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The distribution of values of $$L(1,\chi_d)$$. (English) Zbl 1044.11080
Geom. Funct. Anal. 13, No. 5, 992-1028 (2003); errata 14, No. 1, 245-246 (2004).
Let $$d$$ denote a fundamental discriminant, and $$\chi_d$$ the associated primitive real character to the modulus $$| d|$$. In this interesting paper the authors investigate the distribution of values of $$L(1, \chi_d)$$ as $$d$$ varies over all fundamental discriminants with $$| d| \leq x$$.
They compare the distribution of values of $$L(1, \chi_d)$$ with the distribution of “random Euler products” $$\prod_p(1-X(p)/p)^{-1}$$ where the $$X(p)$$’s are independent random variables taking values $$0$$ or $$\pm1$$ with suitable probabilities. Then they give asymptotics for the probability that $$L(1, \chi_d)$$ exceeds $$e^\gamma\tau$$, and the probability that $$L(1, \chi_d)\leq \frac{\pi^2}{6}\frac1{e^\gamma\tau}$$ uniformly in a wide range of $$\tau$$. The method allows to prove (really the authors prove more) a recent conjecture of H. L. Montgomery and R. C. Vaughan [Number theory in progress, Volume 2: Elementary and analytic number theory. Berlin: de Gruyter, 1039–1052 (1999; Zbl 0942.11040)], that {the proportion of fundamental discriminants $$| d| \leq x$$ with $$L(1, \chi_d)\geq e^\gamma\log\log | d|$$ is $$>\exp(-C\log x/\log\log x)$$ and $$<\exp(-c\log x/\log\log x)$$ for appropriate constants $$0<c<C<\infty$$; similar estimates apply to the proportion of fundamental discriminants $$| d| \leq x$$ with $$L(1, \chi_d)\leq \zeta(2)/(e^\gamma\log\log | d| )$$.}
The results are presented in 6 theorems and one proposition.
In the errata the authors correct some notational errors discovered in the paper under review.

##### MSC:
 11M20 Real zeros of $$L(s, \chi)$$; results on $$L(1, \chi)$$
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