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On a problem of P. Erdős. (English) Zbl 0216.03803

In [Ann. Math. (2) 47, 1–20 (1946; Zbl 0061.07902)], P. Erdős conjectured a theorem equivalent to:
(i) If the additive arithmetic function \(f(n)\) \((f(ab)=f(a) + f(b)\), provided \((a,b) =1)\) is such that \(f(n+ 1) - f(n)\) is bounded for all \(n\), then \(f(n)= g(n) + h(n)\), where \(h(n)\) is bounded and \(g(n)\) is unique and completely additive \((g(ab) = g(a) + g(b)\) for all \(a,b)\).
(ii) \(g(n) = A \log n\) with constant \(A\).
Here the author gives a neat and complete proof of the first part of the conjecture and states his inability to decide whether the second part is true or false. Assuming \(\vert f(n) - f(n-1)\vert \le m\) for all \(n\ge 2\), he achieves the proof after establishing three lemmas enumerated below: \begin{align*} \text{If } (s,n -1) = 1, \qquad & \vert f(n^s) - sf(n)\vert \le 2sM. \tag{1} \\ \text{If } k\ge 0, n> 0, \qquad & \vert f(n^{2^k}) - 2^kf(n)\vert \le 4\cdot 2^kM. \tag{2} \\ \text{If } t>s>0, \qquad & \vert f(n^t) - f(n^s)\vert \le 4(t-s) nM. \tag{3} \end{align*}
During the proof, the author twice states: “\(s\) runs over the primes to \(n^{2^k} -1\)” implying “\(s\) runs over all numbers prime to \(n^{2^k} -1\)”.
Reviewer: S. M. Kerawala

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas

Citations:

Zbl 0061.07902
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