On the consecutive k-free values for certain classes of polynomials
In the present paper we propose an asymptotic formula for R ( H , k ), the number of triples of positive integers x , y , z ≤ H such that x 2 + y 2 + z 2 + 1 , x 2 + y 2 + z 2 + 2 are k -free with k ≥ 2 . Especially, in the case of k = 2 we prove that R ( H , 2 ) = σ 2 H 3 + O ( H 9 / 4 + ε ) , wh...
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| Veröffentlicht in: | Periodica mathematica Hungarica 2024-03, Vol.88 (1), p.25-37 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
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| Zusammenfassung: | In the present paper we propose an asymptotic formula for
R
(
H
,
k
), the number of triples of positive integers
x
,
y
,
z
≤
H
such that
x
2
+
y
2
+
z
2
+
1
,
x
2
+
y
2
+
z
2
+
2
are
k
-free with
k
≥
2
.
Especially, in the case of
k
=
2
we prove that
R
(
H
,
2
)
=
σ
2
H
3
+
O
(
H
9
/
4
+
ε
)
,
where
σ
2
is an absolute constant and
ε
is an arbitrary small positive number, which improves the error term
O
(
H
7
/
3
+
ε
)
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. |
|---|---|
| ISSN: | 0031-5303 1588-2829 |
| DOI: | 10.1007/s10998-023-00534-5 |