I was reading
http://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf
How did the counterexample for the equation on page 8 look like ??
Specificly :
(quote)
“Tur´an’s lecture (probably a quite informal thing considering the small group) could not have been later than July 14, since it was before my return. Straus has speeded up events; Erd˝os told me he was trying to prove pn+1 pn → 1 on July 15. He told me he had a proof only late on July 16 or possibly earlier the next day. Straus’ quote is also clearly wrong for the following reasons; first, I needed more than just pn+1 pn → 1 for my first proof of the PNT, second, I only saw how to do it on Sunday, July 18. It is true, however, as Erd˝os’ and Straus’ stories indicate, that when I first was told by Erd˝os that he was trying to prove pn+1 pn → 1 from my formula, I tried to discourage him, by saying that I doubted whether the formula alone implied these things. I also said I had constructed a counterexample showing that the relation in the form
$$f(x) \ln(x) + \int_1^x f(\frac{x}{t}) d f(t) = 2 x \ln(x) + (O(x)) $$
does not imply $f(x)$ is close to $x$.
(end quote)
Note: it is not clear to me and others what is meant by " close " here. I just copied a quote, if anyone knows I would love to know.
What about $f(t)=t+ct^{1-\delta}$ for any small positive $\delta$? Of course, I don’t know what “is close to” means. Can you specify?