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Counting fundamental solutions to the Pell equation with prescribed size

  • Ping Xi (a1)

Abstract

The cardinality of the set of $D\leqslant x$ for which the fundamental solution of the Pell equation $t^{2}-Du^{2}=1$ is less than $D^{1/2+\unicode[STIX]{x1D6FC}}$ with $\unicode[STIX]{x1D6FC}\in [\frac{1}{2},1]$ is studied and certain lower bounds are obtained, improving previous results of Fouvry by introducing the $q$ -analogue of van der Corput method to algebraic exponential sums with smooth moduli.

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[Bom66] Bombieri, E., On exponential sums in finite fields , Amer. J. Math. 88 (1966), 71105.
[Bou15] Bourgain, J., A remark on solutions of the Pell equation , Int. Math. Res. Not. IMRN 2015 (2015), 28412855.
[Fou16] Fouvry, É., On the size of the fundamental solution of the Pell equation , J. Reine Angew. Math. 717 (2016), 133.
[FJ12] Fouvry, É. and Jouve, F., Fundamental solutions to Pell equation with prescribed size , Proc. Steklov Inst. Math. 276 (2012), 4050.
[Hea78] Heath-Brown, D. R., Hybrid bounds for Dirichlet L-functions , Invent. Math. 47 (1978), 149170.
[Hea95] Heath-Brown, D. R., A mean value estimate for real character sums , Acta Arith. 72 (1995), 235275.
[Hoo84] Hooley, C., On the Pellian equation and the class number of indefinite binary quadratic forms , J. Reine Angew. Math. 353 (1984), 98131.
[IK04] Iwaniec, H. and Kowalski, E., Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53 (American Mathematical Society, Providence, RI, 2004).
[Ten95] Tenenbaum, G., Introduction to analytic and probabilistic number theory, Cambridge Studies in Advanced Mathematics, vol. 46 (Cambridge University Press, Cambridge, 1995).
[TW03] Tenenbaum, G. and Wu, J., Moyennes de certaines fonctions multiplicatives sur les entiers friables , J. Reine Angew. Math. 564 (2003), 119166.
[Wei48] Weil, A., On some exponential sums , Proc. Natl. Acad. Sci. USA 34 (1948), 204207.
[WX16] Wu, J. and Xi, P., Arithmetic exponent pairs for algebraic trace functions and applications, Preprint (2016), arXiv:1603.07060 [math.NT].
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Counting fundamental solutions to the Pell equation with prescribed size

  • Ping Xi (a1)

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