Weighted quadratic partitions modulo $P^m$- a new formula and a new demonstration

Main Article Content

Ali Hafiz Hakami

Abstract

Let $Q({\bf{x}}) = Q(x_1 ,x_2 ,...,x_n )$ be a quadratic form over $\mathbb{Z}$, $p$ be an odd prime. Let $V = V_Q = V_{p^m } $ denote the set of zeros of $Q({\mathbf{x}})$ in $\mathbb{Z}_{p^m }$ and $\left| V \right|$ denotes the cardinality of $V$. Set $ \phi (V_{p^m } ,{\mathbf{y}}) = \sum _{{\mathbf{x}} \in V} e_{p^m } ({\mathbf{x}} \cdot {\mathbf{y}})$ for ${\mathbf{y}} \ne {\mathbf{0}}$ and $\phi (V_{p^m } ,{\mathbf{y}}) = \left| {V_{p^m } } \right| - p^{m(n - 1)}$ for ${\mathbf{y}} = {\mathbf{0}}.$ In this paper we shall give a formula for the calculation of the function $\phi (V,{\mathbf{y}}).$

Article Details

How to Cite
Hakami, A. H. (2012). Weighted quadratic partitions modulo $P^m$- a new formula and a new demonstration. Tamkang Journal of Mathematics, 43(1), 11–19. https://doi.org/10.5556/j.tkjm.43.2012.753
Section
Papers
Author Biography

Ali Hafiz Hakami, Department ofMathematics, King Khalid University, P.O.Box 9004, Abha, Postal Code: 61431, Saudi Arabia.

Ali Hakami ,

Analytic number theory- Distribution of solutions of congruences- Exponential sums- Quadratic forms and trigonometric sums

B. Sc. in mathematics, King Abdulaziz University 1996, SA
M. Sc. in mathematics, Kansas State University 2004, USA
Ph. D. in mathematics, Kansas State University 2009, USA

References

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T. Cochrane and Z. Zheng, Pure and mixed exponential sums, Acta Arithmetica, XCI.3. (1999), 249--278.

A. Hakami, Small zeros of quadratic congruences to a prime power modulus. PhD thesis, Kansas State University, 2009.