Abstract
The value of sums of the type
where G is a linear polynomial, a quadratic irreducible polynomial, a sequence connected with primes, etc., has been largely studied. We give here a first result concerning the distribution modulo 1 of such sequences for the case of polynomials of arbitrary degree.
Dedicated to Professor Kumar Murty for his 60th birthday.
1 Introduction
The values of sums of the type
where G is a linear polynomial, a quadratic irreducible polynomial, a sequence connected with primes, etc., have been largely studied; we simply mention the references [8, 10]. More recently, the distribution modulo one of those sequences has been approached in the cases when G is a linear polynomial [3, 5], a quadratic irreducible polynomial [1, 4] or the values of a linear polynomial at shifted primes [2]. We give here a first result concerning the distribution modulo 1 of such sequences for the case of polynomials of arbitrary degree.
Theorem 1
Let \(\varphi \) denote the Euler function and G be a non-constant polynomial with integral coefficients and taking positive values at positive arguments. The sequence
is dense modulo 1.
A first remark is that, without loss of generality, we may assume that the polynomial G has no quadratic irreducible rational factor. We keep this assumption throughout the paper.
The proof is somewhat close to that of [1], in that we construct convenient consecutive values of G of the type \(G(Qx+n_0+m)\) for \(1 \le m \le M\), by forcing some medium-range factors and forbidding the other ones. However, two difficulties occur. First, the medium-range factors used for some m can also be present for some \(m'\): a combinatorial argument in Lemma 1 takes care of this phenomenon. Second, we have no precise control on the small prime factors: this is dealt with by a mean value argument in Lemma 2.
We’ll come back to similar questions for different arithmetical functions as well as for considering the values of polynomials at prime arguments.
2 Notation
The following notation will be consistently used in the whole paper and not always repeated.
The letters p and q are restricted to denote prime numbers.
For a polynomial F with integral coefficients, we let
We say that a prime p is a fixed divisor of F if \(\omega (F, p) = p\).
We denote by G a non-constant polynomial with integral coefficients which takes positive values at positive arguments; we moreover assume that the polynomial G has no square irreducible rational factor; its degree is denoted by g, and we denote by h an integer which is at least equal to the minimum of the degree of G and the maximal fixed prime divisor of G; in other words we have
3 Preliminary Lemmas
Lemma 1
For M large enough (in term of G), one can find a finite set of primes \(\mathcal {Q}\) and a positive integer \(n_0\) such that any \(q \in \mathcal {Q}\) is larger than \(2(h+1)M\) and
Proof
Due to Nagell [9], we know that the series of the inverses of prime numbers p, such that the equation \(G(y) \equiv 0 \, ({\text {mod}}\, p)\) has a solution, is divergent.
For a prime number p and a positive integer M, we call a subset C of \([-M, +M]\) a (p, M)-pattern of the roots of G modulo p if it contains 0 and if there exists an integer \(\ell _p\) such that \(\forall x \in [-M, +M] :G(x+\ell _p) \equiv 0 \, ({\text {mod}}\, p) \Leftrightarrow x\in C\). We first notice that to a prime p we can associate a (p, M)-pattern if and only if the equation \(G(y) \equiv 0 \, ({\text {mod}}\, p)\) has a solution. Secondly, if p is not a fixed divisor of G, then any (p, M)-pattern contains at most g elements.
Since, for a given M, the number of possible (p, M)-patterns is finite, we can find a subset \(C_M\) of \([-M, +M]\) which contains 0 and at most g elements, which is a (p, M)-pattern for a set of primes, say \(\mathcal {R}\), such that the series of the inverses of those primes diverges; thus, for each \(m \in [1, M]\), we can find a set of primes \(\mathcal {Q}_m\) in \(\mathcal {R}\) having the following properties
We now define \(\mathcal {Q} \) as the union of the sets \(\mathcal {Q}_m\) for \(1 \le m \le M\). By definition, for each \(q \in \mathcal {Q}_m\) there exists an integer \(\ell _q\) such that \(\forall x \in [-M, +M] :G(\ell _q+x) \equiv 0 \, ({\text {mod}}\, q) \Leftrightarrow x\in C_M\); we define \(n_q = \ell _q -m\). By the Chinese remainder theorem, we can find \(n_0\) which is congruent to any \(n_q\) modulo q, for any \(q \in \mathcal {Q}\). Since \(0 \in C_M\), all the elements in \(\mathcal {Q}_m\) divide \(G(n_0+m)\); the upper bound in (3.1) comes from the upper bound in (3.5).
