Approximate formulae for \(L(1,\chi)\) (Q2773290)

Let \(\chi\) be a primitive character of conductor \(q>1\). The paper presents upper bounds on \(L(1,\chi)= \sum_{n\geq 1}\chi (n)/n\). Theorem: Let \(F:\mathbb{R} \to\mathbb{R}\) be such that \(f(t)=F(t)/t\) is \(C^2\) over \(\mathbb{R}\), vanishes at \(t=\pm \infty\), and its first and second derivatives belong to \({\mathcal L}^1 (\mathbb{R})\). Moreover, \(F\) is assumed to be odd or even according as \(\chi\) is even or odd. Then for any \(\delta>0\), NEWLINE\[NEWLINEL(1,\chi) =\sum_{n\geq 1}\chi (n){1-F(\delta n)\over n}+ {\chi(-1) \tau(\chi) \over q}\sum_{m\geq 1}\overline \chi(m) \int^\infty_{-\infty} {F(t)\over t}e^{mt\over \delta q}dt,NEWLINE\]NEWLINE where \(\tau(\chi)= \sum_{a \text{mod} q} \chi(a)e^{a/q}\). Special choices of \(F\), as e.g. \(F(t) =1-{\sin (\pi t)\over \pi t}\) in the case when \(\chi\) is odd, yield NEWLINE\[NEWLINEL(1,\chi)= \sum_{n\geq 1}{\chi(n) \sin(\pi \delta n)\over\pi \delta n^2}-{i\pi \tau(\chi) \over q}\sum_{1\leq m\leq\delta q/2}\overline \chi(m)\left(1-{2m\over \delta q} \right).NEWLINE\]NEWLINE Taking then \(\delta\) close to \({1\over\sqrt q}\) gives, for large \(q\), \(|L(1,\chi)|\leq{1\over 2}\log q+0.9\). The strongest result is NEWLINE\[NEWLINE|L(1,\chi) |\leq {1\over 2}\log q+ \begin{cases} 0 & \chi\text{ even }\\ {5\over 2}-\log 6 & \chi\text{ odd }\end{cases}.NEWLINE\]NEWLINE For related work see, for instance, \textit{S. Louboutin} [C. R. Acad. Sci., Paris, Sér. I 316, 11-14 (1993; Zbl 0774.11051); 323, 443-446 (1996; Zbl 0864.11042) and J. Math. Soc. Japan 50, 57-69 (1998; Zbl 1040.11081)] and \textit{J. D. Vaaler} [Bull. Am. Math. Soc., New Ser. 12, 183-216 (1985; Zbl 0575.42003)].