Can anybody suggest a good (e.g. "non-technical") introduction to estimating bounds for logarithmic sums of the form

$$\sum_{i=1}^{r}{{\alpha_i}{\log(q_i)}}$$

where the $$\alpha_i$$ are positive integers (not necessarily distinct) and the $$q_i$$ are odd primes.

The reason why I ask this question is because I am trying to estimate (both lower and upper) bounds for the logarithm of a number-theoretic function, specifically $$\sigma_{1}(N)$$

I was able to show (in 2008) that

$$\sigma_{1}({q_i}^{\alpha_i}) \le \frac{2}{3}\frac{N}{{q_i}^{\alpha_i}}$$

for all $i = 1, 2, ..., r = \omega(N)$,

where $N$ is an OPN (i.e. Odd Perfect Number) and ${q_i}^{\alpha_i} || N$ for all $i$. You will be able to get an upper bound for the logarithmic sum given above, by first taking logarithms of both sides of the inequality, then summing over all $i$.

Unfortunately, for the "numbers" $N$ that I am considering, the current literature (on OPNs) do not point to an approach "strong enough" to prove nonexistence of such "numbers" $N$. (This is because the upper bound alluded to in the previous paragraph is still (of course) dependent on $r = \omega(N)$).

(And that is the reason why --) I'd be particularly interested in an analytic-number-theoretic approach. =) (Thanks to Gerry Myerson for encouraging this "clarification".)