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An application of a product formula for the cubic Gauss sum

โœ Scribed by Ito, Hiroshi


Book ID
122131777
Publisher
Elsevier Science
Year
2014
Tongue
English
Weight
240 KB
Volume
135
Category
Article
ISSN
0022-314X

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โœ Tsuyoshi Takagi ๐Ÿ“‚ Article ๐Ÿ“… 1997 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 916 KB

Let | be a prime in the quadratic field Q(e 2?iร‚3 ), and let G 3 (|) be the cubic Gauss sum. Matthews [Invent. Math. 52 (1979), 163 185; 54 (1979), 23 52] determined the product formula of G 3 (|) using Weierstrass' ^function. In this paper, we establish an analogous result for the cubic Gauss sum m

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We obtain a representation formula for the trigonometric sum f (m, n) and deduce from it the upper bound f(m, n) < (4/p 2 ) m log m+ (4/p 2 )(c -log(p/2)+2C G ) m+O(m/`log m), where C G is the supremum of the function G(t) :=; . k=1 log |2 sin pkt|/(4k 2 -1), over the set of irrationals. The coeffi