Abstract
We show that the determinant of a Hankel matrix of odd dimension n whose entries are the enumerators of the Jacobi symbols which depend on the row and the column indices vanishes if and only if n is composite. If the dimension is a prime p, then the determinant evaluates to a polynomial of degree p − 1 which is the product of a power of p and the generating polynomial of the partial sums of Legendre symbols. The sign of the determinant is determined by the quadratic character of −1 modulo p. The proof of the evaluation makes use of elementary properties of Legendre symbols, quadratic Gauss sums, and orthogonality of trigonometric functions.
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References
Borevich Z.I., Shafarevich I.R.: Number Theory. Academic Press, New York-London (1966)
Chapman R.: Determinants of Legendre symbol matrices. Acta Arith. 115(3), 231–244 (2004)
Davenport H.: On certain exponential sums. J. Reine Angew. Math. 169, 158–176 (1933)
Gradshteyn I.S., Ryzhik I.M.: Table of Integrals, Series, and Products. Academic Press, Orlando (1980)
Hardy G.H., Wright E.M.: An Introduction to the Theory of Numbers. Oxford University Press, New York (1980)
Ireland K., Rosen M.: A Classical Introduction to Modern Number Theory. Springer-Verlag, New York (1990)
Moon J.W.: Counting Labelled Trees. Canadian Mathematical Congress, Montreal, Que. (1970)
Stanley R.P.: Enumerative Combinatorics, Volume 2. Cambridge University Press, Cambridge (1999)
Williams K.S.: Finite transformation formulae involving the Legendre symbol. Pacific J. Math. 34, 559–568 (1970)
Acknowledgment
I would like to thank the anonymous referee who suggested an alternate proof of Lemma 2.5 and whose comments greatly improved the presentation of this paper.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Eğecioğlu, Ö. A Prime Sensitive Hankel Determinant of Jacobi Symbol Enumerators. Ann. Comb. 14, 443–456 (2010). https://doi.org/10.1007/s00026-011-0069-6
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DOI: https://doi.org/10.1007/s00026-011-0069-6