论文标题
欧拉（Euler
Explicit results for Euler's factorial series in arithmetic progressions under GRH
论文作者
论文摘要
在本文中，我们研究了Euler的阶乘系列$ f_p（t）= \ sum_ {n = 0}^\ infty n！首先，我们表明，如果我们考虑$kφ（m）/（k+1）$降低的残留系统模型$ m $中的残留类别，那么在某些明确的额外条件下，我们必须具有$λ_0+λ_1f_p（α__1）+\ \ \ \ \ ldots+λ_kf_p（α_k）\ neq 0 $ yeq 0 $ prime。我们还证明了以前的线性表单明确的$ p $ - adic下限。其次，我们考虑了从$kφ（m）/（k+1）$残留类中进行算术进展中的素数的情况。然后是一个无限的间隔集合，每个间隔都至少包含一个素数，该间隔是在那些算术进程中，并且我们的$λ_0+λ_1f_p（α_1）+\ ldots+λ_kf_p（α__k）\ neq 0 $。我们还为先前的线性表单得出了明确的$ p $ -adic下限。
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 $kφ(m)/(k+1)$ residue classes in the reduced residue system modulo $m$, then under certain explicit extra conditions we must have $λ_0+λ_1F_p(α_1)+\ldots+λ_kF_p(α_k) \neq 0$ 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 $kφ(m)/(k+1)$ 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 $λ_0+λ_1F_p(α_1)+\ldots+λ_kF_p(α_k) \neq 0$. We also derive an explicit $p$-adic lower bound for the previous linear form.
