On quotients of Riemann zeta values at odd and even integer arguments. (English) Zbl 1290.11118

The author gives (see Theorem 1.3, for the precise statement) an explicit formula for the ratio \[ \zeta(n+1)/\zeta(n) \] where \(\zeta\) is the Riemann zeta-function and \(n\) is an even natural number, in terms of a polynomial \(p_n\) (see Theorem 1.2 for its definition and some properties), linked with some other other polynomial \(q_n\) (see again Theorem 1.2), that the author (after Theorem 1.5, proving that \(q_n\) is an Eisenstein polynomial, hence irreducible, when \(n-1\) is an odd prime) conjectures to be irreducible for all \(n\geq 4\) (see Conjecture 1.6).
The proof of Theorem 1.3 (which, regards more general linear combinations of zeta values ratios) is rather straightforward, from the functional equation for \(\zeta\), together with Euler’s formulae, containing Bernoulli numbers, linked to the Stirling numbers of the first kind and of the second kind.
Also, the author provides a table for three instances of Theorem 1.3, with the corresponding values of \(p_n\) (see Table 1.4) and a rich wealth of congruence and combinatorial properties of the quantities involved in the proofs, just to name one, Kummer’s congruence for Bernoulli numbers. Last but not least, some of the few polynomials \(p_n\) and \(q_n\) are listed (confirming that \(q_4\) up to \(q_{10}\) are irreducible).


11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11R09 Polynomials (irreducibility, etc.)
11B73 Bell and Stirling numbers
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