Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

New pseudorandom sequences constructed by quadratic residues and Lehmer numbers

Author(s): Huaning Liu
Journal: Proc. Amer. Math. Soc. 135 (2007), 1309-1318.
MSC (2000): Primary 11A07, 11K45
Posted: November 14, 2006
MathSciNet review: 2276639
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Abstract | References | Similar articles | Additional information

Abstract: Let be an odd prime. Define

$\displaystyle e_n=\left\{\begin{array}{ll}\displaystyle (-1)^{n+\overline{n}}, ... ...e{n}+1}, & \hbox{if $n$ is a quadratic nonresidue mod $p$}, \end{array}\right.$

where $\overline{n}$ is the multiplicative inverse of

modulo

such that $1\leq \overline{n}\leq p-1$ . This paper shows that the sequence $\{e_n\}$ is a ``good" pseudorandom sequence, by using the properties of exponential sums, character sums, Kloosterman sums and mean value theorems of Dirichlet

-functions.

References:

1.: T. M. Apostol, Introduction to analytic number theory, Springer-Verlag, New York, 1976. MR 0434929 (55:7892)
2.: J. Cassaigne, S. Ferenczi, C. Mauduit, J. Rivat and A. Sárközy, On finite pseudorandom binary sequencs III: the Liouville function, I, Acta Arithmetica, 87 (1999), pp. 367-390. MR 1671629 (2000c:11126)
3.: J. Cassaigne, S. Ferenczi, C. Mauduit, J. Rivat and A. Sárközy, On finite pseudorandom binary sequencs IV: the Liouville function, II, Acta Arithmetica, 95 (2000), pp. 343-359. MR 1785199 (2002c:11087)
4.: J. Cassaigne, C. Mauduit and A. Sárközy, On finite pseudorandom binary sequencs VII: the measures of pseudorandomness, Acta Arithmetica, 103 (2002), pp. 97-108. MR 1904866 (2004c:11139)
5.: E. Fouvry, P. Michel, J. Rivat and A. Sárközy, On the pseudorandomness of the signs of Kloosterman sums, Journal of the Australian Mathematical Society, 77 (2004), pp. 425-436. MR 2099811 (2005h:11165)
6.: T. Funakura, On Kronecker's limit formula for Dirichlet series with periodic coefficients, Acta Arithmetica, 55 (1990), pp. 59-73.MR 1056115 (91j:11064)
7.: R. K. Guy, Unsolved problems in number theory, Springer-Verlag, New York, 1994.MR 1299330 (96e:11002)
8.: S. R. Louboutin, J. Rivat and A. Sárközy, On a problem of D. H. Lehmer, Proceedings of the American Mathematical Society, to appear.
9.: C. Mauduit and A. Sárközy, On finite pseudorandom binary sequences I: measure of pseudorandomness, the Legendre symbol, Acta Arithmetica, 82 (1997), pp. 365-377. MR 1483689 (99g:11095)
10.: C. Mauduit and A. Sárközy, On the measures of pseudorandomness of binary sequences, Discrete Mathematics, 271 (2003), pp. 195-207. MR 1999543 (2004e:11081)
11.: A. Weil, Sur les courbes algébriques et les variétés qui s'en déduisent, Act. Sci. Ind., Vol. 1041, Hermann, Paris, 1948.MR 0027151 (10:262c)
12.: A. Weil, On some exponential sums, Proc. Nat. Acad. Sci., 34 (1948), pp. 204-207. MR 0027006 (10:234e)
13.: W. Zhang, A problem of D. H. Lehmer and its generalization (I), Compositio Mathematica, 86 (1993), pp. 307-316.MR 1219630 (94f:11104)
14.: W. Zhang, A problem of D. H. Lehmer and its generalization (II), Compositio Mathematica, 91 (1994), pp. 47-56.MR 1273925 (95f:11079)
15.: W. Zhang, On the difference between a D. H. Lehmer number and its inverse modulo , Acta Arithmetica, 68 (1994), pp. 255-263.MR 1308126 (96a:11100)
16.: W. Zhang, On a problem of D. H. Lehmer and Kloosterman sums, Monatshefte für Mathematik, 139 (2003), pp. 247-257.MR 1994384 (2004e:11088)
17.: W. Zhang, On a problem of D. H. Lehmer and general Kloosterman sums, Acta Mathematica Sinica English Series, 20 (2004), pp. 515-524.MR 2084715 (2005h:11226)
18.: W. Zhang, Z. Xu and Y. Yi, A problem of D. H. Lehmer and its mean square value formula, Journal of Number Theory, 103 (2003), pp. 197-213.MR 2020268 (2004m:11128)

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