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I was wondering if anyone is aware of any existing literature on the arithmetic function defined as $$f(n):=2^{\omega(n)}\tau(n).$$ Here $\omega(n)$ is the number of distinct prime divisors of $n$ and $\tau(n)$ is the number of divisors of $n$. In particular, I am interested in the behavior of $$\sum_{d \mid n}f(d)$$

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Likely there is no reference dealing with exactly this function, but there are many things which deal with the general properties of multiplicative functions. What behavior are you interested in? What would you like to know, and/or prove? –  Eric Naslund Apr 22 '12 at 9:27
I am looking for relatively simple bounds. Not the best possible necessarily, just something that might give one an idea of how big or small this thing is. For example, I think it's fairly easy to show that $$\sum_{d \mid n}f(d)<2^{\omega(n)}\tau^2(n)<2^{\omega(n)+2}n.$$ However, I ran a few values with Mathematica (a simple command to do this is: t[d_] := DivisorSum[d, (2^(PrimeNu[#]))*DivisorSigma[0, #] &]) and this bound is probably far too large. –  the_fox Apr 22 '12 at 9:46
Actually, the above inequality holds trivially. –  the_fox Apr 22 '12 at 9:59
When $n=p$, $p$ prime, $2^{\omega(n)}\tau^2(n)=\sum_{d \mid n}{f(d)}+3$, so a better question is: what is the mean value of $\sum_{d \mid n}f(d)$? –  the_fox Apr 22 '12 at 10:22
I think I can calculate the asymptotics for $$\sum_{n\leq x}\sum_{d|n} f(d),$$ but it is a bit messy. –  Eric Naslund Apr 23 '12 at 9:20

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