On some new congruences for generalized Bernoulli numbers (Q2919685)

Let \(n>3\) be an odd number, and let \(\chi_n\) be the trivial Dirichlet character modulo \(n\). Let \(\varphi\) denote the Euler function, \(E_n\) the Euler numbers, and \(B_n\) the Bernoulli numbers. This paper proves that NEWLINE\[NEWLINE\begin{aligned} &\sum_{0<i<n/4}\frac{\chi_n(i)}{i^2}\equiv 8\left(nB_{n\varphi(n)-2}\prod_{p\mid 4n}\left(1-p^{n\varphi(n)-3}\right)\right.\\ &\qquad\left.+\frac{1}{2}(-1)^{(n-1)/2}E_{n\varphi(n)-2}\prod_{p\mid n}\left(1-(-1)^{(p-1)/2}p^{n\varphi(n)-2}\right)\right) \pmod{n^2}. \end{aligned}NEWLINE\]