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BOUNDS OF MULTIPLICATIVE CHARACTER SUMS WITH FERMAT QUOTIENTS OF PRIMES

Part of: Multiplicative number theory Elementary number theory

Published online by Cambridge University Press:  07 February 2011

IGOR E. SHPARLINSKI
IGOR E. SHPARLINSKI*
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia (email: igor.shparlinski@mq.edu.au)
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Abstract

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Given a prime p, the Fermat quotient qp(u) of u with gcd (u,p)=1 is defined by the conditions We derive a new bound on multiplicative character sums with Fermat quotients qp(ℓ) at prime arguments ℓ.

Keywords

Fermat quotientscharacter sumsVaughan identity

MSC classification

Secondary: 11A07: Congruences; primitive roots; residue systems 11N25: Distribution of integers with specified multiplicative constraints
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 3 , June 2011 , pp. 456 - 462
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Bourgain, J., Ford, K., Konyagin, S. V. and Shparlinski, I. E., ‘On the divisibility of Fermat quotients’, Michigan Math. J. 59 (2010), 313–328.CrossRefGoogle Scholar
[2]Chen, Z., Ostafe, A. and Winterhof, A., ‘Structure of pseudorandom numbers derived from Fermat quotients’, in: Arithmetic of Finite Fields, Lecture Notes in Computer Science, 6087 (eds. Anwar Hasan, M. and Helleseth, Tor) (Springer, Berlin, 2010), pp. 73–85.CrossRefGoogle Scholar
[3]Davenport, H., Multiplicative Number Theory, 2nd edn (Springer, New York, 1980).CrossRefGoogle Scholar
[4]Ernvall, R. and Metsänkylä, T., ‘On the p-divisibility of Fermat quotients’, Math. Comp. 66 (1997), 1353–1365.CrossRefGoogle Scholar
[5]Fouché, W. L., ‘On the Kummer–Mirimanoff congruences’, Q. J. Math. Oxford 37 (1986), 257–261.CrossRefGoogle Scholar
[6]Garaev, M. Z., ‘An estimate of Kloosterman sums with prime numbers and an application’, Mat. Zametki 88(3) (2010), 365–373 (in Russian).Google Scholar
[7]Gomez, D. and Winterhof, A., ‘Multiplicative character sums of Fermat quotients and pseudorandom sequences’, Period. Math. Hungar., to appear.Google Scholar
[8]Granville, A., ‘Some conjectures related to Fermat’s last theorem’, in: Number Theory (W. de Gruyter, New York, 1990), pp. 177–192.Google Scholar
[9]Granville, A., ‘On pairs of coprime integers with no large prime factors’, Expo. Math. 9 (1991), 335–350.Google Scholar
[10]Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers (Oxford University Press, Oxford, 1979).Google Scholar
[11]Heath-Brown, R., ‘An estimate for Heilbronn’s exponential sum’, in: Analytic Number Theory: Proc. Conf. in Honor of Heini Halberstam (Birkhäuser, Boston, 1996), pp. 451–463.Google Scholar
[12]Ihara, Y., ‘On the Euler–Kronecker constants of global fields and primes with small norms’, in: Algebraic Geometry and Number Theory, Progress in Mathematics, 850 (Birkhäuser, Boston, 2006), pp. 407–451.CrossRefGoogle Scholar
[13]Iwaniec, H. and Kowalski, E., Analytic Number Theory (American Mathematical Society, Providence, RI, 2004).Google Scholar
[14]Lenstra, H. W., ‘Miller’s primality test’, Inform. Process. Lett. 8 (1979), 86–88.CrossRefGoogle Scholar
[15]Ostafe, A. and Shparlinski, I. E., ‘Pseudorandomness and dynamics of Fermat quotients’, SIAM J. Discrete Math. 25 (2011), 50–71.CrossRefGoogle Scholar
[16]Shparlinski, I. E., ‘Character sums with Fermat quotients’, Q. J. Math., to appear.Google Scholar
[17]Vaughan, R. C., ‘An elementary method in prime number theory’, Acta Arith. 37 (1980), 111–115.CrossRefGoogle Scholar