View source for Kummer hypothesis

A hypothesis concerning the behaviour of the cubic [[Gauss sum|Gauss sum]]

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055970/k0559702.png" /> is a cubic character modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055970/k0559703.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055970/k0559704.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055970/k0559705.png" /> is a prime number. It is known that

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Therefore <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055970/k0559707.png" /> lies either in the first, third or fifth sextant. Accordingly, E. Kummer divided all primes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055970/k0559708.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055970/k0559709.png" />) into three classes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055970/k05597010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055970/k05597011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055970/k05597012.png" />. The Kummer hypothesis is that each of the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055970/k05597013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055970/k05597014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055970/k05597015.png" /> contains infinitely many primes, and that their respective asymptotic densities are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055970/k05597016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055970/k05597017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055970/k05597018.png" />. 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 [[#References|[3]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Hasse,   "Vorlesungen über Zahlentheorie" , Springer  (1950)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Davenport,   "Multiplicative number theory" , Springer  (1980)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D.R. Heath-Brown,   S.I. Patterson,   "The distribution of Kummer sums at prime arguments"  ''J. Reine Angew. Math.'' , '''310'''  (1979)  pp. 111–130</TD></TR></table>