Let $\{ f_{n}(x) \}$ be a sequence of piecewise linear functions of $x$. If the order of growth of $\sum_{n \leq x} f_{n}(x)$ is $O(g(x))$ for some smooth $g(x)$, then what can be said about the order of $\sum_{n \leq x} f_{n}(x) \log n$? In particular, if $\sum_{n \leq x} a_{n} = O(1)$, then is $\sum_{n \leq x} a_{n} \log n$ at most $O(\log x)$, where $a_n$ is a sequence depending only on $n$?
Summatory Asymptotics
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real-analysis
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0I thought the generalization was obvious--see Edit. – 2012-09-30