# About sign changes of Li(x)-π(x)

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 $1 \leq x \leq C$?

• – Alexey Ustinov Jan 5 '14 at 9:22
• 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. – Stefan Kohl Jan 10 '14 at 11:49
• Are you familiar with the results of Kaczorowski? See Theorem 1 of matwbn.icm.edu.pl/ksiazki/aa/aa45/aa4517.pdf – so-called friend Don Jan 10 '14 at 16:27
• The result @so-calledfriendDon mentions is a lower bound. (That OP wants upper bounds only was not really visible when that comment was made.) – user9072 Jan 10 '14 at 16:41
• @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(?) – Stefan Kohl Jan 10 '14 at 17:09