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On the Number of Divisors of the Quadratic Form m2 + n2

  • Gang Yu (a1)
Abstract

For an integer n, let d(n) denote the ordinary divisor function. This paper studies the asymptotic behavior of the sum

It is proved in the paper that, as x → ∞,

where A 1 and A 2 are certain constants and ε is any fixed positive real number.

The result corrects a false formula given in a paper of Gafurov concerning the same problem, and improves the error claimed by Gafurov.

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References
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[1] Davenport, H., Multiplicative Number Theory. Graduate Texts in Math. 74, Revised by L, H.. Montgomery Springer, 1980.
[2] Friedlander, J. B. and Iwaniec, H., The polynomial x2 + y4 captures its primes. preprint.
[3] Fouvry, E. and Iwaniec, H., Gaussian primes. Acta Arith. 129(1997), 249287.
[4] Gafurov, N.,On the number of divisors of a quadratic form. Proc. Steklov Inst.Math. (1993), 137–148.
[5] Hooley, C., On the number of divisors of quadratic polynomials. ActaMath. 110(1963), 97114.
[6] Iwaniec, H. and Mozzochi, C. J., On the divisor and circle problems. J.Number Theory 29(1988), 6093.
[7] Vaaler, J. D., Some extremal problems in Fourier analysis. Bull. Amer.Math. Soc. (2) 12(1985), 183216.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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