Note on the quadratic Gauss sums
Let p be an odd prime and {χ(m)=(m/p)}, m=0,1,...,p−1 be a finite arithmetic sequence with elements the values of a Dirichlet character χ modp which are defined in terms of the Legendre symbol (m/p), (m,p)=1. We study the relation between the Gauss and the quadratic Gauss sums. It is shown that th...
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| Format: | Article |
| Language: | English |
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Wiley
2001-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S016117120100480X |
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| author | George Danas |
| author_facet | George Danas |
| author_sort | George Danas |
| collection | DOAJ |
| description | Let p be an odd prime and
{χ(m)=(m/p)}, m=0,1,...,p−1 be a finite arithmetic sequence with elements the values of a
Dirichlet character χ modp which are defined in terms of
the Legendre symbol (m/p), (m,p)=1. We study the relation
between the Gauss and the quadratic Gauss sums. It is shown that
the quadratic Gauss sums G(k;p) are equal to the Gauss sums
G(k,χ) that correspond to this particular Dirichlet
character χ. Finally, using the above result, we prove that
the quadratic Gauss sums G(k;p), k=0,1,...,p−1are
the eigenvalues of the circulant p×p matrix X with
elements the terms of the sequence {χ(m)}. |
| format | Article |
| id | doaj-art-9472d72932f342b2bb791977b9ccdce9 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2001-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-9472d72932f342b2bb791977b9ccdce92025-02-03T00:59:10ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252001-01-0125316717310.1155/S016117120100480XNote on the quadratic Gauss sumsGeorge Danas0Technological Educational Institution of Thessaloniki, School of Sciences, Department of Mathematics, P.O. Box 14561, Thessaloniki GR-54101, GreeceLet p be an odd prime and {χ(m)=(m/p)}, m=0,1,...,p−1 be a finite arithmetic sequence with elements the values of a Dirichlet character χ modp which are defined in terms of the Legendre symbol (m/p), (m,p)=1. We study the relation between the Gauss and the quadratic Gauss sums. It is shown that the quadratic Gauss sums G(k;p) are equal to the Gauss sums G(k,χ) that correspond to this particular Dirichlet character χ. Finally, using the above result, we prove that the quadratic Gauss sums G(k;p), k=0,1,...,p−1are the eigenvalues of the circulant p×p matrix X with elements the terms of the sequence {χ(m)}.http://dx.doi.org/10.1155/S016117120100480X |
| spellingShingle | George Danas Note on the quadratic Gauss sums International Journal of Mathematics and Mathematical Sciences |
| title | Note on the quadratic Gauss sums |
| title_full | Note on the quadratic Gauss sums |
| title_fullStr | Note on the quadratic Gauss sums |
| title_full_unstemmed | Note on the quadratic Gauss sums |
| title_short | Note on the quadratic Gauss sums |
| title_sort | note on the quadratic gauss sums |
| url | http://dx.doi.org/10.1155/S016117120100480X |
| work_keys_str_mv | AT georgedanas noteonthequadraticgausssums |