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Zbl 1153.11318
Chowla, S.
On series of the Lambert type which assume irrational values for rational values of the argument. (English)
[J] Proc. Natl. Inst. Sci. India, Part A 13, 171-173 (1947). ISSN 0370-0860

The author shows that the function $$ g(x)=x/(1-x)-x^3/(1-x^3)+x^5/(1-x^5)-\cdots $$ has an irrational value when $x$ has the form $1/t$, where $t\geq 5$ is a positive integer. To this end he considers the sum $S=\sum_{n=1}^\infty r(n)/t^n$, where $r(n)$ is the number of representations of $n$ as a sum of two squares. Using the ``decimal'' scale with basis $t$ he shows by means of well-known arithmetical theorems that $S$ is irrational for an integer $t\geq 5$. The result stated above then follows on account of the relation $4g(x)=\sum_{n=1}^\infty r(n)x^n$.
[J. Popken (MR0024472)]

MSC 2000:: *11J72 Irrationality

Cited in: Zbl 1153.11034

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