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Counting square-free integers represented by binary quadratic forms of a fixed discriminant

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Abstract

In this article, we prove a result concerning the infinitude of square-free integers represented by a class of polynomials in two variables. More precisely, we prove that infinitely many square-free positive integers are represented by a primitive integral positive-definite binary quadratic form of a given discriminant D. We obtain our result by deriving an asymptotic formula for the summatory function associated to it using some known L-functions.

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References

  1. Bourgain, J.: Decoupling, exponential sums and the Riemann zeta function. J. Amer. Math. Soc. 205–224 (2017)

  2. Buell, D.A.: Binary Quadratic Forms: Classical Theory and Modern Computations. Springer, New York (1989)

    Book  MATH  Google Scholar 

  3. Cox, D.A.: Primes of the Form \(x^2+n y^{22}\). Fermat, Class Field Theory, and Complex Multiplication. Second edition. Pure and Applied Mathematics (Hoboken). John Wiley & Sons, Inc., Hoboken, NJ (2013)

  4. Dimitrov, S.I.: Square-free values of \(n^{2}+n+1\). Georgian Math. J. 30(3), 333–348 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dimitrov, S.I.: On the number of pairs of positive integers \(x, y \le H\) such that \(x^{2}+y^{2}+1\), \(x^{2}+y^{2}+2\) are square-free. Acta Arith. 194(3), 281–294 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dimitrov, S.I.: Pairs of square-free values of the type \(n^{2}+1\), \(n^{2}+2\). Czechoslovak Math. J. 74(4), 991–1009 (2021)

    Article  MATH  Google Scholar 

  7. Dimitrov, S. I.: Square-free pairs \(n^{2}+n+1\), \(n^{2}+n+2\). Preprint (2023). https://hal.science/hal-03735444

  8. Estermann, T.: Einige Sätze über quadratfreie Zahlen. Math. Ann. 105, 653–662 (1931)

  9. Friedlander, J.B., Iwaniec, H.: Square-free values of quadratic polynomials. Proc. Edinburgh Math. Soc. 53(2), 385–392 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Friedlander, J.B., Iwaniec, H.: The polynomial \(x^2 + y^4\) captures its primes. Ann. of Math. (2) 148(3), 945–1040 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Heath-Brown, D.R.: The growth rate of the Dedekind zeta-function on the critical line. Acta Arith. 49(4), 323–339 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. Heath-Brown, D.R.: Primes represented by \(x^3+2y^3\). Acta Math. 186(1), 1–84 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Heath-Brown, D.R.: Square-free values of \(n^2+1\). Acta Arith. 155, 1–13 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Heath-Brown, D.R.: Power-free values of polynomials. Quart. J. Math. 64, 177–188 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ivíc, A.: Exponent pairs and the zeta function of Riemann. Studia Sci. Math. Hungar. 15(1–3), 157–181 (1980)

    MathSciNet  MATH  Google Scholar 

  16. Iwaniec, H.: Primes represented by quadratic polynomials in two variables. Acta. Arith. 24, 435–459 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  17. Iwaniec, H.: Topics in Classical Automorphic Forms. Graduate Studies in Mathematics, 17. American Mathematical Society, Providence, RI (1997)

  18. Jiang, Y., Lü, G.: Sum of coefficients of \(L\)- functions and applications. J. Number Theory 171, 56–70 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jing, M., Liu, H.: Consecutive square-free numbers and square-free primitive roots. Int. J. Number Theory 18, 205–226 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ricci, G.: Ricerche aritmetiche sui polinomi. Rend. Circ. Mat. Palermo 57, 433–475 (1933)

    Article  MATH  Google Scholar 

  21. Tolev, D.I.: On the number of pairs of positive integers \(x, y \le H\) such that \(x^{2}+y^{2}+1\) is square-free. Monatsh. Math. 165, 557–567 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Vaishya, L.: Signs of Fourier coefficients of cusp forms at integers represented by an integral binary quadratic form. Proc. Indian Acad. Sci. Math. Sci. 131(2), Paper No. 41, 14 pp. (2021)

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Authors and Affiliations

  1. The Institute of Mathematical Sciences (A CI of Homi Bhabha National Institute), CIT Campus, Taramani, Chennai, 600 113, Tamilnadu, India

    Lalit Vaishya

  2. Department of Mathematics, SRM University AP, Andhra Pradesh, India

    Manish Kumar Pandey

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  1. Lalit Vaishya
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  2. Manish Kumar Pandey
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Correspondence to Manish Kumar Pandey.

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Vaishya, L., Pandey, M.K. Counting square-free integers represented by binary quadratic forms of a fixed discriminant. Arch. Math. 121, 385–395 (2023). https://doi.org/10.1007/s00013-023-01915-5

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  • DOI: https://doi.org/10.1007/s00013-023-01915-5

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