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For $X\ge 1$, the divisor function obeysOne proof compares it with the generalized divisor function $d_4(n)$, the number of ordered factorizations $n=abcd$ into four positive integers. At a prime power $p^a$, the two counts are $(a+1{)}^2$ and $\left(\genfrac{}{}{0ex}{}{a+3}{3}\right)$. The inequality $(a+1{)}^2\le (\genfrac{}{}{0ex}{}{a+3}{3})$ holds for every nonnegative integer $a$, and multiplicativity gives $\tau (n{)}^2\le d_4(n)$. Hence, with $H_m$ the harmonic number,Applying partial summation to this upper bound also yields