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Upper bounds for \(|L(1,\chi)|\). (English) Zbl 1022.11041
Let \(\chi\) be a non-principal Dirichlet character to the modulus \(q\). The well-known bounds for the size of \(L(1,\chi)\) are \(q^{-\varepsilon} \ll_{\varepsilon} |L(1,\chi)|\ll \log q\), and if the Generalized Riemann Hypothesis is assumed \((\log\log q)^{-1} \ll |L(1,\chi)|\ll \log\log q\) which is best possible in terms of order of magnitude. It is elementary to observe that if \(\sum_{n\leq x}\chi(n) = o(x)\) for all \(x>X\), then \(L(1,\chi) = \sum_{n\leq X}{\chi(n)\over n} + o(\log q)\). Burgess’ character sum estimates with Heath-Brown’s improvement allow the smallest values for \(X\) that can be used here, specifically \(X=q^{{1\over 4}+o(1)}\) if \(q\) is cube-free or if \(\chi\) has order \(q^{o(1)}\), and \(X=q^{{1\over 3}+o(1)}\) otherwise. These values of \(X\) used in the bound \(|\sum_{n\leq X}{\chi(n)\over n}|\leq \log X +O(1)\) yield what the authors call the trivial bounds for \(|L(1,\chi)|\). For the Legendre symbol Stephens proved \(L(1,(\cdot/p)) \leq (2-2/\sqrt{e} + o(1)) {1\over 4}\log p\), which is best possible in the sense that completely multiplicative functions \(f\) with range \(\pm 1\), \(\sum_{n\leq x}f(n) = o(x)\) for \(x>X\), and \(\sum_{n\leq X}{f(n)\over n} \sim (2-{2\over \sqrt{e}})\log X\) can be constructed. Pintz extended Stephens’ result to all quadratic characters.
In this paper the authors establish for the first time improvements over the trivial bounds for complex characters starting from a more general setting as follows. For a subset \(S\) of the unit disc \(U\), denote by \({\mathcal F}(S)\) the class of completely multiplicative functions \(f\) such that \(f(p)\in S\) for all primes \(p\). Consider \[ \gamma(S;A) := \limsup_{x\to\infty}\max_{f\in {\mathcal F}(S)}{1\over \log X} \Biggl|\sum_{n\leq X}{f(n)\over n}\Biggr|\quad (A\geq 1) \] where \(\sum_{n\leq x}f(n) = o(x)\) for \(X\leq x \leq X^{A}\) is assumed, and write \(\gamma(S)= \lim_{A\to\infty}\gamma(S;A)\). Then Burgess’ estimates give \(|L(1,\chi)|\leq ({1\over 4 \text{ or} 3})(\gamma(S_k)+ o(1))\log q\), (whether 1/4 or 1/3 is determined as described aboved), where \(S_k = \{0\}\cup \{\xi: \xi^{k} =1\}\). The authors prove that \(\gamma(S_{k};1)\leq c_k \) for \(k=2,3,4\) with \(c_2 = 2- 2e^{-{1\over 2}}\), \(c_3 = {4\over 3}-e^{-{2\over 3}}\), \(c_4 = 0.829\dots\), and \(\gamma(S_k) \leq \gamma(U;\sqrt{e})\leq {34\over 35}\) for \(k\geq 5\). To this end they first show, drawing upon some of their earlier results on multiplicative functions [in Proc. Conf. on Topics in number theory (University Park, PA, 1997), Kluwer Acad. Publ., Dordrecht, 1-15 (1999; Zbl 0935.11033); Ann. Math. (2) 153, 407-470 (2001; Zbl 1036.11042)], that without loss of generality the additional restriction \(f(p)=1\) for \(p\leq \exp((\log X)^{1/4})\) can be assumed, and then they go through further analytic arguments to bound the sums and integrals which crop up in order to reach the result.
The paper also contains some results for lower bounds. For this the authors use the equivalence between problems on the distribution of multiplicative functions and problems on certain integral equations that they established in their former work referred to above. The approach through integral equations is remarked to be particularly useful for the treatment of lower bounds. One of the theorems states \(\gamma(U)\geq \gamma(S_{2k})\geq \gamma(S_2) = 2-2/\sqrt{e}\), and for odd \(k\), \(\gamma(S_k)\geq (1+\cos{\pi\over k})(1-e^{-1/(1+\cos{\pi\over k})})\). Furthermore the authors take into consideration the fact that when \(\chi\) is of order \(k\), \(\chi^j \) is non-principal for \(j=1,2,\ldots,k-1\), so that Burgess’ estimates are also applicable to \(\chi^j \). Defining \[ \gamma_{k}(A) := \limsup_{x\to\infty} \max_{f\in {\mathcal F}(S_{k})}{1\over \log X}\Biggl|\sum_{n\leq X}{f(n)\over n}\Biggr|\quad (A\geq 1) \] subject to the condition \(\sum_{n\leq x}f(n)^j = o(x)\) for \(X\leq x \leq X^A \) for each \(j=1,\ldots,k-1\), and writing \(\gamma_k = \lim_{A\to\infty}\gamma_{k}(A)\), one has just as described above \(|L(1,\chi)|\leq ({1\over 4 \text{ or} 3})(\gamma_{k}+o(1))\log q\) for \(\chi(\bmod q)\) of order \(k\). The other theorem on lower bounds states that for \(k\) large, \(\gamma_{k}\geq (e^{\gamma}+o_{k}(1)){\log\log k\over \log k}\) (a more precise bound is given in the proof), and this is expected to be the correct size of \(\gamma_{k}\). Some numerical data for small \(k\) is provided at the end of the paper. Moreover an argument is devised to obtain the upper bound for averages, \[ \biggl(\prod_{\chi(\bmod q) \text{ of order} k}|L(1,\chi)|\biggr)^{{1\over \phi(k)}} \leq \biggl({43\over 60 \text{ or } 45}e^{\gamma} + o_{k}(1)\biggr){\log\log k\over \log k}\log q, \] where \(60\) or \(45\) (resp.) is determined in the same way as 1/4 or 1/3 (resp.) is determined as described above.
All told, this is a paper rich with strong results. The intermediate results given as lemmas and propositions, and the remarks on the methods are quite interesting as well, and are bound to be effective in future research.

11M20 Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\)
11L40 Estimates on character sums
11K65 Arithmetic functions in probabilistic number theory
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