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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.

##### MSC:
 11M41 Other Dirichlet series and zeta functions 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$
##### Citations:
Zbl 0609.10032; Zbl 0736.11047
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