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In mathematics, Kummer sum is the name given to certain cubic Gauss sums for a prime modulus ''p'', with ''p'' congruent to 1 modulo 3. They are named after Ernst Kummer, who made a conjecture about the statistical properties of their arguments, as complex numbers. These sums were known and used before Kummer, in the theory of cyclotomy. ==Definition== A Kummer sum is therefore a finite sum : taken over ''r'' modulo ''p'', where χ is a Dirichlet character taking values in the cube roots of unity, and where ''e''(''x'') is the exponential function exp(2π''ix''). Given ''p'' of the required form, there are two such characters, together with the trivial character. The cubic exponential sum ''K''(''n'',''p'') defined by : is easily seen to be a linear combination of the Kummer sums. In fact it is 3''P'' where ''P'' is one of the Gaussian periods for the subgroup of index 3 in the residues mod ''p'', under multiplication, while the Gauss sums are linear combinations of the ''P'' with cube roots of unity as coefficients. However it is the Gauss sum for which the algebraic properties hold. Such cubic exponential sums are also now called Kummer sums. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Kummer sum」の詳細全文を読む スポンサード リンク
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