Kummer sum
Encyclopedia
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.
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(2πix). 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 3P where P is one of the Gaussian period
s 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.
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 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.
Kummer made a statistical conjecture about θp 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).
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 θp. This work involved automorphic form
s for the metaplectic group
, and Vaughan's lemma in analytic number theory
.
, 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.)
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Kummer sum is the name given to certain cubic Gauss sum
Gauss sum
In mathematics, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typicallyG := G= \sum \chi\cdot \psi...
s for a prime modulus p, with p congruent to 1 modulo 3. They are named after Ernst Kummer
Ernst Kummer
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.-Life: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 sumtaken over r modulo p, where χ is a Dirichlet character
Dirichlet character
In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z...
taking values in the cube roots of unity, and where e(x) is the exponential function exp(2πix). 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 3P where P is one of the Gaussian period
Gaussian period
In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis . They are basic in the classical theory called cyclotomy...
s for the subgroup of index
Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
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.
Statistical questions
It is known from the general theory of Gauss sums that- |G(χ)| = √p.
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
- θp
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
Eisenstein integer
In mathematics, Eisenstein integers , also known as Eulerian integers , are complex numbers of the formz = a + b\omega \,\!where a and b are integers and...
s, but its argument is determined by that of the Eisenstein prime dividing p, which splits in that field.
Kummer made a statistical conjecture about θp 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
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
and Herman Goldstine extended Kummer's computations, on ENIAC
ENIAC
ENIAC was the first general-purpose electronic computer. It was a Turing-complete digital computer capable of being reprogrammed to solve a full range of computing problems....
(written up in John von Neumann and H.H. Goldstine, A Numerical Study of a Conjecture of Kummer 1953).
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
Tomio Kubota
is a Japanese mathematician, who studies number theory. His contributions include works on p-adic L functions and real-analytic automorphic forms.His work on p-adic L-functions, later recognised as an aspect of Iwasawa theory, was jointly with Leopoldt....
, S. J. Patterson and Roger Heath-Brown
Roger Heath-Brown
David Rodney "Roger" Heath-Brown F.R.S. , is a British mathematician working in the field of analytic number theory. He was an undergraduate and graduate student of Trinity College, Cambridge; his research supervisor was Alan Baker...
in 1978 proved a modified form of Kummer conjecture. In fact they showed that there was equidistribution of the θp. This work involved automorphic form
Automorphic form
In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...
s for the metaplectic group
Metaplectic group
In mathematics, the metaplectic group Mp2n is a double cover of the symplectic group Sp2n. It can be defined over either real or p-adic numbers...
, and Vaughan's lemma in analytic number theory
Analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...
.
Cassels' conjecture
A second conjecture on Kummer sums was made by J. W. S. CasselsJ. W. S. Cassels
John William Scott Cassels , FRS is a leading English mathematician.-Biography:Educated at Neville's Cross Council School in Durham and George Heriot's School in Edinburgh, Cassels graduated from the University of Edinburgh with an MA in 1943.His academic career was interrupted in World War II...
, again building on previous ideas of Tomio Kubota. This was a product formula in terms of elliptic function
Elliptic function
In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...
s with complex multiplication
Complex multiplication
In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense In mathematics, complex multiplication is the...
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.)