Timeline for Does $L(-1+it,f)\ll_f \log^c q(f)t$ hold ture?

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Oct 20, 2013 at 1:50	comment	added	GH from MO		@Houfei: The bound I stated is stronger and more general than (5.20) that you quote. First, the implied constant only depends on $\sigma$ (not on $\pi$). In fact this dependence is continuous in $\sigma$. Second, I stated it for all $\sigma$ (not just $0\leq\sigma\leq 1$).
Oct 17, 2013 at 8:04	comment	added	H.Flip		Brilliant! Thanks for your reply. In IK'book, $(5.20)$ states $L(\sigma+it,f)\ll_{\epsilon,f}C(\frac{1}{2}+it,f)^{\frac{1-\sigma}{2}+\epsilon}$, when $0\le\sigma\le 1$ and $t$ is large. $C^\epsilon$ factor may be not omitted in the convexity bound above.
Oct 14, 2013 at 19:02	comment	added	GH from MO		@Lucia: Thanks for this comment. Let us mention that Heath-Brown requires some control on the Langlands parameters at the various primes, but these are known by the work of Luo-Rudnick-Sarnak. The bound I displayed is more general, it relies on the work of Molteni (which in turn needs the Luo-Rudnick-Sarnak bounds if I recall correctly).
Oct 14, 2013 at 18:25	comment	added	Lucia		Actually the convexity bound on the $1/2$ line does hold without the $C^{\epsilon}$ factor. This is due to Heath-Brown arxiv.org/abs/0809.1752 , which appeared in Acta Arithmetica.
Oct 14, 2013 at 11:27	history	edited	GH from MO	CC BY-SA 3.0	deleted 9 characters in body
Oct 14, 2013 at 11:16	history	answered	GH from MO	CC BY-SA 3.0