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Chebyshev's $\psi$ function is defined for primes $p$ as

$$\psi(x)=\sum _{p^k\leq x} \log (p)$$

von Mangoldt found an explicit formula for this, with the exception that the function takes half-values at each 'step':

$${\psi}^{}_{0} (x)=x-\frac{\zeta '(x)}{\zeta(x)}-\frac{1}{2}\ln\bigl(1-x^2 \bigr)-\sum_{\rho}\frac{x^{\rho}}{\rho}$$

where $\rho$ denotes the non-trivial zeroes of the zeta function. I have two questions:

  1. Does this formula assume that the Riemann hypothesis is correct, or does it remain valid if instances of $\rho$ exist that lie away from the critical line but still within the critical strip?
  2. Given that $\frac{\zeta '(x)}{\zeta(x)}=\ln(2\pi)$, and given that $\frac{1}{2}\ln\bigl(1-x^2 \bigr)$ is defined only in the half-plane $\ge 1$, is always negative with a pole at $x=1$, and rapidly converges towards $0$ from below, the expression $x-\frac{\zeta '(x)}{\zeta(x)}-\frac{1}{2}\ln\bigl(1-x^2 \bigr)$ is asymptotic to $x-\ln(2\pi)$. Is it therefore possible to write von Mangoldt's formula using big or little O notation for the expression $x-\frac{\zeta '(x)}{\zeta(x)}-\frac{1}{2}\ln\bigl(1-x^2 \bigr)$? If so, how?
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The answer to your first question, in a nutshell, is no. Depending on whether the Riemann Hypothesis is true or not, $\psi$ will take certain or other values. In particular, if the Riemann Hypothesis is true, $|\psi(x)-x|$ will be bounded by $O(\sqrt(x)\log^2(x))$, and conversely. This was proved by Shoenfeld. In addition, if the Riemann Hypothesis is true, then $\psi(x)-x$ will “oscillate” logarithmically (that is, argument of a sine function describing the oscillation will be a function of log x). The “dominant” log frequency will the absolute value of the imaginary part of the first zeta zero. If the Riemann Hypothesis is false, then the dominant frequency will be the absolute value of the smallest imaginary part of a zeta zero out of those zeta zeros with the largest real part. All this can be derived from con Mangoldt’s formula. Two good references to get started with this are Edwards’ book on the Riemann Zeta function, or Mazur and Stein’s “Prime numbers and the Riemann Hypothesis”. But perhaps your best resource is Dittrich’s “Reassessing Riemann’s paper: on the number of primes less than a given magnitude”, chapter 6, which deals with the derivation of von Mangoldt’s formula.

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