Sign changes of the Liouville function on some quadratic irreducible polynomials Anitha Srinivasan

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Abstract Let λ(n) be the Liouville function. Chowla conjectured that if 2 f (x) ∈ PxZ[x] and f (x) 6= b(g(x)) for any integer b and g(x) ∈ Z[x], then n=1 λ(f (n)) = o(x). Cassaigne et al. made a weaker conjecture, that states that under the same assumptions on f (x) as above, we have that λ(f (n) changes sign infinitely often. Both these conjectures are still unproven. We extend some existing results in the literature on quadratic polynomials for the weaker conjecture above, by providing infinite families of irreducible quadratic polynomials for which this conjecture is true. For instance, we prove that the weaker conjecture above holds for monic quadratic polynomials with odd prime discriminants. 2000 Mathematics Subject Classification: Primary 11D09, 11C08; Secondary 11Y65.

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Introduction

The Liouville function is defined as λ(n) = (−1)Ω(n) where Ω(n) denotes the number of prime factors of the integer n counted with multiplicity. In other words λ is a completely multiplicative function where for each prime p λ(p) = −1 and λ(1) is taken as 1. Chowla in [3, p. 96] made the following conjecture and comment. ∗

Department of Mathematics, Saint Louis University-Madrid campus, Avenida del Valle 34, 28003 Madrid, Spain

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“ Conjecture: Let f (x) be an arbitrary polynomial with integer coefficients, which is not, however, of the form c(g(x))2 where c is an integer and g(x) is a polynomial with integer coefficients. Then x X

λ(f (n)) = o(x).

n=1

If f (x) = x this is equivalent to the Prime Number Theorem. If the degree of f (x) is at least 2, this seems an extremely hard conjecture. ” Cassaigne et al. in [2] made the following weaker conjecture. Conjecture 1.1. (Cassaigne et al.) If f (x) ∈ Z[x] and f (x) 6= b(g(x))2 for any integer b and g(x) ∈ Z[x], then λ(f (n) changes sign infinitely often. It is easy to see that Chowla’s conjecture implies the above conjecture. Despite the fact that Conjecture 1.1 is weaker than Chowla’s conjecture, it still remains hard to prove for polynomials of degrees greater than 1. Cassaigne et al. proved their conjecture in some special cases. For quadratic polynomials they prove the following results. Theorem 1.2. ([2, Theorem 4]) If f (n) = (n + a)(bn + c) where a, b, c ∈ Z with a > 0 and ab 6= c, then λ(f (n) changes sign infinitely often. Theorem 1.3. ([2, Theorem 3]) Let a, b, c ∈ Z with a > 0 and let f (n) = an2 + bn + c with d = b2 − 4ac < 0. If 2a|b and there is a positive integer k such that   d 2 λ − k + 1 = −1 4 then λ(f (n) changes sign infinitely often. Borwein, Choi and Ganguly made further progress on Conjecture 1.1 by the following elegant result that simplifies the problem in the case of quadratic polynomials. Theorem 1.4. ([1, Theorem 2.2 ]) Let f (x) = ax2 +bx+c with a > 0 and let l be a positive integer such that al is not a perfect square. Then if f (n) = lm2 has one solution (n0 , m0 ) ∈ Z2 , then it has infinitely many solutions in N2 . As a result of the above theorem, to prove Conjecture 1.1 for quadratic polynomials, it is sufficient to find a pair of integers n1 and n2 such that λ(n1 ) 6= λ(n2 ). More specifically the authors prove the following theorem. 2

Theorem 1.5. ([1, Theorem 2.3]) Let f (x) = ax2 + bx + c with a ∈ N and b, c ∈ Z. Let d = b2 − 4ac and   |b| + (|d| + 1)/2 A= + 1. 2a Then {λ(f (n))}∞ n=A is either a constant sequence or changes sign infinitely often. As observed by the authors in [1] the solvability of the equation x2 − 4ldy 2 = d

