Estimates for lattice points of quadratic forms with integral coefficients modulo a prime number square
© Hakami; licensee Springer. 2014
Received: 30 June 2014
Accepted: 18 July 2014
Published: 18 August 2014
Let be a nonsingular quadratic form with integer coefficients, n be even. Let denote the set of zeros of in , p be an odd prime, and denote the cardinality of V. In this paper, we are interested in giving an upper bound of the number of integer solutions of the congruence in small boxes of the type centered about the origin, where , and for .
MSC:11E04, 11E08, 11E12, 11P21.
The final result of this paper is stated in the following theorem.
Equations (6) and (7) express the ‘incomplete’ sum as a fraction of the ‘complete’ sum plus an error term. In general so that the fractions in the two equations are about the same. In fact, if V is defined by a ‘nonsingular’ quadratic form then . (That is, .)
for . If we take then it is clear from the definition that is the number of ways of expressing x as a sum with and . Moreover, is nonempty if and only if .
The last identity is Parseval’s equality.
2 Fundamental identity
Using identities for the Gauss sum , one obtains the following.
Lemma 1 ([, Lemma 2.3])
where is the quadratic form associated with the inverse of the matrix for Q modp.
Inserting the value in Lemma 1 yields (see ) the following.
Lemma 2 (The fundamental identity)
3 Auxiliary lemma on estimating the sum
The lemma is established. □
4 Proof of Theorem 1
We next determine which of the terms , , and in (23) is the dominant term. We consider two cases:
This leads to the proof of the lemma. □
The author would like to thank the anonymous referee for his helpful and constructive comments and suggestions. He would also like to thank the Editors for their generous comments and support during the review process. Finally, he would like to thank the VTEX Typesetting Services for their assistance in formatting and typesetting this paper.
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