Abstract. It has been conjectured that the real parts of the zeros of a linear combination of two or more L-functions are dense in the interval [1, σ * ], where σ * is the least upper bound of the real parts of such zeros. In this paper we show that this is not true in general. Moreover, we describe the optimal configuration of the zeros of linear combinations of orthogonal Euler products by showing that the real parts of such zeros are dense in subintervals of [1, σ * ] whenever σ * > 1.
It is well known that the Riemann zeta function, as well as several other Lfunctions, is universal in the strip 1/2 < σ < 1; this is certainly not true for σ > 1. Answering a question of Bombieri and Ghosh, we give a simple characterization of the analytic functions approximable by translates of L-functions in the half-plane of absolute convergence. Actually, this is a special case of a general rigidity theorem for translates of Dirichlet series in the halfplane of uniform convergence. Our results are closely related to Bohr's equivalence theorem.
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