradov, An improvement of the estimation of sums with primes, Izvestia Akad. Nauk SSSR 7 943) 7 34 Russian). W. D. Banks Department of Mathematics University of Missouri Columbia, MO 652 USA bbanks@math missouri edu D. R. Heath-Brown Mathematical Institute 24-29, St Giles Oxford OX 3LB UK rhb@maths ox ac uk M. Z. Garaev Instituto de Matemáticas Universidad Nacional Autónoma de México C.P Morelia Michoacán México garaev@matmor unam mx I. E. Shparlinski Department of Computing Macquarie University Sydney NSW 209 Australia igor@ics mq edu au