Let us now prove that for \(q \in \mathcal {Q}\), the condition \(q | G(n_0 + m)\) implies that q belongs to a \(\mathcal {Q}_{m'}\), with \(m-m' \in C_M\). By the definition of \(n_0\), the relation \(q | G(n_0 + m)\) is equivalent to \(q | G(n_q + m)\). Since \(q \in \mathcal {Q}\), there exists \(m'\) such that \(q \in \mathcal {Q}_{m'}\). Then \(q | G(n_q + m)\), which can be written as \(q | G(n_q + m' + (m-m'))\), where \((m-m') \in [-M, M]\) is equivalent to \((m-m') \in C_M\), as asserted. We thus have
and the lower bound in (3.1) comes from the lower bound in 3.5 and the fact that \(C_M\) has at most g elements. \(\square \)
Lemma 2
Let a, b, c, k be positive integers with \(bk>4a\) and u(n, j) be real numbers with
We have
Remark The value of the constant in the right-hand side of (3.7) has no importance. What is indeed important for our application is that for a, b, c given, the right-hand side of (3.7) tends to infinity with k, even if it is not linearly in k.
Proof
Let us call S the sum which appears in (3.7). We denote as “bad guys” those integers j for which the product is at most \(\lambda = \exp (-4c/b)\) and “good guys” the other ones. A lower bound for S is \(\lambda \) times the number of good guys, and so it is enough to find an upper bound for the number of bad guys. The bad guys should concentrate a rather large part of the u(n, j), and since we have an upper bound for the total number of the u(n, j), there should not be too many bad guys. Let us show it.
It will be always implicit that the integers j are always limited to \(a \le j \le bk\) and the integers n to \(n \ge 2\); let us define \(\mathcal {B} = \{j :\sum _n u(n,j) \log (1-1/n) \le \log \lambda \}\) and \(\mathcal {G} = [a, bk] \backslash \mathcal {B}\). We have
In the other direction, we have
Using the fact that \(\log \left( 1 - \frac{1}{n}\right) \ge \frac{-2}{n}\), we get
whence
and so, using (3.6)
which implies
\(\square \)
4 A Sieve Argument
We start this section with a preliminary technical lemma.
Lemma 3
Let M, \(\mathcal {Q}\) and \(n_0\) satisfy Lemma 1 and let \(\Gamma \) be the polynomial defined by
Let p be a prime number which does not belong to \(\mathcal {Q}\) and is larger than \(2(h +1) M\). Then
Proof
We first show that p cannot be a fixed divisor of \(\Gamma \). Let us assume that p is a fixed divisor of \(\Gamma \), and consider the polynomial \(U(x) = \prod _{m=1}^{M} G(x + n_0 +m)\), so that \(\Gamma (x) = U(Qx)\). Since p does not divide Q, we can find (Bézout) integers a and b such that \(aQ+bp=1\); since for all x we have \(U(Qx) \equiv 0 ({\text {mod}}\, p)\), we have for all x : \(U(Qax+bpx) \equiv 0 ({\text {mod}}\, p)\), and so p is a fixed divisor of U. Since \(p > h\), p is not a fixed divisor of any \(G(x+n_0+m)\) and so the number of solutions of \(U(x) \equiv 0 ({\text {mod}}\, p)\) is at most \(gM < p\). Thus, p is not a fixed divisor of U and thus not a fixed divisor of \(\Gamma \). We thus have \(\omega (\Gamma , p) \le gM\) which is easily seen to be less than p / 2, whence the lemma. \(\square \)
Proposition 1
Let \(\Gamma \) be the polynomial introduced in Lemma 3. There exist infinitely many integers x such that \(\Gamma (x)\) has no other prime factor in \([2(h + 1) M, x^{1/7gM}]\) than those from \(\mathcal {Q}\).
Proof
This result is fairly classical, although we did not find a direct way to quote it from the literature. The general sieve result which is sufficient for our purpose is Theorem 7.4 of Halberstam and Richert’s book [7]; this result was introduced as Theorem B by those authors in their paper [6] devoted to polynomial values. This is fine for us so far, except that they are dealing with polynomials with no fixed prime divisors—which is not our case—and they are more interested in almost prime values, for which they consider a sophisticated combination of the lower-bound and upper-bound sieves which is not needed in our case. Thus, we shall use Theorem B from [6] and take directly from [6] the treatment of the conditions necessary for applying Theorem B.
