I am having some issue verifying Lemma 2 of K. Soundarajan's paper Moments of the Riemann Zeta function. It states the following:
Assume RH. Let $T \leq t \leq 2T$, $2 \leq x \leq T^2$ and $\sigma \geq 1/2.$ Then $$\left| \sum_{n \leq x, n \neq p} \frac{\Lambda(n)}{n^{\sigma + it} \log n} \frac{\log x/n}{ \log n} \right| \ll \log \log \log T + O(1). $$
He shows this by noting that the terms where $n = p^k, k \geq 3$ contribute a bounded amount, hence we only need to worry about the prime squares. By definition of the Von Mangholdt function $\Lambda(n)$, we don't have to consider any other possible $n$s. Next he uses an estimate from Davenport's Multiplicative number theory,
$$\sum_{p \leq z} (\log p) p^{-2it} \ll z/T + \sqrt z \log ^2 zT,$$
and uses partial summation to get the final estimate. Reading the cited chapters of Davenport, I can't seem to deduce the latter estimate, and how do you use partial summation to get the desired result? Thanks!