(1.1)

plays a crucial role in solving the problem for quadratic polynomials ax2 + bx + c. Indeed they prove the following result where they are able to solve the above equation (1.1). Theorem 1.6. ([1, Theorem 2.5] Let f (x) = px2 + bx + x with p prime. Suppose the discriminant d = b2 − 4pc is a non-zero perfect square. Then λ(f (n)) changes sign infinitely often. The solvability of (1.1) is intimately related to the sign of the norm √ of the fundamental unit of the ring of integers in the quadratic field Q( d), which in turn is related √ to the parity of the period length of the continued fraction expansion of d. We use this connection to extend the results above by giving families of quadratic polynomials for which Conjecture 1.1 holds. In all the theorems above, those that provide definitive results (Theorems 1.2 and 1.6), in the sense that they apply to a class of polynomials, apply only to reducible polynomials (with square discriminants). The remaining results provide certain conditions under which Conjecture 1.1 holds. While these conditions may be verified for any given fixed polynomial, it is not clear that they apply to any family of polynomials. Indeed, from the results above one cannot conclude that Conjecture 1.1 holds for an infinite number of irreducible quadratic polynomials. Our main results below, however, enable us to conclude this as they apply to a family of polynomials. Theorem 1.9 and Corollary 1.8 provide infinite sets of irreducible quadratic polynomials for which Conjecture 1.1 is true, where Theorem 1.9 applies to polynomials of negative discriminant and Corollary 1.9 to those with positive disciminant. We would like to mention that our theorems use Theorems 1.5 and 1.3 above, thereby exhibiting infinite families of quadratic polynomials to which they apply. 3

Theorem 1.7. Let f (n) = n2 + bn + c where b, c are integers and d = b2 −√4c is odd. If λ(d) is odd and the period length of the continued fraction of d is odd, then λ(f (n)) changes sign infinitely often. The following result is an immediate corollary using the well known fact √ that the length of the period of p, where p is prime is odd. Corollary 1.8. Let f (n) = n2 + bn + c where b, c are integers and d = b2 − 4c is prime. Then λ(f (n)) changes sign infinitely often. Theorem 1.9. Let f (n) = an2 + bn + c where a, b, c are integers with d = b2 −4ac < 0 and 2a|b. If −d is a prime congruent to 3 modulo 4, then λ(f (n)) 4 changes sign infinitely often.

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A lemma by Legendre

This section is devoted to a lemma by Legendre that is used in one of our main theorems. Even though this lemma was proved way back in 1830, it has been revisited by several authors until some years ago. As was often the case with authors prior to the twentieth century, results were proved and then stated in a somewhat descriptive manner. Legendre wrote “Given any positive integer A that is not a square, it is always possible to decompose this number into two factors M and N such that one of the following two equations M x2 −N y 2 = ±1, M x2 −N y 2 = ±2 is satisfied, where the signs are to be taken appropriately.” We re-phrase the above statement of Legendre including the main ingredients in his proof in the following lemma. Lemma 2.1. (Legendre [5, Pp. 68-69] ) Let A be any positive integer that is not a square. √ Let l be the length of the period of the continued fraction expansion of A and let m be defined as ( l , if l is even m= 2 l if l is odd . √ If fg is the (m − 1)th convergent in the continued fraction of A with f 2 − Ag 2 = (−1)m−2 D, then there exist integers M, N and h such that A = M N and f = M h and one of the following holds. 1. D = M = 1 and h2 − Ag 2 = −1. 4

