# Limit connected with a periodic function

I am posting the following question from Math.Stackexchange:

Let $f$ be a $1$-periodic function, i.e., $f(x+1)=f(x)$, defined on the interval $(0, 1)$ by the formula $$f(x)=2x-1.$$ For a real number $x$ consider the series $$\sum_{n=1}^\infty\frac{f(nx)}{n}.$$ It is easily seen that the series converges if $x$ is rational and if $f$ is assumed to vanish at all integers.

My question is: does the convergence hold for irrational $x$?

I would also be grateful for references to any results about kindred objects.

I am also interested in a description of the set of all $x$ for which the series converges, and in properties of the function determined by the series.

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No, convergence does not hold for all irrational $x$. I posted a full answer to the question at math.stackexchange and I'll summarize the result here.

There are uncountably many values of $x$ for which the partial sums $\sum_{n=1}^N\frac{f(nx)}n$ are unbounded. In fact, the set of values of $x$ for which the partial sums are bounded is meagre. Consequently, the set of $x$ for which the sum diverges is nonempty and, furthermore, is uncountable in the neighbourhood of any point. We can say rather more than this though. Choosing any function $\theta\colon\mathbb{Z}^+\to\mathbb{R}^+$ such that $\theta(n)=o(\log n)$ (i.e., $\theta(n)/\log n\to0$ as $n\to\infty$) then the set of $x$ such that the partial sums are bounded by a multiple of $\theta$ is meagre. This statement is, in a sense, optimal, because it can be shown that the partial sums do grow at a rate $o(\log N)$ for every $x$.

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In the paper

J. Rosenblatt, Convergence of Series of Translations, Math. Ann. 230 (1977) 245-272

it is proved a general Theorem (Theorem 2.5) that in particular proves the convergence a. e. . Also the unconditional convergence in L^2[0,1].

The result in this particular case maybe was known before. Rosenblatt points to works of Hardy Littlewood and Khinchin.

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Many thanks. I would like to see the proof, I plan to visit our library on Monday. –  kap44 Jul 19 '12 at 12:08
As far as I understand, you say that for any $x$ the series converges. This expression appeared in my calculations, and I would like to see what is already known. –  kap44 Jul 19 '12 at 12:07