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Transactions of the American Mathematical Society

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The Riemann hypothesis and the Turán inequalities

Authors: George Csordas, Timothy S. Norfolk and Richard S. Varga
Journal: Trans. Amer. Math. Soc. 296 (1986), 521-541
MSC: Primary 11M26; Secondary 11N30, 30D10, 30D15
MathSciNet review: 846596
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Abstract: A solution is given to a fifty-eight year-old open problem of G Pólya, involving the Taylor coefficients of the Riemann $ \xi $-function.

References [Enhancements On Off] (What's this?)

  • [B] Ralph Philip Boas Jr., Entire functions, Academic Press Inc., New York, 1954. MR 0068627
  • [Br] R. P. Brent, A FORTRAN multiple-precision arithmetic package, Assoc. Comput. Mach. Trans. Math. Software 4 (1978), 57-70.
  • [G1] E. Grosswald, Generalization of a formula of Hayman and its application to the study of Riemann’s zeta function, Illinois J. Math. 10 (1966), 9–23. MR 0186797
  • [G2] Emil Grosswald, Correction and completion of the paper “Generalization of a formula of Hayman”, Illinois J. Math. 13 (1969), 276–280. MR 0240290
  • [Hav] E. K. Haviland, On the asymptotic behavior of the Riemann 𝜉-function, Amer. J. Math. 67 (1945), 411–416. MR 0012627,
  • [Hay] W. K. Hayman, A generalisation of Stirling’s formula, J. Reine Angew. Math. 196 (1956), 67–95. MR 0080749,
  • [K] Samuel Karlin, Total positivity. Vol. I, Stanford University Press, Stanford, Calif, 1968. MR 0230102
  • [L] Yudell L. Luke, The special functions and their approximations. Vol. II, Mathematics in Science and Engineering, Vol. 53, Academic Press, New York-London, 1969. MR 0249668
  • [P] G. Pólya, Über die algebraisch-funktionentheoretischen Untersuchungen von J. L. W. V. Jensen, Kgl. Danske Vid. Sel. Math.-Fys. Medd. 7 (1927), 3-33.
  • [PS] G. Pólya and J. Schur, Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, J. Reine Angew. Math. 144 (1914), 89-113.
  • [PSz] G. Pólya and Szegö, Problems and theorems in analysis, Vol. 1, Springer-Verlag, New York, 1972.
  • [SB] J. Stoer and R. Bulirsch, Introduction to numerical analysis, Springer-Verlag, New York-Heidelberg, 1980. Translated from the German by R. Bartels, W. Gautschi and C. Witzgall. MR 557543
  • [T] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford, at the Clarendon Press, 1951. MR 0046485
  • [W] A. Wintner, A note on the Riemann $ \xi $-function, J. London Math. Soc. 10 (1935), 82-83.

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Keywords: The Riemann Hypothesis, the Riemann $ \xi $-function, Turán inequalities, concavity, moments
Article copyright: © Copyright 1986 American Mathematical Society

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