For every prime p ≥ 5, some p-adic zeta value ζ_p(i) with odd i ≤ p + p/log p + 5 is irrational.
Beukers,Irrationality of somep-adicL-values, Acta Math
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On the irrationality of certain $p$-adic zeta values
For every prime p ≥ 5, some p-adic zeta value ζ_p(i) with odd i ≤ p + p/log p + 5 is irrational.