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Remark on Smith’s result on a divisor problem in arithmetic progressions

Published online by Cambridge University Press:  22 January 2016

Kohji Matsumoto
Kohji Matsumoto*
Affiliation:
Department of Mathematics, Rikkyo University, Nishi-Ikebukuro, Toshima-ku, Tokyo 171, Japan
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Let dk(n) be the number of the factorizations of n into k positive numbers. It is known that the following asymptotic formula holds:

where r and q are co-prime integers with 0 < r < q, Pk is a polynomial of degree k − 1, φ(q) is the Euler function, and Δk (q; r) is the error term. (See Lavrik [3]).

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 98 , June 1985 , pp. 37 - 42
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1985

References

[ 1 ] Deligne, P., Séminaire géométrie algébrique 4½, Lecture notes in Math., 569, Springer, 1977.Google Scholar
[ 2 ] Heath-Brown, D. R., The fourth power moment of the Riemann zeta function, Proc. London Math. Soc, (3) 38 (1979), 385–422.Google Scholar
[ 3 ] Lavrik, A. F., A functional equation for Dirichlet L-series and the problem of divisors in arithmetic progressions, Izv. Akad. Nauk SSSR Ser. Mat., 30 (1966), 433–448. = Amer. Math. Soc. Transl., (2) 82 (1969), 47–65.Google Scholar
[ 4 ] Matsumoto, K., Master Thesis, Rikkyo Univ., 1983.Google Scholar
[ 5 ] Smith, R. A., The generalized divisor problem over arithmetic progressions, Math. Ann., 260 (1982), 255–268.Google Scholar
[ 6 ] Titchmarsh, E. C., The theory of the Riemann zeta-function, Oxford, 1951.Google Scholar
[ 7 ] Weinstein, L., The hyper-Kloosterman sum, Enseignement Math., (2) 27 (1981), 29–40.Google Scholar