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On the distribution (mod 1) of polynomials of a prime variable

Published online by Cambridge University Press:  22 January 2016

Ming-Chit Liu  and
Kai-Man Tsang
Ming-Chit Liu
Affiliation:
Department of Mathematics, University of Hong Kong, Hong Kong
Kai-Man Tsang
Affiliation:
Department of Mathematics, University of Hong Kong, Hong Kong
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Throughout, ε is any small positive number, θ any real number, n, nj, k, N some positive integers and p, pj any primes. By ‖θ‖ we mean the distance from θ to the nearest integer. Write C(ε), C(ε, k) for positive constants which may depend on the quantities indicated inside the parentheses.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 85 , March 1982 , pp. 241 - 249
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1982

References

[ 1 ] Davenport, H. and Heilbronn, H., On indefinite quadratic forms in five variables, J. London Math. Soc, 21 (1946), 185–193.CrossRefGoogle Scholar
[ 2 ] Davenport, H. and Roth, K.F., The solubility of certain diophantine inequalities, Mathematika, 2 (1955), 81–96.Google Scholar
[ 3 ] Davenport, H., On a theorem of Heilbronn, Quart. J. Math. Oxford, (2), 18 (1967), 339–344.Google Scholar
[ 4 ] Hardy, G.H. and Wright, E.M., An Introduction to the Theory of Numbers, 4th ed. Oxford, 1965.Google Scholar
[ 5 ] Hartman, S. and Knapowski, S., Bemerkungen über die Bruchteile von pa, Ann. Polon. Math., 3 (1957), 285–287.Google Scholar
[ 6 ] Heilbronn, H., On the distribution of the sequence n20(modl), Quar Math t. J.. Oxford, 19 (1948), 249–256.Google Scholar
[ 7 ] Hua, L.K., Additive Theory of Prime Numbers, Translations of Mathematical Monographs Vol. 13, Amer. Math. Soc, Providence, R.I. 1965.Google Scholar
[ 8 ] Liu, M.C., Approximation by a sum of polynomials involving primes, J. Math. Soc. Japan, 30 (1978), 395–412.CrossRefGoogle Scholar
[ 9 ] Schmidt, W.M., Small Fractional Parts of Polynomials, Regional Conference Series in Mathematics No. 32, Amer. Math. Soc. Providence, R.I. 1977.Google Scholar
[10] Schmidt, W.M., On the distribution modulo 1 of the sequence an2 + βn, Canacl. J. Math., 29 (1977), 819–826.Google Scholar
[11] Vaughan, R.C., On the distribution of ap modulo 1, Mathematika, 24 (1977), 135–141.Google Scholar
[12] Vinogradov, I.M., Analytischer Beweis des Satzes über die Verteilung der Bruchteile eines ganzen Polynoms, (in Russian), Bull. Acad. Sci. USSR (6), 21 (1927), 567–578.Google Scholar
[13] Vinogradov, I.M., A new estimate of a trigonometric sum containing primes, (in Russian with English summary), Bull. Acad. Sci. USSR ser. Math., 2 (1938), 3–13.Google Scholar
[14] Vinogradov, I.M., The Method of Trigonometrical Sums in the Theory of Numbers, New York: Interscience 1954.Google Scholar