\(p\)-adic \(L\)-functions and higher dimensional magic cubes (Q1893468)

An \(N\)-dimensional magic cube is a natural generalization of a magic square. The authors consider the case \(N= p\), a prime, and construct such cubes by using \(p\)-adic \(L\)-functions \(L_p (s, \chi)\). A crucial problem is to find coefficients prime to \(p\) in the Iwasawa power series attached to these \(L\)-functions. The authors' solution is based on the theory of uniform distribution mod 1; their argument is similar to one applied by the second author and \textit{B. Ferrero} in the original proof of the vanishing of the Iwasawa \(\mu\)-invariant [Ann. Math., II. Ser. 109, 377-395 (1979; Zbl 0443.12001)]. In particular, they take \(\chi\) a \(k\)th power of the cyclotomic character \(\omega\), where \(k\) is even, \(0\leq k\leq p-2\), and the Bernoulli number \(B_k\) has numerator prime to \(p\). The \(p\)-dimensional magic cube thus obtained has edges of length \(p^{ap^m}\), where \(m\) is sufficiently large and \(a\) is any integer \(\geq 1\). There is also a result of independent interest about coefficients prime to \(p\) in the power series expansions of certain rational functions. A predecessor of this article is the first author's paper in [Math. Intell. 14, No. 3, 14-23 (1992; Zbl 0770.05021)] dealing with the case \(p=3\).