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A hypothesis concerning the behaviour of the cubic [[Gauss sum|Gauss sum]]
 
A hypothesis concerning the behaviour of the cubic [[Gauss sum|Gauss sum]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055970/k0559701.png" /></td> </tr></table>
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$$\tau(\chi)=\sum_{n=0}^{p-1}\chi(n,p)e^{\pi in/p},$$
  
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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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055970/k0559706.png" /></td> </tr></table>
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$$\tau(\chi)=\sqrt pe^{i\arg\tau(\chi)}.$$
  
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]]]).
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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 [[#References|[3]]]).
  
 
====References====
 
====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>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  H. Hasse,  "Vorlesungen über Zahlentheorie" , Springer  (1950)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  H. Davenport,  "Multiplicative number theory" , Springer  (1980)</TD></TR>
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<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>
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</table>
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[[Category:Number theory]]

Latest revision as of 20:25, 31 October 2014

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
How to Cite This Entry:
Kummer hypothesis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kummer_hypothesis&oldid=13609
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article