Comptes Rendus
Number theory/Mathematical analysis
On a conjecture of Faulhuber and Steinerberger on the logarithmic derivative of ϑ4
Comptes Rendus. Mathématique, Volume 356 (2018) no. 5, pp. 457-462.

Faulhuber and Steinerberger conjectured that the logarithmic derivative of ϑ4 has the property that y2ϑ4(y)/ϑ4(y) is strictly decreasing and strictly convex. In this small note, we prove this conjecture.

Faulhuber et Steinerberger ont conjecturé que la dérivée logarithmique de ϑ4 possède la propriété selon laquelle y2ϑ4(y)/ϑ4(y) est strictement décroissant et strictement convexe. Dans cette courte note, nous démontrons cette conjecture.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.04.006

Anne-Maria Ernvall-Hytönen 1; Esa V. Vesalainen 1

1 Matematik och Statistik, Åbo Akademi University, Domkyrkotorget 1, 20500 Åbo, Finland
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Anne-Maria Ernvall-Hytönen; Esa V. Vesalainen. On a conjecture of Faulhuber and Steinerberger on the logarithmic derivative of ϑ4. Comptes Rendus. Mathématique, Volume 356 (2018) no. 5, pp. 457-462. doi : 10.1016/j.crma.2018.04.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.04.006/

[1] M. Coffey; G. Csordas On the log-concavity of a Jacobi theta function, Math. Comput., Volume 82 (2013), pp. 2265-2272

[2] A. Dixit; A. Roy; A. Zaharescu Convexity of quotients of theta functions, J. Math. Anal. Appl., Volume 386 (2012), pp. 319-331

[3] A.-M. Ernvall-Hytönen; E.V. Vesalainen On the secrecy gain of -modular lattices | arXiv

[4] M. Faulhuber Extremal Bounds of Gaussian Gabor Frames and Properties of Jacobi's Theta Functions, University of Vienna, 2016 (Doctoral dissertation)

[5] M. Faulhuber Properties of logarithmic derivatives of Jacobi's theta functions on a logarithmic scale | arXiv

[6] M. Faulhuber; S. Steinerberger Optimal Gabor frame bounds for separable lattices and estimates for Jacobi theta functions, J. Math. Anal. Appl., Volume 445 (2017), pp. 407-422

[7] H.L. Montgomery Minimal theta functions, Glasg. Math. J., Volume 30 (1988), pp. 75-85

[8] K. Schiefermayr Some new properties of Jacobi theta functions, J. Comput. Appl. Math., Volume 178 (2005), pp. 419-424

[9] A.Y. Solynin Harmonic measure of radial segments and symmetrization, Sb. Math., Volume 189 (1998), pp. 1701-1718

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This work was supported by the Academy of Finland project 303820, and E. V. V. was supported by the Magnus Ehrnrooth Foundation.

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