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Claim. The number of integers up to $x$ that are sums of two squares is asymptotic to $cx/\sqrt{\log x}$ for some constant $c>0$: in the notation of Problem 323, $f_{2,2}(x)\sim cx/\sqrt{\log x}$. The theorem is E. Landau, Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate, Arch. Math. Phys. (3) 13 (1908), 305--312, whose Jahrbuch record is linked above; the proof proceeds by complex integration over Dirichlet series. The site's commentary credits Landau with resolving the case $k=2$ in this form.

Covers. The first question at $k=2$: the answer is yes, since $cx/\sqrt{\log x}\gg_\epsilon x^{1-\epsilon}$ for every $\epsilon>0$. At $k=2$ the second question concerns $m=1$ alone, the count of squares up to $x$, which is trivially of order $x^{1/2}$. Values $k\ge3$ are not addressed.

Depends on. No page of this wiki.

Acceptance. Refereed: Arch. Math. Phys. (3) 13 (1908), 305--312, as the Jahrbuch record (JFM 39.0264.03) gives it. The site credits the case but labels the problem OPEN, so no curator acceptance is listed.

Date. The paper carries a year only; the page name uses the first day of 1908 for want of an issue date.