We know that there are some approximation like Abel's identity. If $\lambda_n$ is increasing and $$ C(x)=\sum_{\lambda_n\le x}c_n,\qquad(c_n\in\mathbb{C}) $$ Then if $X\ge\lambda_1$ and $\phi(x)$ has continuous derivative, we have $$ \sum_{\lambda\le X}c_n\phi(\lambda_n)=C(X)\phi(X)-\int_{\lambda_1}^X C(x)\phi'(x)dx $$ If $C(X)\phi(X)\rightarrow0$ as $X\rightarrow\infty$, then $$ \sum_{\lambda\le X}c_n\phi(\lambda_n)=-\int_{\lambda_1}^X C(x)\phi'(x)dx $$ where both sides are convergent.
do we have similar estimate (for example alternating harmonic series) with integral?