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In Schoenfeld's (1976) Paper:

"Sharper bounds for the Chebyshev functions $\theta(x)$ and $\psi(x)$. II",

it is shown in Corollary 1. (6.18) that if the Riemann Hypothesis holds, then :

$$|\pi (x) - \text{Li}(x)| < \frac{\sqrt x \ln(x)}{8\pi},~~~~~\text{for}~~~~~x \geq 2657$$

Right under it, (6.19) :

$$\pi (x) - \text{Li}(x) < \frac{\sqrt x\ln(x)}{8\pi},~~~~~\text{for}~~~~~ x\geq \frac{3}{2}$$

My question is, are these both true if and only if the Riemann Hypothesis is true, or does the phrase "If the Riemann Hypothesis holds, then" only apply for the first (6.18) statement? That is, if either one of them is proven or debunked, then does it follow that the Riemann Hypothesis is also proven or debunked?

PseudoNeo
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stefan
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  • The first statement is an if and only if relation; it is equivalent to the Riemann hypothesis. This is completely standard. The second statement does not remove the possibility that $\mathrm{Li}(x) - \pi(x)$ is often much bigger than $\sqrt{x} \ln x$, and if this were to happen particularly badly (for example, if $\mathrm{Li}(x) - \pi(x)$ were larger than $x^{1/2 +\delta}$ infinitely often for some fixed $\delta > 0$), then the Riemann hypothesis would be false. So the Riemann hypothesis implies the second statement, but not vice versa. – Peter Humphries Oct 15 '15 at 21:44
  • By any chance is there a way I can confirm this? I understand what both you and martini are saying, so is there proof that the first statement is -equivalent- to RH? – stefan Oct 16 '15 at 13:08

1 Answers1

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The phrase "If the Riemann hypothesis is true, then" applies as an assumption for both (6.18) and (6.19). But: Note that you only have "if", not "only if", that is, we have, if ($\sf RH$) denotes Riemanns' hypothesis, that $$ ({\sf RH}) \implies (6.18), \quad ({\sf RH}) \implies (6.19) $$ which means that if $(\sf RH)$ is proven, both (6.18) and (6.19) are, and, by contraposition $$ \neg (6.18) \implies \neg ({\sf RH}), \quad \neg (6.19) \implies \neg ({\sf RH}) $$ That is, if (6.18) or (6.19) is disproven, also ($\sf RH$) is. Note that proving (6.18) or (6.19) does not imply anything about the truth of $(\sf RH)$.

martini
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  • I see! I have one little question left then. If the two statements are stated to only be valid for x>=2657 for the first, and x>=3/2 for the second, then does proving that an x value outside the specified region exists that satisfies the statements imply that RH is not true? – stefan Oct 15 '15 at 12:50
  • No. Both statements do not say anything about the numbers $x$ outside the specified region. Again only a "if $x \ge 2657$, then ..." (not a if and only if). – martini Oct 15 '15 at 12:51
  • Thank you for your reply :D – stefan Oct 15 '15 at 12:54
  • I am now quite confused. I kept on searching on some consequences of Riemann's hypothesis, trying to find an "if and only if" consequence. I stumbled upon this link: http://math.stackexchange.com/questions/614009/riemann-hypothesis-and-the-prime-counting-function Here the OP is told that yes, one of the statements, at least, is an if and only if. – stefan Oct 15 '15 at 15:24
  • Your last sentence is false; (6.18) is equivalent to RH. – Peter Humphries Oct 15 '15 at 21:45