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If $f(n)$ is an arithmetic function with $|f(n)|=1$, and $$\lim_{s\to+1} (s-1)\sum_{n=1}^\infty\frac{f(n)}{n^s}=0$$ Can I deduce that $$\lim_{s\to +1}(s-1)^2\sum_{n=1}^\infty\frac{f(n)\ln(n)}{n^s}=0$$ I tried abel summation but I am still having trouble, I would appreciate some help

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Maybe I am misunderstanding something, but $(s-1)^2$ is smaller than $s-1$ near $1$? –  Alex Youcis Apr 17 '13 at 4:35
    
@AlexYoucis Yes, I made a mistake there should be a logarithmic factor lol –  Ethan Apr 17 '13 at 4:53

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