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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2$\begingroup$ See mathoverflow.net/questions/48461/… $\endgroup$ – Alexey Ustinov Jan 5 '14 at 9:22

1$\begingroup$ 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 Jan 10 '14 at 11:49

1$\begingroup$ Are you familiar with the results of Kaczorowski? See Theorem 1 of matwbn.icm.edu.pl/ksiazki/aa/aa45/aa4517.pdf $\endgroup$ – socalled friend Don Jan 10 '14 at 16:27

$\begingroup$ The result @socalledfriendDon mentions is a lower bound. (That OP wants upper bounds only was not really visible when that comment was made.) $\endgroup$ – user9072 Jan 10 '14 at 16:41

1$\begingroup$ @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 Jan 10 '14 at 17:09