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Given a constant $C$, which are the best known upper bounds for the number of sign changes of the function $$ f: \mathbb{N} \rightarrow \mathbb{R}, \ \ x \mapsto {\rm Li}(x)-\pi(x) $$ in the range $1 \leq x \leq C$?

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    $\begingroup$ See mathoverflow.net/questions/48461/… $\endgroup$
    – Alexey Ustinov
    Commented Jan 5, 2014 at 9:22
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    $\begingroup$ The OP has clarified that the question is not asking for bounds on the first sign change of ${\rm Li}(x) - \pi(x)$, and is hence not a mere duplicate. I have tried to improve the formulation of the question, and voted to reopen. $\endgroup$
    – Stefan Kohl ♦
    Commented Jan 10, 2014 at 11:49
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    $\begingroup$ Are you familiar with the results of Kaczorowski? See Theorem 1 of matwbn.icm.edu.pl/ksiazki/aa/aa45/aa4517.pdf $\endgroup$
    – so-called friend Don
    Commented Jan 10, 2014 at 16:27
  • $\begingroup$ The result @so-calledfriendDon mentions is a lower bound. (That OP wants upper bounds only was not really visible when that comment was made.) $\endgroup$
    – user9072
    Commented Jan 10, 2014 at 16:41
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    $\begingroup$ @quid: o.k. -- I noticed this, but have chosen the more general formulation since upper bounds may be delicate. -- At least I guess that once there is a sign change, there might be a whole lot of sign changes which are very close together(?) $\endgroup$
    – Stefan Kohl ♦
    Commented Jan 10, 2014 at 17:09

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