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Halász, Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen, Acta Math. Acad. Sci. Hungar. 19 (1968), no. 3–4, 365–403, determines the asymptotic behavior of $\sum_{n\le N}f(n)$ for every multiplicative $f$ with $|f(n)|\le1$: the sum is $o(N)$ unless $f(p)$ stays close to $p^{it}$ on average over the primes for some real $t$, and in that case

$$\sum_{n\le N}f(n)=\frac{N^{1+it}L(\log N)}{1+it}+o(N)$$

for a slowly varying $L$ (the main term may itself be $o(N)$). The theorem is stated in this form in Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Chapter III.4 (Mean values of multiplicative functions), and Elliott, Probabilistic Number Theory I, Chapter 6 (Theorems of Delange, Wirsing, and Halász). A real-valued $f$ can only be close to $p^{it}$ for $t=0$, so for $f:\mathbb{N}\to\{-1,1\}$ the mean value exists, which answers Problem 239 in the affirmative and contains Wirsing's theorem (Wirsing 1967). The theorem also explains the site's remark that the limit can fail to exist for complex values of modulus one, with $f(n)=n^{i}$ as the example. The page's date is the journal issue's month, September 1968, as Crossref records it; the issue gives no day, so the first of the month stands in for it.

Depends on. Nothing in this wiki; the result rests on the refereed paper linked above.

Acceptance. Refereed: the paper appeared in Acta Mathematica Academiae Scientiarum Hungaricae, volume 19, issue 3–4 (1968). Reviewed: the site's curator, Thomas F. Bloom, records it in the problem's commentary as the generalization of Wirsing's answer. The paper is not carded in the library; the statement follows the site's commentary and the textbook accounts cited above. No formal proof is held or audited, so no formalized evidence is listed.