×

Hybrid bounds for Dirichlet’s \(L\)-function. (English) Zbl 0970.11033

Let \(\chi\) be a non-principal Dirichlet character modulo a prime \(r\). The authors prove a variety of so-called “hybrid bounds” (i.e. bounds with non-trivial exponents of \(r\) and \(t\)) for \(L({1\over 2}+it)\). One of the consequences of their main theorem, whose formulation is too complicated to be reproduced here, is that \[ L({\textstyle{1\over 2}} + it) \ll_\varepsilon r^{31\over 190}t^{89\over 570}\log^At (1) \] for \(0 < \alpha = {\log r\over\log t} \leq {2\over 753} - \varepsilon\), where \(A>0\) is an absolute constant. One of the merits of (1) is that the exponent of \(t\) is the same as the best published exponent of Huxley for \(\zeta({1\over 2}+it)\). The main theorem of the authors is a consequence of a deep result on exponential sums of the form \[ \sum_{M\leq m\leq M_2\leq 2M}\chi(m)\exp(2\pi if(m)), \] where \(f(x)\) is a real-valued function whose derivatives satisfy certain conditions restricting their size. The new bound is obtained by a variation of the well-known Bombieri-Iwaniec method. The novelty of the approach used here is to write the above sum in term of \(r\) sums (one for each residue class mod\( r\)) and then to bound the average of the absolute values of the \(r\) different sums. The results may be compared to classical hybrid bounds for \(L(\frac 12+ it)\) of D. A. Burgess [Proc. Lond. Math. Soc., III. Ser. 13, 524-536 (1963; Zbl 0123.04404) and ibid. 52, 215-235 (1986; Zbl 0586.10020)] and D. R. Heath-Brown [Invent. Math. 47, 149-170 (1978; Zbl 0383.10024) and Recent progress in analytic number theory, Symp. Durham 1979, Vol. 1, 121-126 (1981; Zbl 0457.10021)].

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11L40 Estimates on character sums
PDFBibTeX XMLCite
Full Text: DOI