On the irrationality of certain 2-adic zeta values (Q6658795)

Let \(\zeta_2\) denote the Kubota-Leopoldt \(2\)-adic zeta function. It was proved by \textit{F. Calegari} [Int. Math. Res. Not. 2005, No. 20, 1235--1249 (2005; Zbl 1070.11027)] that \(\zeta_2(3)\notin\mathbb Q\), and Calegari-Dimitrov-Tang announced in 2020 that \(\zeta_2(5)\notin\mathbb Q\). It is proved in this paper for every integer \(s\ge 0\), there exists an odd integer \(j\) in the interval \([s + 3, 3s + 5]\) such that \(\zeta_2(j)\) is irrational. In particular, at least one of \(\zeta_2(7), \zeta_2(9), \zeta_2(11), \zeta_2(13)\) is irrational.\N\NThe author's approach is inspired by a recent work of \textit{L. Lai} and \textit{J. Sprang} [``Many $p$-adic odd zeta values are irrational'', Preprint, \url{arXiv:2306.10393}] who proved that, for every fixed prime \(p\), infinitely many values \(\zeta_p(j)\) are irrational when \(j\) is odd. The author constructs an explicit sequence of rational functions such that the Volkenborn integrals of high order derivatives of these rational functions produce a sequence of good \(\mathbb Z\)-linear combinations of 1 and of \(2\)-adic Hurwitz zeta values \(\zeta_2(j,1/4)=2\zeta_2(j)\) (when \(j\ge 3\) is odd). By ``good'', it is meant that the sequence of archimedean absolute values of the coefficients do not grow too fast while the sequence of linear combinations tend \(2\)-adically fast to 0, so that the irrationality of \(\zeta_2(j)\) is obtained for at least one odd \(j\) in the above mentioned interval (by means of an elementary irrationality criterion). The most difficult step is the proof that certain of these integrals are nonzero.\N\NThese results are similar to those obtained by Zudilin in 2001 for the values of the Riemann zeta function: for every odd integer \(s\ge 1\), there exists an odd integer \(j\) in the interval \([s + 2, 8s -1]\) such that \(\zeta(j)\) is irrational and moreover at least one of \(\zeta(5), \zeta(7), \zeta(9), \zeta(11)\) [\textit{W. Zudilin}, Izv. Math. 66, No. 3, 489--542 (2002; Zbl 1114.11305); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 66, No. 3, 49--102 (2002); Russ. Math. Surv. 56, No. 4, 774--776 (2001; Zbl 1047.11072); translation from Usp. Mat. Nauk 56, No. 4, 149--150 (2001)] respectively). The main difference is that Apéry proved in 1978 that \(\zeta(3)\notin \mathbb Q\) but the arithmetic nature of \(\zeta(5)\) is still unknown.