Exponential sums and the abelian group problem
Let \(A(x)\) be the number of isomorphism classes of abelian groups of orders \(\leq x\). Using the method of exponential sums, the author proves that for \(x>10\) \[ A(x)=C_1x+C_2x^{\frac 12}+C_3x^{\frac 13}+O(x^{\frac14}e^{V(x)}), \] with explicit constants \(C_1,C_2,C_3\) and \[ V(x)=\frac{1}{\sqrt{3}}(\log x\log\log x)^{\frac12}+O((\log x)^{\frac12}(\log\log x)^{-\frac12}). \]
scientific article; zbMATH DE number 5758459
scientific article