By von Oniscik A.L., Sulanke R.

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Let k1 be the greatest prime factor of N1 N2 · · · Nh or h if N0 N1 · · · Nh = 1. Let C be an absolute constant such that 1 log z < C log for all z > 2y 2. p log y c c √ y p

C If α = 1, one obtains b = a β , in spite of the conditions assumed. Hence, α > 1 and h | 2(δ , δ , . . , δ ). c there exists a rational prime l | α. Choose a positive integer h so that l / 1 2 s As the numbers qi are primes, it follows from the last formula that c is neither of the form h h/2 h nl nor of the form 2l nl , where n is a rational integer. By Trost’s theorem ([5]) there exists, therefore, an infinite set P of rational primes p, for which c is not a residue of l h -th degree. As l | α, l /| β for any rational integer x: l /| αx − β.

This completes the proof. We conclude from Theorems 2 and 3 that if an additive function f satisfies conditions 1, 2, (1/p) is divergent and f (p) /p convergent, then the distribution f (p)=0 function of {f (m + 1), . . , f (m + h)} exists, is continuous and strictly decreasing on some half straight-line, thus the sequence of integers n for which inequality (15) holds has a positive density. Similarly we can prove the following: 1 f (p) 2 = ∞ and that < ∞ then p f (p)=0 p f (n + 1) − f (n), f (n + 2) − f (n + 1), .