 sum in (\[bigt\]) is bounded above by $$\begin{aligned} \sum_{{{\rm N}}({\mathfrak p}) \leq \sqrt{x}} \sum_{\substack{k \geq 0\\ {{\rm N}}({\mathfrak p})^k > x}}  \frac{d}{k{{\rm N}}{\mathfrak p}^{k\sigma}} 