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The Least Common Multiple of a Quadratic Sequence

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The Least Common Multiple of a Quadratic Sequence The least common multiple of a quadratic sequence Javier Cilleruelo Abstract For any irreducible quadratic polynomial f(x) in Z[x] we obtain the estimate log l.c.m. (f(1); : : : ; f(n)) = n log n + Bn + o(n) where B is a constant depending on f. 1. Introduction The problem of estimating the least common multiple of the ¯rstPn positive integers was ¯rst investigated by Chebyshev [Che52] when he introduced the function ª(n) = pm6n log p = log l.c.m.(1; : : : ; n) in his study on the distribution of prime numbers. The prime number theorem asserts that ª(n) » n, so the asymptotic estimate log l.c.m.(1; : : : ; n) » n is equivalent to the prime number theorem. The analogous asymptotic estimate for any linear polynomial f(x) = ax + b is also known [Bat02] and it is a consequence of the prime number theorem for arithmetic progressions: q X 1 log l.c.m.(f(1); : : : ; f(n)) » n ; (1) Á(q) k 16k6q (k;q)=1 where q = a=(a; b). We address here the problem of estimating log l.c.m.(f(1); : : : ; f(n)) when f is an irreducible quadratic polynomial in Z[x]. When f is a reducible quadratic polynomial the asymptotic estimate is similar to that we obtain for linear polynomials. This case is studied in section x4 with considerably less e®ort than the irreducible case. We state our main theorem. Theorem 1. For any irreducible quadratic polynomial f(x) = ax2 + bx + c in Z[x] we have log l.c.m. (f(1); : : : ; f(n)) = n log n + Bn + o(n) where B = Bf is de¯ned by the formula X (d=p) log p 1 X ³ r ´ B = γ ¡ 1 ¡ 2 log 2 ¡ + log 1 + (2) f p ¡ 1 Á(q) q p 16r6q (r;q)=1 X ³1 + (d=p) X s(f; pk)´ + log a + log p ¡ : p ¡ 1 pk pj2aD k>1 In this formula γ is the Euler constant, D = b2 ¡ 4ac = dl2, where d is a fundamental discriminant, (d=p) is the Kronecker symbol, q = a=(a; b) and s(f; pk) is the number of solutions of f(x) ´ 0 (mod pk) which can be easily calculated using Lemma 2. 2 For the simplest case, f(x) = x + 1, the constant Bf in Theorem 1 can be written as log 2 X (¡1=p) log p B = γ ¡ 1 ¡ ¡ ; (3) f 2 p ¡ 1 p6=2 2000 Mathematics Subject Classi¯cation 2000 Mathematics Subject Classi¯cation: 11N37. Keywords: least common multiple, quadratic sequences, equidistribution of roots of quadratic congruences This work was supported by Grant MTM 2008-03880 of MYCIT (Spain) Javier Cilleruelo p¡1 where (¡1=p) is the Kronecker symbol (or Legendre symbol since p is odd) de¯ned by (¡1=p) = (¡1) 2 when p is odd. In section x3 we give an alternative expression for the constant Bf which is more convenient for numerical computations. As an example we will see that the constant Bf in (3) can be written as log 2 X1 ³0(2k) X1 L0(2k; ) X1 log 2 B = γ ¡ 1 ¡ + + ¡4 ¡ f k k 2k 2 ³(2 ) L(2 ;¡4) 2 ¡ 1 k=1 k=0 k=1 = ¡0:066275634213060706383563177025::: It would be interesting to extend our estimates to irreducible polynomials of higher degree, but we have found a serious obstruction in our argument. Some heuristic arguments and computations allow us to conjecture that the asymptotic estimate log l.c.m. (f(1); : : : ; f(n)) » (deg(f) ¡ 1)n log n (4) holds for any irreducible polynomial f in Z[x] of degree deg(f) > 3. In subsection 2.4 we explain the obstruction to prove this conjecture. There we also proof that log l.c.m. (f(1); : : : ; f(n)) » n log n (5) holds for any irreducible quadratic polynomial f(x). Although this estimate is weaker than Theorem 1, the proof is easier. To obtain the linear term in Theorem 1 we need a more involved argument. An important ingredient in this part of the proof is a deep result about the distribution of the solutions of the quadratic congruences f(x) ´ 0 (mod p) when p runs over all the primes. It was proved by Duke, Friedlander and Iwaniec [DFI95] (for D < 0), and by Toth (for D > 0). Actually we need a more general statement of this result, due to Toth. Theorem 