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On the comparative theory of primes. (English) Zbl 0548.10027

Let \(\ell_ 1\) and \(\ell_ 2\) be two integers relatively prime to q and incongruent (mod q) and let \(\epsilon (n)=\epsilon_ 1(n)-\epsilon_ 2(n)\) where \(\epsilon_ i(n)=1\) if \(n\equiv\ell_ i(q)\) and 0 otherwise. Knapowski and Turán investigated in a series of papers sign changes of the functions defined by summing \(\epsilon\) (n) and \(\epsilon\) (n)\(\Lambda\) (n) over \(n\leq x\) and \(\epsilon\) (p) over \(p\leq x\). The authors consider the second of these functions and the function defined by summing \(\epsilon\) (p)log p over \(p\leq x.\)
Using the same assumptions as in the earlier papers mentioned, namely the so-called Haselgrove condition and the finite Riemann-Piltz conjecture, the authors improve the earlier results in the cases of \(\epsilon\) (n)\(\Lambda\) (n) for general \(\ell_ 1\) and \(\ell_ 2\) and obtain a similar result for \(\epsilon\) (p)log p if \(\ell_ 1\) and \(\ell_ 2\) are both quadratic non-residues. These results then give intervals in which each function must have at least one sign change.
Reviewer: W.E.Briggs

MSC:

11N37 Asymptotic results on arithmetic functions
11N30 Turán theory
11N13 Primes in congruence classes
11N05 Distribution of primes
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References:

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