Are there effective, explicit upper bounds on $L'(1,\chi)/L(1,\chi)$ in the literature, valid for all Dirichlet characters $\chi$? Due to the possibility of Siegel zeros, I don't imagine one can do much better than $O(\sqrt{q})$, and I don't believe getting $O(\sqrt{q})$ is at all hard -- I was just wonder what the best available result with fully explicit constants is. (Perhaps there is something very clean and simple?)
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asked Sep 16, 2019 at 17:10
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1$\begingroup$ I think one does get something rather easily, as when $\chi$ is exceptional then $L(s,\chi)$ is rather much like $\zeta(2s)/\zeta(s)$. Looking at the imaginary quadratic case, if the class number is $1$ we thus have $L'(1,\chi)\sim\pi^2/6$, with in general there additionally being a product by a reciprocal sum over minima. One can work analogously in the real quadratic case as done by Goldfeld and Schinzel. Whether or not this can be done cleanly and explicitly is a different question. But one should get $(\pi^2/6)/(\pi/\sqrt q)$. archive.numdam.org/item/ASNSP_1975_4_2_4_571_0 $\endgroup$– MyNinthAccountCommented Sep 16, 2019 at 18:24
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$\begingroup$ Another way to go about this could be: consider $\sum_{p^v} {\chi(p^v)\Lambda(p^v)\over p^v}e^{-p^v/X}=\int X^s \Gamma(s) {L'\over L}(s+1){ds\over2\pi i}$ and move the contour left. You are willing to accept $O(\sqrt q)$ and so can grandiosely take $X\approx e^{\sqrt q}$ if desired to truncate the sum, while the poles contribute $(L'/L)(1)+\Gamma(\beta-1)X^{\beta-1}+O(...)$ where bounding the other zeros shouldn't be too hard. The $\Gamma(\beta-1)$ then gives $O(\sqrt q)$ again. $\endgroup$– MyNinthAccountCommented Sep 16, 2019 at 21:27
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$\begingroup$ Interesting, but I want $\leq c \sqrt{q}$ with a nice, clean constant, not just $O(\sqrt{q})$. $\endgroup$– H A HelfgottCommented Sep 16, 2019 at 21:49
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1 Answer
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Self-answer: yes, the answer is Proposition 1.12 in Bennett-Martin-O'Bryant-Rechnitzer: https://projecteuclid.org/euclid.ijm/1552442669
They show that, for $\chi$ a Dirichlet character modulo $q\geq 10^5$, the constant term in the Laurent expansion of $L'(s,\chi)/L(s,\chi)$ at $s=0$ has absolute value $\leq 0.2515 q \log q$.
Two quick comments:
- the case $q<10^5$ can be done computationally (is it out there somewhere?)\
- for $q>1$, that constant term equals $\log(2\pi/q) + \gamma - L'(1,\chi)/L(1,\chi)$, by the functional equation.
Their proof (which is based, not to our surprise, on a (good) explicit bound of the "trivial" kind on how close a zero can be to $s=1$) actually gives an upper bound of $\leq c \sqrt{q} \log q$, with $c$ close to $1/40$, under the additional assumption that $\chi$ is primitive.
answered Sep 22, 2019 at 12:21