论文标题
$ \ ell $ -genocchi数字和Ramanujan风格的普遍存在的Prime除数
Prime divisors of $\ell$-Genocchi numbers and the ubiquity of Ramanujan-style congruences of level $\ell$
论文作者
论文摘要
令$ \ ell $为任何固定质数。我们通过$ g_n：= \ ell（1- \ ell^n）b_n $定义$ \ ell $ -genocchi号码，带有$ b_n $ the $ n $ -th bernoulli编号。他们是整数。我们介绍并研究了库默（Kummer）规律素质概念的一种变体。我们说，如果$ \ ell $ -genocchi不规则，则奇怪的prime $ p $如果$ \ ell $ -genocchi数字$ g_2，g_4，\ ldots，g_ {p-3} $和$ \ ell $ regular-regular。在研究Artin原始根源猜想中使用的技术的帮助下，我们对规定的算术进展中的$ \ ell $ genocchi不规则数量进行渐近估计，以防$ \ ell $奇怪。 CASE $ \ ell = 2 $已经由Hu，Kim，Moree和Sha（2019）处理。 使用类似的方法，我们研究$（1- \ ell^n）b_ {2n}/2n $和$（1+ \ ell^n）b_ {2n}/2n $的主要因素。这使我们能够估算存在Eisenstein系列的Fourier系数与Prime Level Level Level Level $ \ Ell $之间的Primes $ P \ LEQ X $的数量。
Let $\ell$ be any fixed prime number. We define the $\ell$-Genocchi numbers by $G_n:=\ell(1-\ell^n)B_n$, with $B_n$ the $n$-th Bernoulli number. They are integers. We introduce and study a variant of Kummer's notion of regularity of primes. We say that an odd prime $p$ is $\ell$-Genocchi irregular if it divides at least one of the $\ell$-Genocchi numbers $G_2,G_4,\ldots, G_{p-3}$, and $\ell$-regular otherwise. With the help of techniques used in the study of Artin's primitive root conjecture, we give asymptotic estimates for the number of $\ell$-Genocchi irregular primes in a prescribed arithmetic progression in case $\ell$ is odd. The case $\ell=2$ was already dealt with by Hu, Kim, Moree and Sha (2019). Using similar methods we study the prime factors of $(1-\ell^n)B_{2n}/2n$ and $(1+\ell^n)B_{2n}/2n$. This allows us to estimate the number of primes $p\leq x$ for which there exist modulo $p$ Ramanujan-style congruences between the Fourier coefficients of an Eisenstein series and some cusp form of prime level $\ell$.
