Comptes Rendus
no. 5
Number theory/Mathematical analysis
On a conjecture of Faulhuber and Steinerberger on the logarithmic derivative of ϑ4
[De la conjecture de Faulhuber et Steinerberger sur la dérivée logarithmique de ϑ4]

Présenté par : the Editorial Board

Anne-Maria Ernvall-Hytönen1  ; Esa V. Vesalainen1
1 Matematik och Statistik, Åbo Akademi University, Domkyrkotorget 1, 20500 Åbo, Finland
Comptes Rendus. Mathématique, Volume 356 (2018) no. 5, pp. 457-462.

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.

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.

Reçu le :
Accepté le :
Publié le :
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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     title = {On a conjecture of {Faulhuber} and {Steinerberger} on the logarithmic derivative of \protect\emph{\ensuremath{\vartheta}}\protect\textsubscript{4}},
     journal = {Comptes Rendus. Math\'ematique},
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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/

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[3] A.-M. Ernvall-Hytönen; E.V. Vesalainen On the secrecy gain of ℓ-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

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Cité par Sources :

☆ 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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