On the best possible exponent for the error term in the lattice point counting problem on the first Heisenberg group

2017/10/09 by Gath, Yoav A.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1710.02995

Abstract

We use classical methods from analytic number theory to resolve the lattice point counting problem on the first Heisenberg group, in the case where the gauge function is taken to be the Cygan-Kor\acuteanyi Heisenberg-norm N_4,1(z,w)=(|z|⁴+w²)^1/4. In this case, our main theorem establishes the estimate E(x)=Ω_±(x^(1)/(2)), where E(x)=S(x)-\fracπ²2x is the error term arising in the lattice point counting problem, S(x) is given by S(x)=∑_0≤ m²+n²

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