Result: A Series of New Congruences for Bernoulli Numbers and Eisenstein Series: A series of new congruences for Bernoulli numbers and Eisenstein series

Let \(E_k\), for \(k\) even and \(\geq 4\), denote the normalized Eisenstein series given through its \(q\)-expansion in the form \(E_k = 1 -(2k/B_k)\sum_{n\geq 1}\sigma_{k-1}(n)q^n\). Here \(B_k\) is the \(k\)th Bernoulli number and \(\sigma_u(n)\) denotes the sum of \(u\)th powers of the divisors of \(n\). Regard \(E_k\) as a formal power series in \(q\) and consider congruences between various \(E_k\), to be understood coefficientwise. From Kummer's congruences for \(E_k\) it follows, for example, that \(E_{k+12} \equiv E_k\cdot E_{12} \pmod {3^2\cdot 5\cdot 7\cdot 13}\). The author proves that this congruence in fact holds modulo \(2^7\cdot 3^4\cdot 5^3\cdot 7^2\cdot 13\). This result, already announced in the author's previous article [Arch.\ Math. 77, 5-21 (2001; Zbl 0994.11019)], is a typical example of a series of congruences for \(E_k\) proved in the present work. These congruences also produce new congruences for the numbers \(2k/B_k\). The author's method is to employ ``\(p\)-smoothed'' versions of \(E_k\), i.e., normalized \(p\)-adic Eisenstein series \(E_k^\star\), particularly the fact that the \(n\)th coefficient of \(E_k^\star\), as a function of \(k\), is an Iwasawa function (\(n\geq 0\)), provided \(p\) does not divide the \(B_k\); see \textit{J.-P.\ Serre}'s article in Modular Functions of one Variable III [Lect.\ Notes Math.\ 350, 191-268 (1973; Zbl 0277.12014)]. This leads to a procedure allowing one to verify congruences of the above type with a modest amount of numerical calculation.