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| In [[mathematics]], '''Kummer sum''' is the name given to certain cubic [[Gauss sum]]s 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]].
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| ==Definition==
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| A Kummer sum is therefore a finite sum
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| :<math>\Sigma \chi(r)e(r/p) = G(\chi)</math>
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| 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.
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| The cubic exponential sum ''K''(''n'',''p'') defined by
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| :<math>K(n,p)=\sum_{x=1}^p e(nx^3/p)</math>
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| 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 period]]s for the subgroup of [[Index of a subgroup|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.
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| ==Statistical questions==
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| It is known from the general theory of Gauss sums that
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| :|''G''(χ)| = √''p''.
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| In fact the prime decomposition of ''G''(χ) in the cyclotomic field it naturally lies in is known, giving a stronger form. What Kummer was concerned with was the [[argument of a complex number|argument]]
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| :θ<sub>''p''</sub>
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| of ''G''(χ). Unlike the quadratic case, where the square of the Gauss sum is known and the precise square root was determined by Gauss, here the cube of ''G''(χ) lies in the [[Eisenstein integer]]s, but its argument is determined by that of the Eisenstein prime dividing ''p'', which splits in that field.
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| Kummer made a statistical conjecture about θ<sub>''p''</sub> and its distribution modulo 2π (in other words, on the argument of the Kummer sum on the unit circle). For that to make sense, one has to choose between the two possible χ: there is a distinguished choice, in fact, based on the [[cubic residue symbol]]. Kummer used available numerical data for ''p'' up to 500 (this is described in the 1892 book ''Theory of Numbers'' by [[George B. Mathews]]). There was, however, a 'law of small numbers' operating, meaning that Kummer's original conjecture, of a lack of uniform distribution, suffered from a small-number bias. In 1952 [[John von Neumann]] and [[Herman Goldstine]] extended Kummer's computations, on [[ENIAC]] (written up in John von Neumann and H.H. Goldstine, ''A Numerical Study of a Conjecture of Kummer'' 1953).
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| In the twentieth century, progress was finally made on this question, which had been left untouched for over 100 years. Building on work of [[Tomio Kubota]], [[S. J. Patterson]] and [[Roger Heath-Brown]] in 1978 proved a modified form of Kummer conjecture. In fact they showed that there was equidistribution of the θ<sub>''p''</sub>. This work involved [[automorphic form]]s for the [[metaplectic group]], and [[Vaughan's lemma]] in [[analytic number theory]].
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| ==Cassels' conjecture==
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| A second conjecture on Kummer sums was made by [[J. W. S. Cassels]], again building on previous ideas of Tomio Kubota. This was a product formula in terms of [[elliptic function]]s with [[complex multiplication]] by the Eisenstein integers. (J. W. S. Cassels, On Kummer sums, ''Proc. London Math. Soc.'', (3) 21 (1970), 19–27.) The conjecture was proved in 1978 by Charles Matthews. (C. R. Matthews, Gauss sums and elliptic functions: I. The Kummer sum. ''Invent. Math.'', 52 (1979), 163–185.)
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| ==References==
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| * {{citation|last=Branson|first=Mark|title=The solutions of ''X''<sup>3</sup>+''Y''<sup>3</sup>+''Z''<sup>3</sup>=0 and Kummer's conjecture|url=http://www.math.sunysb.edu/~mbranson/papers/kummer.pdf}}
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| *{{springer|id=k/k055970|first=B.M.|last= Bredikhin|title=Kummer hypothesis}}
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| *{{citation|mr=1815372
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| |last=Heath-Brown|first= D. R.
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| |title=Kummer's conjecture for cubic Gauss sums.
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| |journal=Israel J. Math.|volume= 120 |year=2000|pages= part A, 97–124. |url=http://eprints.maths.ox.ac.uk/archive/00000158/01/kummer.pdf }}
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| *{{citation|first= D.R.|last= Heath-Brown|first2= S.I.|last2= Patterson|title=The distribution of Kummer sums at prime arguments|journal= J. Reine Angew. Math. |volume= 310 |year=1979|pages= 111–130|mr=0546667}}
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| [[Category:Cyclotomic fields]]
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Wilber Berryhill is the name his parents gave him and he completely digs that name. What me and my family adore is performing ballet but I've been taking on new issues recently. Kentucky is where I've always been residing. I am currently a journey agent.
Here is my weblog certified psychics [http://www.sirudang.com]