This paper proves that the real projection of each simple zero of any partial sum of the Riemann zeta function n (s) := P n k=1 1 k s , n > 2, is an accumulation point of the set fRe s : n (s) = 0g.

In this paper, we study the distribution of zeros of the ordinary Dirichlet polynomials which are generated by an equivalence relation introduced by Harald Bohr. Through the use of completely multiplicative functions, we construct equivalent Dirichlet polynomials which have the same critical strip, where all their zeros are situated, and satisfy the same topological property consisting of possessing zeros arbitrarily near every vertical line contained in some substrips inside their critical strip. We also show that the real projections of the zeros of the partial sums of the alternating zeta function, for some particular cases, are dense in their critical intervals.

Abstract. In this paper we give a new characterization of the closure of the set of the real parts of the zeros of a particular class of Dirichlet polynomials which is associated to the set of dimensions of fractality of certain fractal strings. We show, for some representative cases of nonlattice Dirichlet polynomials, that the real parts of their zeros are dense in their associated critical intervals, confirming the conjecture and the numerical experiments made by M. Lapidus and M. van Frankenhuysen in several papers.

We prove a general version of the Lebesgue differentiation theorem where the averages are taken on a family of sets that may not shrink nicely to any point. These families of sets involves the unit ball and its dilated by negative integers of an expansive linear map. We also give a characterization of the Lebesgue measurable functions on R n in terms of approximate continuity associated to an expansive linear map.

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