Kummer hypothesis

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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]).


[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:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098