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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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