Abstract
In the present paper we propose an asymptotic formula for R(H, k), the number of triples of positive integers \(x, y, z\le H\) such that \(x^{2}+ y^{2}+ z^{2}+ 1, x^{2}+ y^{2}+ z^{2}+ 2\) are k-free with \(k\ge 2.\) Especially, in the case of \(k= 2\) we prove that \(R(H, 2)= \sigma _{2} H^{3}+ O(H^{9/4+ \varepsilon }),\) where \(\sigma _{2}\) is an absolute constant and \(\varepsilon \) is an arbitrary small positive number, which improves the error term \(O(H^{7/3+ \varepsilon })\) given by Chen (Indian J Pure Appl Math, 2022. https://doi.org/10.1007/s13226-022-00292-z). The key point of the new result is a refinement of Dimitrov’s method.
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1 Introduction
For a natural number \(k\ge 2,\) we say an integer n is k-free if \(p^{k} \not \mid n\) for any prime p. The distribution of k-free numbers is an important theme for number theory scholars. It is well known that
proved by Montgomery and Vanghan [13], where \(\mu _{k}(n)\) denotes the characteristic function of the k-free integers and \(\zeta (k)\) is the usual Riemann zeta function. There also exist many articles that consider the k-free values of polynomials. In the case of \(k= 2,\) we mention Estermann’s work [8], where it is showed that for an absolute constant a
Heath-Brown [9] obtained (1.1) with the error term replaced by \(O(H^{7/12+ \varepsilon }).\) In 1932, Carlitz [2] considered the polynomial \(x (x+ 1)\) and he proved
Later, the exponent \(2/3+ \varepsilon \) was improved to \(7/11+ \varepsilon \) by Heath-Brown [10] using the square sieve, and to \((26+ \sqrt{433})/81+ \varepsilon \) by Reuss [15].
Evaluating square-free values of the polynomial in multiple variables is an essential generalization that has attracted many authors, including Dimitrov, Tolev, Zhou and Chen. In 2012, Tolev [16] studied the square-free values of the polynomial \(x^{2}+ y^{2}+ 1,\) and he proved
where \(\lambda (q)\) is the number of the integer solutions to the following congruence equation:
Recently, by using Tolev’s method and some estimate for the Salié sum, Zhou and Ding [17] got an asymptotic formula for \(\Sigma _{1\le x, y, z\le H} \, \mu ^{2}(x^{2}+y^{2}+ z^{2}+ k).\) On the other hand, motivated by Carlitz’s result (1.2), consecutive square-free values of polynomials in multiple variables have also been considered. Dimitrov [6, 7] found asymptotic formulas for consecutive square-free numbers of the form \(x^{2}+y^{2}+ 1, x^{2}+y^{2}+ 2\) and respectively of the form \(x^{2}+ 1, x^{2}+ 2.\) Chen [3] considered the consecutive square-free numbers \(x_{1}^{2}+ \cdots + x_{k}^{2}+ 1, x_{1}^{2}+ \cdots + x_{k}^{2}+ 2\) and showed
for any given integer \(k\ge 3\) and an absolute constant c.
For the k-free values of polynomials, the classical problem is to study the following:
In 1932, Carlitz [2] obtained
where the exponent \(2(k+ 1)+ \varepsilon \) may be considered as trivial. Later on, Brandes [1] derived an improvement upon the trivial exponent which is of order \(1/k^{2}\) as \(k\rightarrow \infty .\) Using the approximative determinant method of Heath-Brown, Dictmann and Marmon [5] obtained the exponent \(14/9k+ \varepsilon ,\) which sharpens previous bound for \(k\ge 6.\) The k-free values of the polynomial \((x+ a_{1})(x+ a_{2})\) were considered by Mirsky [12]. Recently, Chen and Wang [4] studied the r-free values of \(x^{2}+y^{2}+ z^{2}+ k\) and gave an asymptotic formula.
