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Source. The Corollary, p. 344 (section 2), of A. Dvoretzky and P. Erdős, Divergence of random power series, Michigan Math. J. 6 (1959), 343--347, the edition named on the source card.

Read depth. Claims checked: the statement was read clause by clause on the page image of p. 344. The paper gives no separate proof; it presents the Corollary as a special case of the Theorem. Nothing here is independently reviewed.

Statement

Setting as on the Theorem's page: $\phi_n(t)$ are the Rademacher functions and the series (1) of the paper are $\sum_{n=0}^{\infty}\phi_n(t)\,a_n z^n$, $0\le t<1$, with complex $a_n$; "almost all" is in Lebesgue measure in $t$.

Corollary (p. 344). If $\{a_n\}$ satisfies $|a_n|\ge c/\sqrt n$ for $n>N$, for some $c>0$, then almost all series (1) diverge everywhere on $|z|=1$.

The paper calls it "a specially important case" of the Theorem, and its remark that "diverge" may be strengthened to "have unbounded partial sums" covers the Corollary too (p. 344).

Proof pointer

No separate proof is printed; p. 344 introduces the Corollary as a special case of the Theorem and leaves the reduction to the reader.

Dependencies

The Theorem of the same paper (pp. 343--344).

Bears on

• Problem 527: the problem asks whether, for real $a_n$ with $\sum|a_n|^2=\infty$ and $|a_n|=o(1/\sqrt n)$, almost every choice of signs gives a series that converges at some point of $|z|=1$. The Corollary shows that under the size condition $|a_n|\ge c/\sqrt n$ for all large $n$, almost every choice of signs gives divergence at every point of the circle. The paper does not pose the problem.