This remarkable result raised much attention among specialists, and Reich [18], [19], Gonek [4], Good [5], Bagchi [1], [2], and the first author [8]-[12], [14] improved and generalized the Voronin theorem to various other Dirichlet series including Dirichlet L , and Dedekind, Hurwitz, and Lerch zeta-functions. It is the purpose of the present paper to prove the universality theorem for zetafunctions attached to certain cusp forms. Let F (z) be a holomorphic cusp form of weight κ for the full modular group SL(2, Z) , and assume that F (z) is a normalized eigenform. Then F (z) has the Fourier series expansion

/pdf/the-universality-of-zeta-functions-attached-to-certain-cusp-l1zerai5l9.pdf

The universality of zeta-functions attached to certain cusp forms

We survey the results and the methods in the theory of universality for various zeta and $L$-functions, obtained in these forty years after the first discovery of the universality for the Riemann zeta-function by Voronin.

A survey on the theory of universality for zeta and $L$-functions

We apply an effective multidimensional Ω result of Voronin in order to obtain effective universality-type theorems for the Riemann zeta-function. We further use this approach to study approximation properties of linear combinations of derivatives of the zeta-function.

/pdf/effective-uniform-approximation-by-the-riemann-zeta-function-46ky35fq5p.pdf

Effective uniform approximation by the Riemann zeta-function

Let 0 0, once T is large enough. This was refined by Bagchi who showed that the measure of such t∈[0,T] is (c(e)+o(1))T, for all but at most countably many e>0. Using a completely different approach, we obtain the first effective version of Voronin's Theorem, by showing that in the rate of convergence one can save a small power of the logarithm of T. Our method is flexible, and can be generalized to other L-functions in the t-aspect, as well as to families of L-functions in the conductor aspect.

/pdf/an-effective-universality-theorem-for-the-riemann-zeta-4l3lyxv5rt.pdf

An effective universality theorem for the Riemann zeta function

Universality of the Riemann zeta-function in short intervals

/pdf/on-the-distribution-of-the-values-of-riemann-s-zeta-function-5ninz28mr9.pdf

On the distribution of the values of Riemann's Zeta-function

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On the distribution of the values of Riemann's Zeta-function