On the log-concavity of a Jacobi theta function

Coffey, Mark; Csordas, George

doi:10.1090/S0025-5718-2013-02681-6

On the log-concavity of a Jacobi theta function
HTML articles powered by AMS MathViewer

by Mark W. Coffey and George Csordas PDF

Math. Comp. 82 (2013), 2265-2272 Request permission

Abstract:

A new proof of the log-concavity of the Jacobi theta function, appearing in the Fourier representation of the Riemann $\Xi$ function, is presented. An open problem, involving the normalized moments of log-concave kernels, is investigated. In particular, several Turán-type inequalities are established.

References

Ralph Philip Boas Jr., Entire functions, Academic Press, Inc., New York, 1954. MR 0068627
Sergey Bobkov and Mokshay Madiman, Concentration of the information in data with log-concave distributions, Ann. Probab. 39 (2011), no. 4, 1528–1543. MR 2857249, DOI 10.1214/10-AOP592
Richard E. Barlow, Albert W. Marshall, and Frank Proschan, Properties of probability distributions with monotone hazard rate, Ann. Math. Statist. 34 (1963), 375–389. MR 171328, DOI 10.1214/aoms/1177704147
Mark W. Coffey, Asymptotic estimation of $\xi ^{(2n)}(1/2)$: on a conjecture of Farmer and Rhoades, Math. Comp. 78 (2009), no. 266, 1147–1154. MR 2476576, DOI 10.1090/S0025-5718-08-02167-4
Mark W. Coffey, Relations and positivity results for the derivatives of the Riemann $\xi$ function, J. Comput. Appl. Math. 166 (2004), no. 2, 525–534. MR 2041196, DOI 10.1016/j.cam.2003.09.003
George Csordas, Timothy S. Norfolk, and Richard S. Varga, The Riemann hypothesis and the Turán inequalities, Trans. Amer. Math. Soc. 296 (1986), no. 2, 521–541. MR 846596, DOI 10.1090/S0002-9947-1986-0846596-4
George Csordas and Richard S. Varga, Moment inequalities and the Riemann hypothesis, Constr. Approx. 4 (1988), no. 2, 175–198. MR 932653, DOI 10.1007/BF02075457
George Csordas and Richard S. Varga, Necessary and sufficient conditions and the Riemann hypothesis, Adv. in Appl. Math. 11 (1990), no. 3, 328–357. MR 1061423, DOI 10.1016/0196-8858(90)90013-O
E. K. Haviland, On the asymptotic behavior of the Riemann $\xi$-function, Amer. J. Math. 67 (1945), 411–416. MR 12627, DOI 10.2307/2371955
Samuel Karlin, Total positivity. Vol. I, Stanford University Press, Stanford, Calif., 1968. MR 0230102
B. Ja. Levin, Distribution of zeros of entire functions, American Mathematical Society, Providence, R.I., 1964. MR 0156975
Albert W. Marshall and Ingram Olkin, Inequalities: theory of majorization and its applications, Mathematics in Science and Engineering, vol. 143, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 552278
G. Pólya, Über die algebraisch-funktionentheoretischen Untersuchungen von J. L. W. V. Jensen, Klg. Danske Vid. Sel. Math.-Fys. Medd. 7 (1927), 3–33.
G. Pólya, Collected Papers, Vol. II, Location of Zeros (R. P. Boas, ed.), MIT Press, Cambridge, MA, 1974. MR 0505094 (58:21342)
G. Pólya and J. Schur, Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, J. Reine Angew. Math. 144 (1914), 89–113.
E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550