On zeros of some composite functions

. We obtain an estimate of the number of zeros for the function F ( ζ ( s + imh )) , where ζ ( s ) is the Riemann zeta-function, and F : H ( D ) → H ( D ) is a continuous function, D = { s ∈ C : 12 < σ < 1 } .

The distribution of zeros of zeta and L-functions is the central problem of analytic number theory, and the results in the field allow to solve many other important problems. For example, the location of non-trivial zeros of the Riemann zeta-function ζ(s), s = σ + it, has a direct relation to the distribution of prime numbers. The best result in this direction asserts that ζ(s) = 0 in the region σ > 1 − c (log |t|) 2 3 (log log |t|) 1 3 , |t| t 0 > 0, where c > 0 is an absolute constant. We remind that the Riemann hypothesis says that all non-trivial zeros of ζ(s) lie on the critical line σ = 1 2 , thus by this hypothesis, ζ(s) = 0 in the half-plane σ > 1 2 . There are the zeta-functions for which the Riemann hypothesis is not true. For example, this holds for the Hurwitz-function ζ(s, α), 0 < α 1, defined, for σ > 1, by and by analytic continuation elsewhere. If α is a trancendental number, than [2] ζ(s, α) has zeros in the strip 1 2 < σ < 1. Also, the derivative ζ ′ (s) has zeros in the strip 0 < σ < 1.
For the investigation of zero-distribution of zeta-functions, universality theorems can be applied. The first universality theorem for the Riemann zeta-function has been proved by S.M. Voronin in [5]. The last version of this theorem is the following: Theorem 1. Suppose that K is a compact subset of the strip D = {s ∈ C: 1 2 < σ < 1} with connected complement, and f (s) is a continuous non-vanishing function on K which is analytic in the interior of K. Then, for every ε > 0, Here meas{A} denotes the Lebesgue measure of a measurable set A ⊂ R. The proof of Theorem 1 is given, for example, in [1].
Also, a discrete version of Theorem 1 is known. Let h > 0 be a fixed number.

Theorem 2.
Suppose that K and f (s) satisfy the hypotheses of Theorem 1. Then, for every ε > 0, In [3], certain discrete universality theorems were obtained for the composite function F (ζ(s)).
We recall some of them. Denote by H(D) the space of analytic functions on D equipped with the topology of uniform convergence on compacte, and set The next theorem is a simplification of Theorem 3.  We note that, differently from Theorem 2, the approximated function in Theorems 3-5 is not necessarily non-vanishing.
The aim of his note is to prove the following statement.
Theorem 6. Suppose that the number exp{ 2πk h } is irrational for all k ∈ Z \ {0}, and that the function F is as in one of Theorems 3-5. Then, for arbitrary σ 1 and σ 2 , 1 2 < σ 1 < σ 2 < 1, there exists a constant c = c(σ 1 , σ 2 ) > 0 such that the function F (ζ(s + imh)) has a zero in the disc for more than cN numbers m, 0 m N .
First we will remind the Rouché theorem. Proof of the lemma can be found, for example, in [4].