A Hilbert-Space Method For Zeros Of Dirichlet Series

Since the pioneering works of J.P.G.L. Dirichlet and B. Riemann, various Dirichlet series and classes of zeta-functions have been introduced and intensively studied. And non-vanishing results for Dirichlet series and zeta-functions play an important role in number theory for the understanding of distributions of primes. From the time of the Riemann zcta-function (?)(s) to the age of automorphic L-functions, mathematicians worked out various methods to deal with non-vanishing problems, among which we are interested in the methods derived from the so-called symmetric Miintz formula.Since C.H. Miintz [24] first proved his formula for a certain class of functions, several researches have been done with this formula. For instance, E.C. Titchmarsh [33] gave another proof:L. Baez-Duarte [2] and J.-F. Burnol [7] explored the validity of the formula for wider classes of functions; C. Ryavec [26,27] and S. Albeverio and C. Cebulla [1] made use of the symmetric version of the formula to obtain explicit zero-free regions for (?)(s), in different directions.In this paper, we are interested in generalizing Ryavec's work in relating zeros of (?)(s) [26.27] and Dirichlet L-functions [28] with spectra of certain Gramian matrices, using a Hilbert-space method.In Section 1. we briefly survey the results obtained in this paper.In Section 2. we extend the approach of Ryavec and build a general theory to treat zeros of Dirichlet series in the critical strip S={s∈C| 0< Re(s)< 1}. In view of the fact that the zeros of a Dirichlet series are not necessarily distributed symmetrically in<S with respect to the critical line Re(s)=1/2, we treat zeros in the left-half of S and those in the right-half separately. The theory for the left part is derived from the generalized symmetric Muntz formula for Dirichlet series Eq. (2.1) in§2.1 while the theory for the right part came from the generalized symmetric co-Muntz formula for Dirichlet series Eq. (2.4) in§2.2. The inequalities in our general results, Theorem 2.1 and Theorem 2.2, provides a relation between the zeros of Dirichlet series in S and information about the spectra of certain Gramian matrices. In both§2.1 and§2.2, the positive-definiteness for these Gramian matrices, though vital, is a rather strong requirement to fulfil in practice. Therefore in§2.3, we offer a recipe to overcome this difficulty. This leads to Theorem 2.3, which also gives similar inequalities linking the zeros of Dirichlet series in S and information about the spectra of the matrices in§2.1 and§2.2.In Section 3, we apply the general theory in Section 2 to a subclass of Dirichlet se-ries with non-principal periodic coefficients and derive concrete zero-free regions in§3.2 and§3.3. Eq. (3.3) in Theorem 3.1 gives inexplicit zero-free regions, while Eq. (3.9) in Theorem 3.2 gives explicit ones. Theorem 3.2 simplifies and properly contains Ryavec's previous result [28]. Finally in§3.4, we give some remarks on the Hilbert-space method for zeros of Dirichlet series.