2022/07/30 by Neea Palojärvi, Palojärvi, Neea
Arts and Humanities · Mathematics · #11J61 #33E50 #41A21 #Analytic Number Theory Research #FOS: Mathematics #French Literature and Criticism #History and Theory of Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2208.00294
openalex publication_date 2022/07/30 · openalex created_date 2022/08/03 · openalex updated_date 2026/07/28
In this article, we study the Euler's factorial series Fp(t)=∑n=0^∞ n!tn in p-adic domain under the Generalized Riemann Hypothesis. First, we show that if we consider primes in kφ(m)/(k+1) residue classes in the reduced residue system modulo m, then under certain explicit extra conditions we must have λ0+λ1Fp(α1)+…+λkFp(αk) ≠ 0 for at least one such prime. We also prove an explicit p-adic lower bound for the previous linear form. Secondly, we consider the case where we take primes in arithmetic progressions from more than kφ(m)/(k+1) residue classes. Then there is an infinite collection of intervals each containing at least one prime which is in those arithmetic progressions and for which we have λ0+λ1Fp(α1)+…+λkFp(αk) ≠ 0. We also derive an explicit p-adic lower bound for the previous linear form.