On Incidences of ϕ and σ in the Function Field Setting
Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 403-415.

Erdős first conjectured that infinitely often we have ϕ(n)=σ(m), where ϕ is the Euler totient function and σ is the sum of divisors function. This was proven true by Ford, Luca and Pomerance in 2010. We ask the analogous question of whether infinitely often we have ϕ(F)=σ(G) where F and G are polynomials over some finite field 𝔽 q . We find that when q2 or 3, then this can only trivially happen when F=G=1. Moreover, we give a complete characterisation of the solutions in the case q=2 or 3. In particular, we show that ϕ(F)=σ(G) infinitely often when q=2 or 3.

Erdős a conjecturé qu’il existe une infinité de nombres n et m tels que ϕ(n)=σ(m), où ϕ est l’indicatrice d’Euler et σ est la fonction somme de diviseurs. Cette conjecture a été prouvée en 2010 par Ford, Luca et Pomerance. De façon analogue, on se demande s’il existe une infinité de polynômes F et G sur un corps fini 𝔽 q tels que ϕ(F)=σ(G). On trouve que si q2 ou 3, c’est vrai seulement dans le cas trivial F=G=1. De plus, on donne une caractérisation des solutions dans les cas q=2 et 3. En particulier, on montre que si q=2 ou 3, on a ϕ(F)=σ(G) pour une infinité de polynômes.

Received : 2018-10-08
Accepted : 2018-11-16
Published online : 2019-10-29
DOI : https://doi.org/10.5802/jtnb.1088
Classification:  11N64
Keywords: Function Fields, Euler Totient Function, Primitive Divisors
@article{JTNB_2019__31_2_403_0,
     author = {Patrick Meisner},
     title = {On Incidences of $\varphi $ and $\sigma $ in the Function Field Setting},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {2},
     year = {2019},
     pages = {403-415},
     doi = {10.5802/jtnb.1088},
     language = {en},
     url={jtnb.centre-mersenne.org/item/JTNB_2019__31_2_403_0/}
}
Meisner, Patrick. On Incidences of $\varphi $ and $\sigma $ in the Function Field Setting. Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 403-415. doi : 10.5802/jtnb.1088. https://jtnb.centre-mersenne.org/item/JTNB_2019__31_2_403_0/

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