Positivity of partial Dirichlet series for a quadratic character?

Question

Let $\chi\colon(\mathbb{Z}/N\mathbb{Z})^\times\rightarrow\{\pm1\}$ be a primitive quadratic Dirichlet character of conductor $N$. For any integer $m=1,2,\cdots,\infty$, consider the partial Dirichlet series $$L_m(\chi)=L_m(1,\chi)=\sum_{n=1}^m\frac{\chi(n)}{n}.$$ It is well-known that $L_\infty(\chi)$ is positive, see chapter 6 of Apostol's 'Introduction to analytic number theory'. My question is about the partial sum $L_N(\chi)$. Here the subscript $N$ is the conductor of $\chi$. I have done some verification in sagemath, and for $N\le1000$, $L_N(\chi)$ is always positive. Is it always the case for any $N$? How to prove this? I asked this question in mathematics stack exchange(here) and got a partial answer. I would like to see if there are more relevant results in this direction.

Edit: note here that the $N$ in $L_N(\chi)$ is the same as the conductor of the character $\chi$. I forgot to emphasise this point in the first version, which led to some misunderstandings.

Answer 1

Theorem 6.18 in the same chapter of Apostol shows, by partial summation, that $$L_m(\chi) = L(1,\chi) + O_{\chi}(m^{-1})$$ where the implied constant is effective and depends on $\chi$. This can be explicated by using the Pólya--Vinogradov inequality, which gives $$L_m(\chi) = L(1,\chi)+O( m^{-1} N^{1/2} \log N)$$ where $N$ is the conductor and the implied constant is effective and absolute (one reference is exercise 11.2.1.3 in p. 374 of Montgomery and Vaughan's book). Taking $m=N$ as you do gives $$L_N(\chi) = L(1,\chi)+O( N^{-1/2} \log N)$$ unconditionally with an absolute and effective constant. Explicit constants were worked out by various authors, see e.g. Pomerance's paper "Remarks on the Pólya-Vinogradov inequality", Integers 11, No. 4, 531-542, A19 (2011).

To conclude that this is positive, one needs a lower bound on $L(1,\chi)$. There are two lower bounds on $L(1,\chi)$: effective and non-effective.

We have $L(1,\chi)\gg N^{-1/2}$ with an effective constant, see Theorem 11.11 in the Montgomery--Vaughan book, following from a result of Page. This falls just short of being useful here, so I do not think your question can be verified unconditionally.

Siegel showed $$L(1,\chi) > C(\varepsilon)N^{-\varepsilon}$$ for any given $\varepsilon>0$. The positive constant $C(\varepsilon)$ is not effective (it is if one shows there are no Siegel zeros). See Theorem 11.14 of the same book. Taking, say, $\varepsilon=1/3$, we find $L_N(1,\chi)$ is positive for $N$ sufficiently large. Assuming no Siegel zeros, you can make this effective. So conditionally, to verify your question one needs to check finitely many values. I wouldn't be surprised if the answer is positive.

On GRH, $L(1,\chi) \gg 1/\log \log N$ (Littlewood). The constant has been made effective by Lamzouri, Li and Soundararajan in "Conditional bounds for the least quadratic non-residue and related problems" (Math. Comput. 84, No. 295, 2391-2412 (2015); corrigendum ibid. 86, No. 307, 2551-2554 (2017)), see their Theorem 1.5. This should allow you to get a quick verification of your question under GRH.