On incidences of ϕ and σ in the function field setting

Patrick Meisner*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 Fq. We find that when q ≠ 2 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.

Original languageEnglish
Pages (from-to)403-415
Number of pages13
JournalJournal de Theorie des Nombres de Bordeaux
Issue number2
StatePublished - 2019


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