We define
Finally, for positive square-free d, we let
The three conditions \((\Omega _1), (\Omega _{2}(\kappa , L))~and~(R(\kappa , \alpha ))\) involved in Theorem B of [6] are defined in pp. 26 and 27, respectively. We must keep in mind that the important parameter is X, which tends toward infinity, and that all the “constants” \(A_1, A_2, L, \alpha , A_4, A_5\) may depend on \(\Gamma \). In our case,
-
condition \((\Omega _1)\) is satisfied with \(A_1 = 2\): this is our Lemma 3,
-
condition \( (\Omega _{2}(\kappa , L))\) is valid: the proof given in [6], p. 40, applies as well in our case, except that the choice of \(\mathcal {P}\) is different, which only implies a modification of the constants, etc., by other constants.
-
condition \((R(\kappa , \alpha ))\) is similarly valid with \(\alpha = 1\) and convenient constants \(A_4\) and \(A_5\).
A last step before being able to apply with success Theorem B is to notice that we have
which directly comes for [6], Relations (3.13) and (3.14).
Collecting the previous information, Theorem B and Relation (4.15) from [6] imply that when X is large enough we have
which implies Proposition 1. \(\square \)
5 Proof of the Main Result
We use the previous notation and let x be sufficiently large an integer such that \(\Gamma (x)\) has no prime factor in the interval \([2(h+1)M, x^{1/10h M}]\), except those from \(\mathcal {Q}\); then, the same is true for any number \(G(Qx+n_0+m)\) and \(m \in [1, M]\). This is what we retain from Sect. 4.
For \(m \in [1, M]\), we let
In order to prove Theorem 1, it is enough to prove
an expression which tends to zero with M, and
The first relation is the easier to obtain, since for any pair of integers r and s, one has \(\varphi (rs)/rs \le \varphi (r)/r\). By our construction, \(G(Qx+n_0+m)\) is divisible by the prime factors of \(\mathcal {Q}_m\) and so (5.1) follows from the upper bound in (3.5).
For proving the second one, we write
where in \(f_1(m)\) (resp. \(f_2(m)\), resp. \(f_3(m)\)) we collect the contribution of the primes which divide \(G(Qx+n_0+m)\) and are at most equal to \(2(h+1)M\) (resp. in \(\mathcal {Q}\), resp. larger than \(x^{1/7Mg}\)). By Proposition 1, we know that there are no other prime factors in \(G(Qx+n_0+m)\).
Since \(G(Qx+n_0+m) = O(x^g)\), the integer \(G(Qx+n_0+m)\) has at most \(8Mg^2\) prime factors larger than \(x^{1/7Mg}\). We thus have \(f_3(m) \ge \left( 1-x^{-1/Mg}\right) ^{8Mg^2} \ge 1/2\) as soon as x is large enough.
A lower bound for \(f_2(m)\) is directly given by Lemma 1: since any q in \(\mathcal {Q}\) divides Q, it is equivalent to say that \(q | G(n_0+m)\) or to say that \(q | G(Qx+n_0+m)\); we thus have \(f_2(m) \ge M^{-1/2}\).
We have so far \(f_2(m) f_3(m) \ge (1/2)M^{-1/2}\), and to prove (5.2), it is sufficient to prove
We have no direct control on the small prime factors, except that we know that they cannot be too many of them on average. We are going to apply Lemma 2. We let \(u(n, j)=1\) when \(n \le 2(h+1)M\) is a prime which divides \(G(Qx+n_0+j)\) and 0 otherwise; we thus have
In order to apply Lemma 2, we take \(a=b=1\), \(k=M\) and we have to show that there exists c such that \(\sum _{j\le M}u(n, j) \le cM/n\). If n is a fixed divisor of G, we recall that \(n \le h\) and we have
If n is not a fixed divisor of G, we have
Thus, the second part of (3.6) is satisfied and we can apply Lemma 2 with \(c= g(2h +3)\), which leads to (with a constant C depending on G only)
when M is large enough. This ends the proof of Theorem 1.
\(\Box \)
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Deshouillers, JM., Nasiri-Zare, M. (2018). Density Modulo 1 of a Sequence Associated with a Multiplicative Function Evaluated at Polynomial Arguments. In: Akbary, A., Gun, S. (eds) Geometry, Algebra, Number Theory, and Their Information Technology Applications. GANITA 2016. Springer Proceedings in Mathematics & Statistics, vol 251. Springer, Cham. https://doi.org/10.1007/978-3-319-97379-1_7
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