# How often does the omega theorem hold?

Write $$\psi(x) = \sum_{n\le x} \Lambda(n)$$. The classical omega theorem says that

$$\psi(x) - x = \Omega_{\pm}(x^{1/2})$$.

Question: How often does this hold? For example, what do we know about the size of the set

$$\{ n\le x: \psi(x) - x > c x^{1/2} \}$$

for some $$c$$? Ditto for $$\psi(x) - x < c x^{1/2}$$. What about replacing e.g. $$x^{1/2}$$ by $$x^{\alpha}$$ for some fixed $$0 < \alpha < 1/2$$? What's a good reference for such results? Thanks!

• Not to be confused with the Omega lemma! Mar 24, 2022 at 0:50

For any $$\varepsilon>0$$, there exist $$c(\varepsilon)>0$$ and $$X_0(\varepsilon)>0$$ such that for any $$X>X_0(\varepsilon)$$ we have $$\sup_{X\leq x\leq X^{1+\varepsilon}}\frac{\psi(x)-x}{\sqrt{x}\log\log x}>c(\varepsilon)\qquad\text{and}\qquad \inf_{X\leq x\leq X^{1+\varepsilon}}\frac{\psi(x)-x}{\sqrt{x}\log\log x}<-c(\varepsilon).$$ More precisely, Ingham (1935) proved a stronger result for the case when the real parts of the zeta zeros have a maximum, while Pintz (1980) proved a stronger result for the case when the real parts of the zeta zeros do not have a maximum. There might be even stronger results in the literature, please check.
• @WSao I recommend that you study Bohr's book "Almost periodic functions". Consider $f(x):=\sum_{n=1}^N\max\left(0,1-q\left\|c_n x\right\|\right)$, where $c_n:=\gamma_n/(2\pi)$. This is a sum of continuous periodic functions, hence it is almost periodic (as shown by Bohr). In particular, every sufficiently long interval contains $U$ such that $\sup_x|f(x+U)-f(x)|<1$. Take such a $U$. From $f(0)=N$ it follows that $f(U)>N-1$, so none of the $N$ terms in the definition of $f(x)$ vanishes at $U$. That is, $\|c_n U\|<1/q$ holds for $n\in\{1,\dotsc,N\}$. This is the claim that Ingham uses. Apr 7, 2022 at 5:19