Lattice points on determinant surfaces and the spectrum of the automorphic Laplacian
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We use classical Fourier analysis along with tools from the spectral theory of Automorphic forms to derive an asymptotic formula with a strong error term for the number of integer solutions $(a, b, c, d)$ inside the expanding box $[-X,X]^4$ to the determinant equation $ad-bc=r$, where $r \neq 0$ is a fixed integer. Furthermore, we apply our method to study sums over these solutions where the variables are weighted by periodic arithmetical functions in two of the variables in one case, and by an arbitrary sequence of complex numbers in another.
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Cited by 2 Pith papers
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Counting solutions to the quadratic determinant equation
Proves asymptotic count of solutions to x1 x2 - x3 x4 = h for xi in [-N, N] with square-root cancellation when h = N^2 + O(N), confirming a prior speculation.
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Counting $2 \times 2$ integer matrices with a given determinant
T(h,N) equals (16/ζ(2)) N² times the sum of 1/d over divisors d of h, plus an error O_ε(N^ε (N + h)).
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