Let
and
For simplicity, we also define
and
Inspired by the above results, we shall study the k-free values of the polynomials \(x^{2}+ y^{2}+ z^{2}+ 1\) and \(x^{2}+ y^{2}+ z^{2}+ 2\) by following the method in Dimitrov [6] and pruning some details referring to Chen and Wang [4]. The key ingredient is still the estimation of \(\lambda (q_{1}, q_{2}; m, n, l)\), which we give with the help of the elementary properties of Gauss sums and Salié sums. We prove the following.
Theorem 1.1
For any given integer \(k\ge 2,\) the asymptotic formula
holds. Here
In the case of \(k= 2,\) we obtain from Theorem 1.1 the following:
Theorem 1.2
If \(\varepsilon > 0\) is an arbitrary positive number, then
2 Notations and preliminary lemmas
Throughout this paper, H is a sufficiently large positive number and m, n, l denote integers. By \(\varepsilon \) we denote an arbitrary positive number which may have different values in different places. As usual, \(\mu (n)\) denotes the Möbius function; \(\tau (n)\) and \(\omega (n)\) represent the number of positive divisors of n and the number of distinct prime factors of n, respectively. Instead of \(m\equiv n \pmod {d}\) we write for simplicity \(m\equiv n \,(d).\) (m, n, l) denotes the greatest common divisor of m, n, l and \(\Vert \xi \Vert \) denotes the distance from \(\xi \) to its nearest integer. We write \(e(t)= \exp (2 \pi it)\) and \(e_{q}(t)= e(t/q).\) For any x and q such that \((x, q)= 1\), we denote by \(\overline{x_{q}}\) the inverse of n modulo q. If we can understand the value of the modulus from the context, then we write for simplicity \(\overline{x}.\) For any odd q, \(\left( \frac{\cdot }{q}\right) \) is the Jacobi symbol.
We define the Gauss sum and the Salié sum as follows:
and, for odd integers q,
In what follows, we present some lemmas used in the proof of the theorems. First we quote some important properties of the Gauss sum.
Lemma 2.1
For the Gauss sum, the following hold:
(1) If \((q_{1}, q_{2})= 1,\) then
(2) If \((q, m)= d,\) then
(3) If \((q, 2 m)= 1,\) then
Proof
See Lemma 3.1 of [6]. \(\square \)
In the next lemma, we present the upper bound result of the Salié sum.
Lemma 2.2
If q is an odd integer, then
Proof
See p. 524 in [11]. \(\square \)
Lemma 2.3
If \((q_{1}' q_{1}'', q_{2}' q_{2}'')= (q_{1}', q_{1}'')= (q_{2}', q_{2}'')= 1,\) then
Proof
Let
where \(1\le x_{1}, y_{1}, z_{1}\le q_{1}' q_{2}'\) and \(1\le x_{2}, y_{2}, z_{2}\le q_{1}'' q_{2}''.\)
From the Chinese remainder theorem, we obtain
By using the substitutions \(q_{1}'' q_{2}'' x_{1}\rightarrow x_{1}, q_{1}'' q_{2}'' y_{1}\rightarrow y_{1}, q_{1}'' q_{2}'' z_{1}\rightarrow z_{1},\) we infer
Similarly,
Thus, Lemma 2.3 follows immediately from (2.3)–(2.5). \(\square \)
Now we give an upper bound estimate of \(\lambda (q_{1}, q_{2}; m, n, l)\) by applying Lemmas 2.1, 2.2 and 2.3.
