Abstract
The attempt at interpreting the Weyl sums as finite theta series (theta-Weyl sums) has been successful only in the case of quadratic polynomials. In this paper we shall present basic ingredients for interpreting cubic Weyl sums as finite theta series, i.e. the cubic continued fraction expansion, the van der Corput reciprocal function, cubic reciprocal and parabolic transformations.
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References
H. Fiedler, W. Jurkat and O. Körner, Asymptotic expansions of finite theta series, Acta Arith. 32 (1977), 129–146
S. W. Graham and G. Kolesnik, Van der Corput’s method of exponential sums, London Mathematical Society Lecture Note Series 126, Cambridge Univ. Press, London, 1991
J.-I. Igusa, Lectures on forms of higher degree, Tata Institute of Fundamental Research Lectures on Mathematics and Physics 59, Tata Inst., Bombay, 1978
T. Kubota, On an analogy to the Poisson summation formula for generalised Fourier transformation, J. Reine angew. Math. 268/269 (1974), 180–189
Y. Kuribayashi, On pseudo-Fourier transform (in Japanese), Sûrikaiseki- kenkyusho Kôkyûroku 1039 (1998), 152–164
W. Maier, Transformation der kubischen Thetafunktionen, Math. Ann. 111 (1935), 183–196
Y.-N. Nakai, On a 9-Weyl sum, Nagoya Math. J., 52 (1973), 163–172.
Y.-N. Nakai, On a 9-Weyl sum, Errata, the same J., 60 (1976), 217.
Y.-N. Nakai, Weyl sums and the van der Corput method (in Japanese), Sûrikaisekikenkyusho Kôkyûroku 456 (1982), 2–18
Y.-N. Nakai, On Diophantine inequalities of real indefinite quadratic forms of additive type in four variables, Advanced Studies in Pure Mathematics 13 (1988), Investigations in Number Theory, 25–170
Y.-N. Nakai, Cubic continued fraction expansions I and II (in Japanese), Mem. Fac. Edu. Yamanashi Univ. 41–2 (1990), 4–6 (Cubic I)
Y.-N. Nakai, Cubic continued fraction expansions I and II (in Japanese), ibid. 44–2 (1993), 1–3 (Cubic II)
Y.-N. Nakai, A strengthening of the conjecture that the functions invariant under Fourier transformation are quadratic polynomials, (in Japanese), Mem. Fac. Edu. Yamanashi Univ. 43–2 (1992), 1–3
Y.-N. Nakai, A candidate for cubic theta-Weyl sums (I) (in Japanese), Mem. Fac. Edu. Yamanashi Univ. 49–2 (1998), 1–4
Y.-N. Nakai, A candidate for cubic theta-Weyl sums (in Japanese), Sûrikaisekikenkyusho Kôkyûroku 1091 (1999), 298–307
W. Raab, Kubische und biquadratische Thetafunktionen I und II, Sizungsber. Osterreich. Akad. Wiss. Mat-Natur. KI. 188 (1979), 47–77 and 231–246
T. Suzuki, Weil type representations and automorphic forms, Nagoya Math. J., 77 (1980), 145–166
E. C. Titchmarsh, The theory of the Riemann zeta-function, Oxford Univ. Press, 1951; second ed., rev. by D.R. Heath-Brown, 1996
R. C. Vaughan, Some remarks on Weyl sums, Colloq. Math. Soc. Janos Bolyai, 34 (II), Topics in Classical Number Theory, Budapest (1987), 1585–1602
R. C. Vaughan, The Hardy-Littlewood method, second edition, Cambridge Tracts in Mathematics 125, Cambridge University Press, London 1997.
A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963
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Nakai, Y. (2002). A Penultimate Step toward Cubic Theta-Weyl Sums. In: Kanemitsu, S., Jia, C. (eds) Number Theoretic Methods. Developments in Mathematics, vol 8. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3675-5_16
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DOI: https://doi.org/10.1007/978-1-4757-3675-5_16
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