2. D = 2M is even and M h2 − N g 2 = 2(−1)m . 3. D = M > 1 is odd and M h2 − N g 2 = (−1)m . Note that (h, g) is the fundamental solution of the equation in each part of the of the continued fraction √ lemma above. It √ is well known that the length 2 of A, denoted by l( A) is odd if and only if x − Ay 2 = −1 √ has a solution. Therefore part 1 √ of the lemma above holds in the case when l( A) is odd. In the case when l( A) is even, one of parts 2 and 3 hold. Nagell [8, Theorem 3] proved this result in the case when A is square-free noting Legendre’s contribution ([8, Remark, p. 60]). He showed that if A is a square-free positive integer, then exactly one of the equations M x2 − N y 2 = ±E, where E = ±1 or ±2 and A = M N has a solution, where M > 1 when E = 1 and xy is odd in the case when E = ±2. He also states that if (r, s) is the 2 +N s2 2rs , |E| ) is the fundamental fundamental solution of this equation, then ( M r |E| 2 2 solution of the Pell equation x − Ay = 1. Nagell’s proof is based on elementary considerations of quadratic surds, while Legendre’s proof uses continued fractions. Recent proofs such as [7] and [6] also use continued fractions. However these authors make no mention of the works of either Legendre or Nagell. Given that Legendre’s proof using continued fractions dates back to 1830 and perhaps is the shortest and slickest of all proofs, we think it is worthwhile to revisit his proof, specially since it seems to have been overlooked by modern authors. We start with some basic facts on the Pell equation and continued fractions. Let A be any positive integer that is not a square. Let (p, q) be the fundamental solution of the Pell equation x2 − Ay 2 = 1, that is p, q are the smallest positive integers such that p2 − Aq 2 = 1. Let

√

(2.1)

A = [a0 , a1 , a2 , · · · , al−1 , 2a0 ] √ √ be the continued fraction expansion of A where l = l( A) is the length of its period and a1 , a2 , · · · al−1 is a palindrome. Let fk = [a0 , a1 , · · · ak ] gk be the k th convergent. 5

If

( m=

l , 2

l

if l is even if l is odd ,

2m−1 then it is well known that pq = fg2m−1 . Also well known (see for example [10, Theorems 2 and 6]) is that for all k≥1 gk−1 = [ak−1 , · · · a1 ] (2.2) gk−2

and for all k ≥ 0 fk gk−1 − fk−1 gk = (−1)k−1 . Proof of Lemma 2.1 From (2.2) (with k = m) we have   p gm−2 = [a0 , · · · a2m−1 ] = a0 , · · · , am−1 , am + q gm−1

(2.3)

(2.4)

using the symmetry of the continued fraction, namely that am+i = am−i for i = 1, · · · m − 1. To facilitate the reading of the proof we denote f = fm−1 , g = gm−1 , f 0 = fm−2 and g 0 = gm−2 . Equation (2.4) allows us to write f (am + p = q g(am +

g0 ) g 0 g ) g

+ f0 + g0

,

(2.5)

so that if we denote h0 = am g + 2g 0 , then p = f h0 + (f 0 g − f g 0 ) and q = gh0 .

(2.6)

From (2.1) and (2.3) we have p2 − Aq 2 = 1 = (f 0 g − f g 0 )2 and using (2.6) we obtain (f 2 − Ag 2 )h0 = 2f (f g 0 − f 0 g). If f 2 − Ag 2 = (f g 0 − f 0 g)D then h0 = and therefore D divides 2f . 6

2f D

(2.7) (2.8)

If D = 2M is even, then f = M h0 and (2.7) gives M 2 h2 − Ag 2 = 2M (f g 0 − f 0 g).

(2.9)

Note that g and M are coprime (as gcd(f, g) = 1) and hence M divides A. If A = M N then from (2.9) using (2.3) we have M h2 − N g 2 = 2(f g 0 − f 0 g) = 2(−1)m . Next consider the case when D = M is odd. It follows from (2.8) that f = M = M h so that from (2.7) using (2.3) we have M 2 h2 − Ag 2 = (−1)m M and hence again M divides A. If A = M N then the above gives h0 2

M h2 − N g 2 = (−1)m . Note that if M = 1 then the sign on the right hand side must be −1 as p and q are the smallest integers such that x2 − Ay 2 = 1 has a solution.

