Volume 25, Issue 1

We establish the universality theorem on some region for automorphic L-functions attached to Maass forms for SL(2, ℤ). As is well known, the Ramanujan conjecture is still open for these forms. The universality theorem for the derivatives of those L-functions and some applications are also established.

The paper deals with an operator extension problem for contractive block Hankel operators which arose in the context of the operator version of the classical Nehari interpolation problem. V.M. Adamjan, D.Z. Arov, and M.G. Krein [5] obtained that the solution set of this operator extension problem is an operator ball. Hereby, they constructed the parameters of this operator ball via a regularization procedure using the corresponding expressions of the first studied nondegenerate case. The main aim of this paper is to derive more explicit formulas for the parameters of this operator ball. Hereby we use Moore-Penrose inverses of bounded linear operators in Hilbert space.

The three basic boundary value problems in complex analysis are of Schwarz, of Dirichlet and of Neumann type. When higher order equations are investigated all kind of combinations of these boundary conditions are proper to determine solutions. However, not all of these conditions are leading to well-posed problems. Some are overdetermined so that solvability conditions have to be found. Some of these boundary value problems are treated here for the inhomogeneous polyanalytic equation.

It is known that for any polynomial p of degree less or equal to n, In this note we establish cases of equality and also improve the above inequality.

In our previous paper [2], we introduced the concept of statistical limit of a measurable function and that of strong Cesàro summability of a locally integrable function. As an application, we proved there that the Fourier integral of a function ƒ ∈ L 1 (ℝ)∩C(ℝ) is strongly Cesàro summable and has statistical limit at every point of ℝ. The purpose of this paper is twofold: (i) We prove that if ƒ ∈ L 1 (ℝ) is locally bounded on ℝ and continuous on an open interval J , then the strong Cesàro summability and the existence of the statistical limit of the Fourier integral of ƒ is uniform on J. (ii) We complete and simplify the proof of [2, Statement (γ) of Theorem 3], which says that if ƒ ∈ L 1 (ℝ)∩C 0 (ℝ), then the conclusion in (i) is uniform on the whole ℝ.

Balog and Harman proved that for any λ in the interval 1/2 ≤ λ < 1 and any real θ there are infinitely many primes p satisfying (with an asymptotic result). In the present paper we prove that for 59/85 = 0-694... < λ < 1 the above expo­nent -(1-λ)/2+ε may be replaced by - min{max{(35-22λ)/129, 1/7}, 5/18-λ/6}+ε. This result in particular contains the Piatetski-Shapiro prime number theorem in the ver­sion given by Liu and Rivat: We have |{n ≤ N : [n c ] prime}) ~ N/(c log N) as N → 8734 if 1 < c < 15/13. For the proof of our result we use exponential sum techniques.