## 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
Full Text:

### References:

 [1] R.J. Anderson - H.M. Stark , Oscillation Theorems, Analytic Number Theory , Proceedings, Philadelphia , 1980 , Ed. F. Knopp, Springer , Lecture Notes in Mathematics N . 899 , pp. 79 - 106 . MR 654520 | Zbl 0472.10044 · Zbl 0472.10044 [2] E. Grosswald , On some generalizations of theorems by Landau and Pòlya , Israel J. Math. , 3 ( 1965 ), pp. 211 - 220 . MR 198145 | Zbl 0144.36803 · Zbl 0144.36803 [3] S. Knapowski - P. Turàn , Comparative prime number theory I-VIII , Acta Math. Acad. Sci. Hungar. , part I: 13 ( 1962 ), pp. 299 - 314 ; part II: 13 ( 1962 ), pp. 315 - 342 ; part III : 13 ( 1962 ), pp. 343 - 346 ; part IV : 14 ( 1963 ), pp. 31 - 42 ; part V: 14 ( 1963 ), pp. 43 - 63 ; part. VI: 14 ( 1963 ), pp. 65 - 78 ; part VII : 14 ( 1963 ), pp. 241 - 250 ; part. VIII: 14 ( 1963 ), pp. 251 - 268 . Zbl 0111.04506 · Zbl 0111.04506 [4] E. Landau , Handbuch der Lehre von der Verteilung der Primzahlen , Teubner , Leipzig und Berlin , 1909 . JFM 40.0232.08 · JFM 40.0232.08 [5] K. Prachar , Primzahlverteilung , Springer , Berlin - Göttingen - Heidelberg , 1957 . MR 87685 | Zbl 0080.25901 · Zbl 0080.25901 [6] R. Spira , Calculation of Dirichlet L-functions , Math. Comput. , 23 , N. 107 ( 1969 ), pp. 484 - 497 . MR 247742 | Zbl 0182.07001 · Zbl 0182.07001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.