Lemma 2.4
If \(8\not \mid q_{1} q_{2}\) and \((q_{1}, q_{2})= 1,\) then
Proof
Case 1. \(2\not \mid q_{1} q_{2}.\)
In view of the orthogonality relations
we obtain from (1.5), (2.1) and Lemma 2.1
Since \((q_{1}, q_{2})= 1, (q_{i}, h_{i})= \frac{q_{i}}{l_{i}}, l_{i}\mid q_{i}\) and \(2\not \mid q_{1} q_{2},\) by Lemma 2.1
It is well known that \(|G(l_{1}, 1)|= \sqrt{l_{1}},\) thus, by Lemma 2.2,
Case 2. \(q_{1}= 2^{h} q_{1}',\) where \(2\not \mid q_{1}', h\le 2\) and \(2\not \mid q_{2}.\)
By Lemma 2.3, we have
A combination of the trivial estimate \(\lambda \left( 2^{h}, 1; m \overline{(q_{1}' q_{2})}_{2^{h}}, n \overline{(q_{1}' q_{2})}_{2^{h}}, l \overline{(q_{1}' q_{2})}_{2^{h}}\right) \ll 8^{h}\) and (2.6) yields
Case 3. \(q_{2}= 2^{h} q_{2}',\) where \(2\not \mid q_{2}', h\le 2\) and \(2\not \mid q_{1}.\)
Similarly to Case 2, we obtain
Combining the estimates for the three cases gives the proof of Lemma 2.4. \(\square \)
Lemma 2.5
If \(8\not \mid q_{1} q_{2}\) and \((q_{1}, q_{2})= 1,\) then for the sums
and
we have the estimates
Proof
By Lemma 2.4,
In a similar way, using Lemma 2.4 we obtain
and
\(\square \)
Lemma 2.6
For any real number \(\sigma \) and positive integers \(N_{1}, N_{2}\) with \(N_{1}< N_{2},\) we have
Proof
See Lemma 4.7 of [14]. \(\square \)
3 Proof of Theorem 1.1
Upon using the well-known identity
we find by (1.4) that
where
\(\xi \) is a parameter to be chosen so that \(H^{1/k}\le \xi \le H^{2/k}.\)
3.1 Estimation of \(R_{1}(H)\)
To estimate the contribution of \(R_{1}(H),\) we suppose that \(q_{1}= d_{1}^{k}, q_{2}= d_{2}^{k},\) where \(d_{1}\) and \(d_{2}\) are square-free, \((q_{1}, q_{2})= 1\) and \(d_{1} d_{2}\le \xi .\)
We first analyze \(S(H;q_{1}, q_{2}).\) Define
By orthogonality, \(\Sigma (H, q_{1}, q_{2}, x)\) may be written as
From the definition of \(S(H;q_{1}, q_{2}),\) we easily see that
which combined with (3.4) yields
where
By exchanging the order of summations and noting the definitions of \(\lambda (q_{1}, q_{2}; m, n, l),\) we obtain
Now we treat \(L_{1}(q_{1}, q_{2}; H).\) Employing Lemma 2.6 we get
Hence by Lemma 2.5, it follows that
The same estimates hold for \(L_{2}(q_{1}, q_{2}; H)\) and \(L_{3}(q_{1}, q_{2}; H).\) Gathering the estimates for \(L_{1}(q_{1}, q_{2}; H), L_{2}(q_{1}, q_{2}; H)\) and \(L_{3}(q_{1}, q_{2}; H)\) and noting (3.6), we arrive at
According to (3.2) and (3.7), \(R_{1}(H)\) can be estimated as follows:
In the last step above, we can check that by using Lemma 2.4
Put
From (1.6), Lemma 2.3 and \((d_{1}, d_{2})= 1,\) we get
Combining (3.9) and (3.10) we obtain
where
Since the function
is multiplicative with respect to \(d_{2},\) we have the Euler product representation:
From (3.11) and (3.12) we infer
3.2 Estimation of \(R_{2}(H)\)
From (3.3), we derive by a splitting argument
where
We therefore obtain
Similarly, one can obtain
Hence from (3.14) to (3.16) we get
Combining (3.1), (3.8), (3.13) and (3.17) gives
where \(\sigma _{k}\) is defined in (3.13).
Now we obtain Theorem 1.1 by choosing \(\xi = H^{\frac{3}{2k}}.\)
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Acknowledgements
The author would like to express the most sincere gratitude to the referee for his patience in refereeing this paper. This work is supported by the National Natural Science Foundation of China (Grant No. 11971476).
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Fan, H., Zhai, W. On the consecutive k-free values for certain classes of polynomials. Period Math Hung 88, 25–37 (2024). https://doi.org/10.1007/s10998-023-00534-5
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DOI: https://doi.org/10.1007/s10998-023-00534-5