# Kummer hypothesis

From Encyclopedia of Mathematics

A hypothesis concerning the behaviour of the cubic Gauss sum

where is a cubic character modulo () and is a prime number. It is known that

Therefore lies either in the first, third or fifth sextant. Accordingly, E. Kummer divided all primes () into three classes, , and . The Kummer hypothesis is that each of the classes , and contains infinitely many primes, and that their respective asymptotic densities are , and . There are various generalizations of the Kummer hypothesis to characters of order higher than 3. A modified version of the hypothesis has been proved (see [3]).

#### References

[1] | H. Hasse, "Vorlesungen über Zahlentheorie" , Springer (1950) |

[2] | H. Davenport, "Multiplicative number theory" , Springer (1980) |

[3] | D.R. Heath-Brown, S.I. Patterson, "The distribution of Kummer sums at prime arguments" J. Reine Angew. Math. , 310 (1979) pp. 111–130 |

**How to Cite This Entry:**

Kummer hypothesis. B.M. Bredikhin (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Kummer_hypothesis&oldid=13609

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098