## Similarities in irrationality proofs for $$\pi$$, $$\ln 2$$, $$\zeta(2)$$, and $$\zeta(3)$$.(English)Zbl 0986.11045

This paper develops a strategy suggested by work of the Borweins [J. M. Borwein and P. B. Borwein, Pi and the AGM. A study in analytic number theory and computational complexity, Wiley, New York (1987; Zbl 0611.10001)] for deducing the irrationality of a given number $$\xi$$. Find a function $$f$$ and a sequence of rational numbers $$R_j$$ and $$S_j$$ such that $$\int_0^1 x^jf(x) dx= R_j+\xi S_j$$. If $$\xi$$ is rational, then any finite linear combination of $$R_j+ \xi S_j$$ with integer multipliers is also rational. Use the coefficients $$p_{nj}$$ of the $$n$$th Legendre polynomial $$P_n(x)$$ as multipliers, and sum on $$j$$ to get $$A_n= B_n\int_0^1 P_n(x)f(x) dx$$ where $$A_n$$ and $$B_n$$ are integers. For each number in the title one can choose a suitable $$f$$ and obtain an inequality of the form $$1\leq|A_n|\leq C_n$$, where $$C_n\to 0$$ as $$n\to\infty$$. This is a contradiction.

### MSC:

 11J72 Irrationality; linear independence over a field

irrationality

### Citations:

Zbl 0903.11001; Zbl 0611.10001
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