# Is there a standard way of obtaining bounds on $\lambda_{k}^{(m)}(0)$ when using the Selberg-Delange method?

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 $$s=1$$.