CERTAIN FORMULAE FOR VALUES

OF THE RIEMANN ZETA FUNCTION AT INTEGRAL POINTS^{1}

O. N. Vasilenko (Moscow)

Set ζ(s) = ∑_{m=1}^{∞} m^{−s}, I_{0,n}(σ) = 1, and

I_{k,n}(σ) = ∫_{0}^{1}⋯∫_{0}^{1} \frac{[x_{1} … x_{k}(1 − x_{1}) … (1 − x_{k})]^{n}[x_{1}(1 − x_{2}(… (1 − x_{k}) …)]^{σ} d\overline{x}}{(1 − x_{1}(1 − x_{2}(1 − … (1 − x_{k−1}(1 − x_{k}) …)^{n}}.

It can be easily shown that J_{2,n}(0) and J_{3,n}(0), n = 0, 1, 2, …, coincide with the sequences from [1] that were used there to prove that ζ(2) and ζ(3) are irrational (see also [2]).

Theorem.

1) I_{2M,0}(σ) = ∑_{k=1}^{M}(−1)^{k−1} ∑_{j_{1},...,j_{k}=1}^{∞} \frac{1}{(σ + j_{1})^{2} … (σ + j_{1} + … + j_{k})^{2}} I_{2M−2k,0}(0);

2) I_{2M+1,0}(σ) = ∑_{k=1}^{M}(−1)^{k−1} ∑_{j_{1},...,j_{k}=1}^{∞} \frac{1}{(σ + j_{1})^{2} … (σ + j_{1} + … + j_{k})^{2}} I_{2M+1−2k,0}(0)

+ (−1)^{M−1} ∑_{j_{1},...,j_{M−1}=1}^{∞} \frac{1}{(σ + j_{1})^{2} … (σ + j_{1} + … + j_{M−1})^{2}} ∫_{0}^{1} \frac{1 − x^{σ+j_{1}+...+j_{M−1}}}{1 − x} dx;

3) I_{2,0}(0) = ζ(2), I_{3,0}(0) = 2ζ(3), I_{4,0}(0) = \frac{ζ(2)^{2} + ζ(4)}{2}, I_{5,0}(0) = 2ζ(5);
4) I_{2M,0}(0) = T_{M}(ζ(2), ζ(4), …, ζ(2M)),  T_{M}(x_{1}, …, x_{M}) ∈ ℚ[x_{1}, …, x_{M}].

References

268–272.

Moscow, 1983, pp. 15–18. (Russian)

1. F. Beukers, A note on the irrationality of ζ(2) and ζ(3), Bull. London Math. Soc. 11 (1979), no. 3, 2. O. N. Vasilenko, On methods for proving the irrationality of ζ(3), Transcendence number theory and

its applications, Reports of the Soviet conference (Moscow, February 2–4, 1983), MSU Publications,

1Number theory and its applications, Proceedings of the science-theoretical conference (Tashkent, September 26–28, 1990), Tashkent, 1990, p. 27. (Russian)