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References [1] Bettin , S. 2010 The second moment of the Riemann zeta-function with unbounded shifts Int. J. Number Theory

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) s . Moreover, the function ζ ( s, α ) can be analytically continued to the whole complex plane, except for a simple pole at the point s = 1 with residue 1. For α = 1, the Hurwitz zeta-function becomes the Riemann zeta-function ζ ( s ): ζ ( s

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Abstract  

A. Beurling introduced harmonic functions attached to measurable functions satisfying suitable conditions and defined their spectral sets. The concept of spectral sets is closely related to approximations by trigonometric polynomials. In this paper we consider spectral sets of the harmonic functions attached to the Riemann zeta-function and its modification.

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Abstract  

We disprove some power sum conjectures of Tur�n that would have implied the density hypothesis of the Riemann zeta-function if true.

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Summary  

A. Beurling introduced the concept of spectral sets of unbounded functions to study the possibility of the approximation of those by trigonometric polynomials. We consider spectral sets of unbounded functions in a certain class which contains the square of the Riemann zeta-function as a typical example.

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] Montgomery , H. L. , Vaughan , R. C. 2007 Multiplicative Number Theory I. Classical Theory Cambridge University Press . [4] Titchmarsh , E. C. , The Riemann Zeta-Function , 2nd ed

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Abstract  

We shall investigate several properties of the integral

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\int_1^\infty {t^{ - \theta } \Delta _k \left( t \right) log^j t dt}$$ \end{document}
with a natural number k, a non-negative integer j and a complex variable θ, where Δk(x) is the error term in the divisor problem of Dirichlet and Piltz. The main purpose of this paper is to apply the “elementary methods” and the “elementary formulas” to derive convergence properties and explicit representations of this integral with respect to θ for k = 2.

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