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Zeros of Dirichlet $$L$$-functions. (English) Zbl 0771.11033
This interesting paper deals with the problem of the frequency with which $$L(s,\chi)\neq 0$$, where $${1\over 2}\leq s<1$$ and $$\chi$$ is a Dirichlet character. The main results are as follows. Let $$q$$ be a sufficiently large prime. Then, for a positive proportion of the characters $$\chi\pmod q$$, one has that (i) $$L({1\over 2},\chi)\neq 0$$ and (ii) there are no real zeros of $$L(s,\chi)$$ with $${1\over 2}+c/\log q\leq\sigma<1$$.
The new idea introduced in the paper is to count the desired characters directly, without the intermediary of moments of $$L$$-functions, which was the method used in previous work.
Reviewer: A.Perelli (Genova)

##### MSC:
 11M20 Real zeros of $$L(s, \chi)$$; results on $$L(1, \chi)$$
##### Keywords:
real zeros; Dirichlet $$L$$-functions
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##### References:
 [1] R. BALASUBRAMANIAN , A Note on Dirichlet’s L-Functions (Acta Arithmetica, Vol. 38, 1980 , pp. 273-283). Article | MR 82k:10049 | Zbl 0379.10026 · Zbl 0379.10026 · eudml:205738 [2] M. B. BARBAN and P. P. VEHOV , On an Extremal Problem (Trans. Moscow Math. Soc., Vol. 18, 1968 , pp. 91-99). MR 38 #4410 | Zbl 0195.33101 · Zbl 0195.33101 [3] H. DAVENPORT , Multiplicative Number Theory , Springer-Verlag, 2nd edition. · Zbl 0159.06303 [4] S. GRAHAM , An Asymptotic Estimate Related to Selberg’s Sieve (J. Number Theory, Vol. 10, 1978 , pp. 83-94). MR 80f:10056 | Zbl 0382.10031 · Zbl 0382.10031 · doi:10.1016/0022-314X(78)90010-0 [5] M. JUTILA , On the Mean Value of L(1/2, \chi ) for Real Characters (Analysis, Vol. 1, 1981 , pp. 149-161). MR 82m:10065 | Zbl 0485.10029 · Zbl 0485.10029 [6] V. KUMAR MURTY , Non-Vanishing of L-Functions and their Derivatives , in Automorphic Forms and Analytic Number Theory, R. MURTY Ed., pp. 89-113, Centre de Recherches Mathématiques, 1990 . Zbl 0745.11033 · Zbl 0745.11033 [7] M. RAM MURTY , Simple Zeroes of L-Functions , in Number Theory (Proceedings of the Banff Conference, R. MOLLIN Ed., de Gruyter, 1989 , pp. 427-439. Zbl 0712.14010 · Zbl 0712.14010 [8] C. L. SIEGEL , On the Zeros of the Dirichlet L-Functions (Annals of Math., Vol. 46, 1945 , pp. 409-422). MR 7,146j | Zbl 0063.07012 · Zbl 0063.07012 · doi:10.2307/1969160
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