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Consider the following finite sum

$$\sum_{p\leq x}f(p) = S(x)+C$$

Here, $f(x)$ is smooth

$p$ is prime

$S(x)$(=smooth+oscillation) is also a 'function';

$C$ is a constant

We also know the following

$$\ \sum_{p\leq x}f(p) = \int_{2}^{x}f(t)d(π(t))\tag{1}$$

Where $π(t)$ is a prime counting function .

Also,

$$\pi(x) = \operatorname{R}(x) - \sum_{\rho}\operatorname{R}(x^{\rho})\tag{2} $$

$${\displaystyle \operatorname {R} (x)=1+\sum _{k=1}^{\infty }{\frac {(\ln x)^{k}}{k!k\zeta (k+1)}}} $$ $$ = \sum_{n=1}^{\infty} \frac{ \mu (n)}{n} \operatorname{li}(x^{1/n}) \tag{3}$$

I want to know what's the general 'explicit' expression for $C$ using equations 1,2,3?

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    $\begingroup$ I don't understand the question. $C$ could be anything by adjusting $S(x)$ by adding a constant. Without telling us what $S$ is, I would just answer $C=f(2)-S(2)$. $\endgroup$ Apr 24 at 11:33

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