Revision as of 23:02, 5 August 2015 edit 2601:445:4001:4514:85a3:3283:72dc:2303 (talk) →Calculating mean values using Dirichlet series ← Previous edit		Revision as of 23:03, 5 August 2015 edit undo 2601:445:4001:4514:85a3:3283:72dc:2303 (talk) →Divisor functions Next edit →
Line 119: \sum_{n\le x}\sigma_{\alpha}(n)= \begin{cases} \;\;\sum_{n\le x}\sigma_{\alpha}(n)=\frac{\zeta(\alpha+1)}{\alpha+1}x^{\alpha+1}+O(x^{\beta}) & \text{if } \alpha>0, \\ \;\;\sum_{n\le x}\sigma_{-1}(n)=\zeta(2)x+O(~~logx~~\log x) & \text{if } \alpha=-1, \\ \;\;\sum_{n\le x}\sigma_{\alpha}(n)=\zeta(-\alpha+1)x+O(x^{\max(0,1+\alpha)}) & \text{otherwise.} \end{cases} </math>

Average order of an arithmetic function: Difference between revisions