Kummer hypothesis

A hypothesis concerning the behaviour of the cubic Gauss sum

$$\tau(\chi)=\sum_{n=0}^{p-1}\chi(n,p)e^{\pi in/p},$$

where $\chi(n,p)=\exp(2\pi i\operatorname{ind}n/3)$ is a cubic character modulo $p\equiv1$ ($\bmod3$) and $p$ is a prime number. It is known that

$$\tau(\chi)=\sqrt pe^{i\arg\tau(\chi)}.$$

Therefore $\arg\tau(\chi)$ lies either in the first, third or fifth sextant. Accordingly, E. Kummer divided all primes $p\equiv1$ ($\bmod\,3$) into three classes, $P_1$, $P_3$ and $P_5$. The Kummer hypothesis is that each of the classes $P_1$, $P_5$ and $P_3$ contains infinitely many primes, and that their respective asymptotic densities are $1/2$, $1/3$ and $1/6$. 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