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Low lying zeros of families of $$L$$-functions. (English) Zbl 1012.11041
This comprehensive and highly interesting paper is devoted to testing the Random Matrix Model, due to N. M. Katz and P. Sarnak [Bull. Am. Math. Soc. 36, 1-26 (1999; Zbl 0921.11047)], for the distribution of zeros of various families of $$L$$-functions. This model is an analogue or generalization of the well-known Pair Correlation Conjecture of H. L. Montgomery for the zeros of Riemann’s zeta-function.
The authors study the first few zeros of $$L$$-functions in the upper half-plane assuming usually the generalized Riemann Hypothesis. For an individual function, not much can be said about such zeros, but certain distribution laws (Density Theorems) can be established, at least conditionally, for the totality of low lying zeros of sufficiently large families of $$L$$-functions. The families considered are “even” and “odd” Hecke $$L$$-functions $$L(s,f)$$ (for a holomorphic cusp form $$f$$ of weight $$k$$ and level $$N$$) and the related symmetric square $$L$$-functions. The limit is taken with respect to $$N$$, $$kN$$, $$K$$, $$N$$, or $$KN$$ tending to infinity; the parameter $$K$$ occurs when $$k$$ is averaged over $$k\leq K$$.
The density theorems are applied to lower estimates for the frequency of nonvanishing of $$L(1/2,f)$$, $$L'(1/2,f)$$, and $$L(1/2,\text{sym}^2(f))$$.
Another interesting application is given to the quasi-Riemann Hypothesis for a Hecke $$L$$-function $$L(s,f)$$ related to a cusp form $$f$$ for the full modular group. Namely, a certain conjecture on a classical exponential sum over primes implies that $$L(s,f)$$ has no real zero $$s > 10/11$$ if the weight of $$f$$ is sufficiently large and it is supposed that the zeros of $$L(s,f)$$ are either real or lie on the critical line. This application clearly indicates the relevance of classical analytic number theory in the theory of automorphic forms and related $$L$$-functions.

##### MSC:
 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11M41 Other Dirichlet series and zeta functions
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