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- W4289551048 abstract "In this article, we study the Euler's factorial series $F_p(t)=sum_{n=0}^infty n!t^n$ in $p$-adic domain under the Generalized Riemann Hypothesis. First, we show that if we consider primes in $kvarphi(m)/(k+1)$ residue classes in the reduced residue system modulo $m$, then under certain explicit extra conditions we must have $lambda_0+lambda_1F_p(alpha_1)+ldots+lambda_kF_p(alpha_k) neq 0$ 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 $kvarphi(m)/(k+1)$ 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 $lambda_0+lambda_1F_p(alpha_1)+ldots+lambda_kF_p(alpha_k) neq 0$. We also derive an explicit $p$-adic lower bound for the previous linear form." @default.
- W4289551048 created "2022-08-03" @default.
- W4289551048 creator A5047785056 @default.
- W4289551048 date "2022-07-30" @default.
- W4289551048 modified "2023-10-01" @default.
- W4289551048 title "Explicit results for Euler's factorial series in arithmetic progressions under GRH" @default.
- W4289551048 doi "https://doi.org/10.48550/arxiv.2208.00294" @default.
- W4289551048 hasPublicationYear "2022" @default.
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