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Proofs of main theorems

The following result is well known, √ Lemma 3.1. Let d be a positive integer that is not a square. Then l( d) is odd if and only if the equation x2 − dy 2 = −1 has a solution in integers x and y. The following lemma is a consequence of Theorem 1.5 wherein the integer A (used in the lemma) is defined. Lemma 3.2. ([1, Theorem 2.4]) Let f (n) = n2 + bn + c with b, c integers. Suppose there exists a positive integer n0 ≥ A such that λ(f (n0 )) = −1. Then λ(f (n)) changes sign infinitely often. Proof of Theorem 1.8 By Lemma 3.2 it is sufficient to find an integer n such that n ≥ A and λ(f (n)) = −1. We will show that there exist integers √ n and m such that 2 f (n) = dm , so that λ(f (n)) = λ(d) = −1. As l( d) is odd, by Lemma 3.1 the equation x2 − dy 2 = −1 has a solution. As d is odd, it follows that x = 2x1 is even and hence 4x21 −dy 2 = −1 which gives d2 y 2 −4dx21 = d. Taking 7

and m = x1 , we have 2n+b = dy. It follows that (2n+b)2 −d = 4dm2 n = dy−b 2 that is f (n) = dm2 . It remains to show that n ≥ A. This follows readily as the negative Pell equation x2 − dy 2 = −1 has infinitely many solutions, so that y and hence n = dy−b may be taken as large as possible. 2 Proof of Theorem 1.9 By Theorem 1.3 it is sufficient to show that there exists an integer y such that   d 2 λ − y + 1 = −1. 4 We will find integers y and x such that − d4 y 2 + 1 = px2 for some prime p so that the above equation holds. In other words, if d = −4d0 it is sufficient to find integers x and y such that px2 − d0 y 2 = 1.

(3.1)

By the Chinese remainder theorem we may choose an integer p0 such that p0 ≡ 1

mod d0 and p0 ≡ −1

mod 4.

(3.2)

By Dirichlet’s theorem on primes in arithmetic progressions, there exists a prime p such that p ≡ p0 mod 4d0 . (3.3)   = 1, which is not possible as from If x2 −pd0 y 2 = −1 has a solution, then −1 p   (3.2) and (3.3) we have p ≡ p0 ≡ −1 mod 4 (and so −1 = −1). Therefore p the equation x2 − pd0 y 2 = −1 has no solution. It follows by Legendre’s Lemma 2.1, that one of the equations px2 − d0 y 2 = ±1 or px2 − d0 y 2 = ±2 has a solution. Given that p and d0 are both congruent to −1 modulo 4, it is clear that px2 − d0 y 2 = ±2 has no solution. Hence one of the equations ) = ( dp0 ) px2 − d0 y 2 = ±1 has a solution. If px2 − d0 y 2 = −1, then we have ( −1 d0 which is not possible, as on one hand ( −1 ) = −1 as d0 ≡ −1 mod 4 (by the d0 assumption in the theorem) and on the other hand from (3.2) and (3.3) we have ( dp0 ) = 1 as p ≡ p0 ≡ 1 mod d0 . Therefore px2 − d0 y 2 = 1 and we have shown that (3.1) holds and hence the result follows.

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References [1] P. Borwein, S.K.K. Choi and H. Ganguly, Sign changes of the Liouville function on quadratics, Canad. Math. Bull. 56 (2013), pp. 251 – 257. [2] J. Cassaigne, S. Ferenczi, C. Mauduit, J. Rivat and A. Sarkozy The Liouville function II, Acta Arithmetica XCV. 4 (2000), 343–359. [3] S. Chowla, The Riemann hypothesis and Hilbert’s tenth problem Gordon and Breach, New York, 1965. [4] C.F. Gauss, Disquisitiones Arithmeticae, English Edition, SpringerVerlag, New York, Berlin, Heidelberg, Tokyo (1986). [5] A. M. Legendre, Th´eorie des nombres, 3rd ed., Vol. 1, Paris, 1830. [6] R.A. Mollin A continued fraction approach to the diophantine equation ax2 − by 2 = ±1, JP journal of Algebra, Number Theory and Apps., 4 (2004), 159–207. [7] P.J. Rippon and H. Taylor, Even and odd periods in continued fractions of square roots, Fibonacci Quart., 42 (2004), no. 2, 170–180. [8] T. Nagell, On a special class of Diophantine equations of the second degree, Arkiv for Matematik, Band 3, nr 2, 1953. [9] I. Niven, H.S. Zuckerma and H.L. Montgomery, An introduction to the theory of numbers, 5th ed., John Wiley & Sons, Inc. [10] A.Ya. Khinchin Continued fractions, Dover publications, Mineola, New York, 1997.

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