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Necessary and sufficient conditions and the Riemann hypothesis. (English) Zbl 0707.11062

From the introduction: The purpose of this paper is (1) to investigate several necessary and sufficient conditions for a real entire function to have only real zeros and to apply these conditions to the Riemann \(\xi\)-function and (2) to prove results concerning the distribution of zeros of entire functions related to the Riemann \(\xi\)-function. The interest in this area of research stems, in part, from the well-known fact that the Riemann hypothesis is equivalent to the statement that all the zeros of the Riemann \(\xi\)-function are real. Since \(\xi(x/2)\) is a real entire function, the Riemann hypothesis is valid if and only if the function \(\xi(x)\) belongs to the Laguerre-Pólya class.

MSC:

11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
30D20 Entire functions of one complex variable (general theory)
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References:

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