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I am currently using the Selberg-Delange method in my research, and I am running into some troubles. I am somewhat following the method presented in Tenenbaum's book where first he shows that

$$\sum_{n\leq x}a_z(n)=x(\log(x))^{z-1}\left[\sum_{j=0}^N\frac{z\lambda_j(z)}{(\log(x))^j}+O_A(R_N(x))\right]$$

and then he concludes that

$$\sum_{n\leq x} c_k(n)=\left[\sum_{j=0}^N\frac{Q_{j,k}(\log_2(x))}{(\log(x))^j}+O_A\left(\frac{(\log_2(x))^k}{k!}R_N(x)\right)\right]$$

Where $c_k(n)$ is the final function we are looking to estimate. In this method, we get that

$$Q_{j,k}(X):=\sum_{m+l=k-1}\frac{1}{m!l!}\lambda_j^{(m)}(0)X^l$$

and I am wondering if there is any standard way to bound $|\lambda_j^{(m)}(0)|$, since when I am doing work with this method not being able to bound $|\lambda_j^{(m)}(0)|$ means that I really don't have any information about the polynomials at all.

I am defining $\lambda_j^{(m)}(0)$ as Tenenbaum does, namely that

$$\lambda_k(z)=\frac{1}{\Gamma(z-k)}\sum_{n+j=k}\frac{1}{n!j!}H^{(n)}(1;z)\gamma_j(z)$$

where $H$ is the function dependent on $c_k(n)$ that appears when applying the SD method, and $\gamma_j(z)$ are the normalized power series coefficients of $s^{-1}((s-1)\zeta(s))^z$ around $s=1$.

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