Does the RH have to be true in order for Riemann's explicit formula for the number of primes <= x to hold? The formula is (copied from wikipedia:https://en.wikipedia.org/wiki/Explicit_formulae_(L-function)): $f(x) = li(x)-\sum_{\rho}li(x^{\rho})-\log 2 +\int_x^\infty\frac{dt}{t(t^2-1)log t}$ where $f(x) = \pi_0(x)+(1/2)\pi_0(x^{1/2})+ \ldots$ where $\pi_0$ is the normalised prime counting function. In other words, does this formula hold even if not all the critical roots are on the critical line x = 1/2?

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    $\begingroup$ Quote the specific formula you refer to in your question, please. $\endgroup$ – Sean Nemetz May 24 at 5:24
  • $\begingroup$ the usual explicit formulas for the number of primes hold unconditionally, but the error bounds are weakish and haven't been improved since the 1950's - RH gives pretty much the strongest possible bound $\endgroup$ – Conrad May 24 at 14:40

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