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Let $\chi$ be a Dirichlet character mod $q$ and $\Lambda(n)$ be the von Mangoldt function. Let $c(\chi)=1$ if $\chi$ is the principal character, and zero otherwise. Let $\Theta_\chi$ be the supremum of real part of the zeros of the associated $L$-function $L(s, \chi)$. Define the generalised Chebyshev psi function $\psi(x, \chi):=\sum_{n\leq x} \Lambda(n)\chi(n)$. Are there infinitely many $x \rightarrow \infty$ such that $$\psi(x, \chi) - c(\chi)x = \Omega(x^{\Theta_{\chi}-\varepsilon})$$ for each $\chi$ mod $q$ and every $\epsilon>0$ ?

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    $\begingroup$ I think this might actually not be known unless one assumes that $L(s,\chi)$ is nonvanishing when $\Im(s) = 0$ and $\Re(s) > 1/2$, in which case the usual proof via Landau's lemma (Lemma 15.1 of Montgomery-Vaughan) works. $\endgroup$
    – Peter Humphries
    Jul 13, 2020 at 0:52

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