# Convergent in $L^1(0,1)$ but not in $L^2(0,1)$ help understanding a paper from arxiv

http://arxiv.org/pdf/math/0205003v1

In around equation (1.1) the author says

"By necessity all authors have been led in one way or another to the natural approximation

$$F(n) := \sum_{a=1}^n \mu(a) \rho_a$$

which tends to $-\chi$ both a.e. and in L1 norm when restricted to (0,1) but which has been shown to diverge in H."

My question is:

How to determine $F$ diverges in $L^2(0,1)$? because $$\int_{0}^{1} |1 + F|^2 dx < \infty$$ where $F \to -1$?

I am new to functional analysis, so little confused on this.

Thanks,

Roupam

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In around the point of confusion, the author makes a reference citation to papers [2] and [3] in the reference list. Have you checked there? The phrasing of the sentence suggest that the proof may not be utterly trivial. –  Willie Wong Jan 26 '11 at 15:38
Yes, [3]. But I couldn't get the paper on [2]. But I couldn't understand from there. –  Roupam Ghosh Jan 26 '11 at 16:39