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Erdős problem 1061

For S(x)=#{(a,b):a+b≤x, σ(a)+σ(b)=σ(a+b)}S(x) = \#\{(a,b) : a + b \le x,\ \sigma(a) + \sigma(b) = \sigma(a+b)\}, is S(x)∼cxS(x) \sim cx? The preprint claims S(x)S(x) grows faster than x(log⁡x)Rx (\log x)^R for every fixed RR, ruling out the linear asymptotic.

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