A modification of Sivaramakrishnan-Venkataraman's inequality (Q2907573)

Let \(d(n)\), \(\sigma(n)\), and \(\psi(n)\) denote the number of distinct positive divisors of \(n\), the sum of of distinct divisors of \(n\), resp. Dedekind's arithmetic function. In 1965 R. Sivaramakrishnan and C. S. Venkataraman [Problem 5326, Am. Math. Mon. 72, 915 (1965)] found the inequality: NEWLINE\[NEWLINE \frac {\sigma_k(n)}{d(n)}\geq n^{k/2}.NEWLINE\]NEWLINE For \(k=1\) this implies \(\sigma(n)\geq d(n)\cdot\sqrt n\). The author improves this inequality as follows: NEWLINE\[NEWLINE\psi(n)\geq d(n)\cdot\sqrt n.NEWLINE\]