Abstract
We give new estimates for multiple exponential sums, which infers $$A(x)=C_1x+C_2x^{1/2}+C_3x^{1/3}+O(x^{1/4}e^{V(x)}), V(x)=\frac{1}{\sqrt{3}}(L\log{L})^{1/2}+O((L\log{L})^{1/2}),$$ where $L=\log{x}$, $A(x)$ is the number of non-isomorphic abelian groups of orders $\leq{x}$, and $x$ is large.
Citation
H.-Q. Liu. "Exponential sums and the abelian group problem." Funct. Approx. Comment. Math. 42 (2) 113 - 129, June 2010. https://doi.org/10.7169/facm/1277811635
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