关于有限Abel群计数的若干问题-基础数学专业毕业论文.pdfVIP

n = pα1 pαs , αj ≥ k(j = 1, . . . , s), n k − full 1 s δk (n) k − full 1 1 1 1 +ε a(n)δ (n) = x 2 P (log x) + x 3 P (log x) + O(x 4 ), (2) 2 1 2 n≤x P (t)(j = 1, 2) t j j 2 1 1 1 0.1876+ε a(n)δ (n) = x 3 Q (log x) + x 4 Q (log x) + x 5 Q (log x) + O(x ), (3) 3 2 4 6 n≤x Q (t)(j = 2, 4, 6) t j j  3 Riemann-zeta Lindel¨of 2k−1 1 1 a(n)δk (n) = x j HP (j )−1(log x) + O(x 2k +ε ), (4) n≤x j =k HP (j )−1(t) t P (j ) − 1 Perron  O156.4 2 Some Questions About Counting Functions of Finite Abelian Groups Shuqian Gao The School of Mathematical Science, Shandong Normal University Jinan, Shandong, 250014, P. R. China ABSTRACT Let the arithmetical function a(n) denote the number on non-isomorphic abelian groups with n elements. It is well known that a(n) is a positive, integer-valued mul- tiplicative function, with the property that a(pα ) = P (α) for every prime p and every integer α ≥ 1. P (α) is the number of unrestricted partitions of α, especially we have 2 3 a(1) = 1, a(p ) = 1, a(p ) = 2, a(p ) = 3, 4 5 6 7