Speiser’s Theorem on the Road
DOI:
https://doi.org/10.48048/wjst.2019.6944Keywords:
Riemann zeta-function, elliptic curve, Riemann hypothesis, Speiser’s Theorem, regular graphs, Gauss-Lucas theoremAbstract
In this note we discuss the Gauss-Lucas theorem (for the zeros of the derivative of a polynomial) and Speiser’s equivalent for the Riemann hypothesis (about the location of zeros of the Riemann zeta-function). We indicate similarities between these results and present there analogues in the context of elliptic curves, regular graphs, and finite Euler products.
Downloads
Metrics
References
EC Titchmarsh, The theory of the Riemann zeta-function, Oxford University Press 1986, 2nd ed., revised
by D.R. Heath-Brown.
A Speiser, Geometrisches zur Riemannschen Zetafunktion, Math. Ann. 1934; 110, 514-21.
A Speiser, Über Riemannsche Flächen, Comment. Math. Helv. 1930; 2, 284-93.
AA Utzinger, Die reellen Züge der Riemannschen Zetafunktion, Dissertation, Zürich, 1934, 31 pp.
J Arias de Reyna, X-Ray of Riemann zeta-function, 2003, Available as arXiv:math/0309433v1.
R Spira, Zero-free regions of (k)(s), J. Lond. Math. Soc. 1965; 40, 677-82.
N Levinson, H.L. Montgomery, Zeros of the derivatives of the Riemann zeta-function, Acta Math. 1974;
, 49-65.
R Šleževičienė, Speiser’s correspondence between the zeros of a function and its derivative in Selberg’s
class of Dirichlet series, Fiz. Mat. Fak. Moksl. Semin. Darb. 2003; 6, 142-53.
R Garunkštis, R Šimėnas, On the Speiser equivalent for the Riemann hypothesis, Eur. J. Math. 2015; 1,
-50.
H Hasse, Beweis des Analogons der Riemannschen Vermutung für die Artinschen und F.K.
Schmidtschen Kongruenzzetafunktionen in gewissen elliptischen Fällen, Nachr. Ges. Wiss. Göttingen,
Math.-Phys. K. 1933, 253-62.
LC Washington, Elliptic Curves, Chapman & Hall / CRCPress, 2003.
A Lubotzky, R Phillips, P Sarnak, Ramanujan graphs, Combinatorica 1988; 8, 261-77.
S Ramanujan, On certain arithmetical functions, Trans. Camb. Phil. Sue. 1916; 22, 159-84.
GA Margulis, Graphs without short cycles, Combinatorica 1982; 2, 71-8.
J Arias de Reyna, Finite fields and Ramanujan graphs, J. Comb. Theory 1997; 70, 259-64.
A Nilli, On the second eigenvalue of a graph, Discrete Math. 1991; 91, 207-10.
P Sarnak, What is an expander?, Notices A.M.S. 2004; 51, 762-3.
HM Stark, A.A. Terras, Zeta functions of finite graphs and coverings, Adv. Math. 1996; 121, 124-65.
Y Ihara, On discrete subgroups of the two by two projective linear group over p-adic fields, J. Math.
Soc. Japan. 1966; 18, 219-35.
T Sunada, L-functions in geometry and some applications, in: Curvature and Topology of Riemannian
Manifolds, Lecture Notes in Mathematics, Springer 1986; 1201, 266-84.
A Terras, HM Stark, Zeta Functions of Graphs. A Stroll through the Garden, Cambridge University
Press, 2011.
CF Gauss, Note appended at end of memoir (186), Werke, Band 3, Göttingen, 1866.
F Lucas, Sur une application de la Mécanique rationnelle à la théorie des équations, Comptes Rendus
de l’Académie des Sciences, 1880; 89, 224-6.
VV Prasolov, Polynomials, Springer, 2010.
EB Van Vleck, On the location of roots of polynomials and entire functions, Bull. A.M.S. 1929; 35,
-83.
B Conrey, Zeros of derivatives of Riemann’s xi-function on the critical line, J. Number Theory 1983;
, 49-74.
T Craven, G Csordas, The Gauss-Lucas theorem and Jensen polynomials, Trans. A.M.S. 1983; 278,
-29.
E Duenez, DW Farmer, S Froehlich, CP Hughes, F Mezzadri, T Phan, Roots of the derivative of the
Riemann-zeta function and of characteristic polynomials, Nonlinearity 2010; 23, 2599-621.
E Landau, Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen
Zetafunktion, Proc. Fifth Internat. Math. Congr. 1913; 1, 93-108.
H Bohr, E Landau, JE Littlewood, Sur la fonction (s) dans le voisinage de la droite = 1
, Bull. de
l’Acad. royale de Belgique 1913, 3-35.
N Levinson, Almost all roots of (s) = a are arbitrarily close to = 1/2, Proc. Nat. Acad. Sci. U.S.A.
; 72, 1322-4
A Selberg, Old and new conjectures and results about a class of Dirichlet series, Proceedings of the
Amalfi Conference on Analytic Number Theory, Maiori 1989, E Bombieri et al. (eds.), Università di
Salerno 1992, 367-85.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2019 Walailak Journal of Science and Technology (WJST)
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.