2 [Tot00]. For any irreducible quadratic polynomial f in Z[x], the sequence fº=p; 0 6 º < p 6 x; p 2 S; f(º) ´ 0 (mod p)g is well distributed in [0; 1) as x tends to in¯nity for any arithmetic progression S containing in¯nitely many primes p for which the congruence f(x) ´ 0 (mod p) has solutions. Acknowledgment. We thank to Arpad Toth for clarifying the statement of Theorem 1.2 in [Tot00], to Guoyou Qian for detecting a mistake in a former version of Lemma 2, to Adolfo Quir¶osfor conversations on some algebraic aspects of the problem, to Enrique Gonz¶alezJim¶enezfor the calculations of some constants and to Fernando Chamizo for some suggestions and a carefully reading of the paper. Finally we thank to the anonymous referee for some valuable suggestions which have improved the quality of the presentation of this paper. 2. Proof of Theorem 1 2.1 Preliminaries For f(x) = ax2 + bx + c we de¯ne D = b2 ¡ 4ac and Ln(f) = l.c.m.(f(1); : : : ; f(n)): Since Ln(f) = Ln(¡f) we can assume that a > 0. Also we can assume that b and c are nonnegative integers. If this is not the case, we consider a polynomial fk(x) = f(k + x) for a k such that fk(x) has nonnegative coe±cients. Then we observe that log Ln(f) = log Ln(fk) + Ok(log n) and that this error term is negligible for the statement of Theorem 1. We de¯ne the numbers ¯p(n) by the formula Y ¯p(n) Ln(f) = p (6) p 2 The least common multiple of a quadratic sequence where the product runs over all the primes p. The primes involved in this product are those for which the congruence f(x) ´ 0 (mod p) has some solution. Except for some special primes (those such that p j 2aD) the congruence f(x) ´ 0 (mod p) has 0 or 2 solutions. We will discus this in detail in Lemma 2. We denote by Pf the set of non special primes for which the congruence f(x) ´ 0 (mod p) has exactly two solutions. More concretely Pf = fp : p - 2aD; (D=p) = 1g where (D=p) is the Kronecker symbol. This symbol is just the Legendre symbol when p is an odd prime. The quadratic reciprocity law shows that the set Pf is the set of the primes lying in exactly '(4D)=2 of the '(4D) arithmetic progressions modulo 4D, coprimes with 4D. As a consequence of the prime number theorem for arithmetic progressions we have x #fp 6 x : p 2 P g » f 2 log x or equivalently, X x 1 » : log x 06º<p6x f(º)´0 (mod p) Let C = 2a + b. We classify the primes involved in (6) in { Special primes: those such that p j 2aD: 8 2=3 >Small primes : p < n : ( <> 2 2=3 bad primes: p j f(i) for some i 6 n: { p 2 Pf : Medium primes: n 6 p < Cn : > good primes: p2 - f(i) for any i 6 n: :> Large primes: Cn 6 p 6 f(n): We will use di®erent strategies to deal with each class. 2.2 Large primes We consider Pn(f) and the numbers ®p(n) de¯ned by Yn Y ®p(n) Pn(f) = f(i) = p : (7) i=1 p Next lemma allow us to analyze the large primes involved in (6). Lemma 1. If p > 2an + b then ®p(n) = ¯p(n). Proof. If ¯p(n) = 0 then ®p(n) = 0. If ®p(n) > ¯p(n) > 1 then there exist i < j 6 n such that p j f(i) and p j f(j). It implies that p j f(j) ¡ f(i) = (j ¡ i)(a(j + i) + b). Thus p j (j ¡ i) or p j a(j + i) + b, which is not possible because p > 2an + b. Since C = 2a + b we can write X log Ln(f) = log Pn(f) + (¯p(n) ¡ ®p(n)) log p: (8) p<Cn Indeed we can take C to be any constant greater than 2a+b. As we will see, the ¯nal estimate of log Ln(f) will not depend on C. The estimate of log Pn(f) is easy: Yn Yn ³ b c ´ log P (f) = log f(k) = log ak2 1 + + (9) n ka k2a k=1 k=1 Xn ³ b c ´ = n log a + log(n!)2 + log 1 + + ka k2a k=1 = 2n log n + n(log a ¡ 2) + O(log n) 3 Javier Cilleruelo and we obtain X log Ln(f) = 2n log n + n(log a ¡ 2) + (¯p(n) ¡ ®p(n)) log p + O(log n): (10) p<Cn 2.3 The number of solutions of f(x) ´ 0 (mod pk) and the special primes The number of solutions of the congruence f(x) ´ 0 (mod pk) will play an important role in the proof of Theorem 1. We write s(f; pk) to denote this quantity. The lemma below resumes all the casuistic for s(f; pk).
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