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Large deviations of Montgomery type and its application to the theory of zeta-functions. (English) Zbl 0818.11032
Let \((\Omega, P)\) be a probability space, and let \(\theta_ 1, \theta_ 2,\dots\) be independent random variables on \((\Omega, P)\) with identical distribution, which is a uniform distribution on \([0, 1]\). We put \(X= \sum_{n=1}^ \infty r_ n \cos(2\pi \theta_ n)\), where \(\{r_ n \}^ \infty_{n=1}\) is a sequence of non-negative real numbers with infinitely many non-zero terms, satisfying \(\sum_{n=1}^ \infty r^ 2_ n <\infty\).
Montgomery has shown \[ P \biggl( X\geq C_ 1 \sum_{n=1}^ N r_ n \biggr)\geq C_ 2\exp \Biggl( -C_ 3 \biggl( \sum_{n=1}^ N r_ n \biggr)^ 2 \biggl( \sum_{n=N+1}^ \infty r^ 2_ n \biggr)^{-1} \Biggr) \tag \(*\) \] for any positive integer \(N\), if \(\{r_ n\}\) is a monotonically decreasing sequence. D. Joyner [Distribution theorems for \(L\)-functions (1986; Zbl 0609.10032)] used this result of Montgomery to obtain a lower bound on the asymptotic probability measures for the value-distribution of the Riemann zeta-function, and K. Matsumoto [Acta Arith. 60, No. 2, 125-147 (1991; Zbl 0736.11047)] generalized Joyner’s result to the case of Dedekind zeta-functions. However, the condition that \(\{r_ n\}\) is monotonically decreasing is too restrictive when we consider more general zeta-functions.
In this paper, the authors find a necessary and sufficient condition for \((*)\) to hold. As an application, they prove a lower bound of the asymptotic probability measures for zeta-functions attached to certain cusp forms.

11M41 Other Dirichlet series and zeta functions
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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