Publisher
Springer Science and Business Media LLC
Subject
General Mathematics,General Mathematics
Reference12 articles.
1. A first systematic treatment of this trend is given in my bookEine neue Methode in der Analysis und deren Anwendungen, Akad. Kiadó (Budapest, 1953). A completely rewritten Chinese edition appeared in 1956. An English edition will appear in the Interscience Tracts series.
2. This sharp form of my first main theorem has been found byE. Makai and a little later but independently byN. G. de Bruijn. SeeE. Makai, The first main theorem of P. Turán,Acta Math. Acad. Sci. Hung.,10 (1959), pp. 405–411;N. G. de Bruijn, On Turán's first main theorem,ibid., Acta Math. Acad. Sci. Hung.,11 (1960), pp. 213–216.
3. The improved form (1.5) of my original form is contained in our paper withVera T. Sós, “On some new theorems in the theory of diophantine approximations”,Acta Math. Acad. Sci. Hung.,6 (1955), pp. 241–255, apart from a remark of Mr.S. Uchiyama; sec his paper “A note on the second main theorem of P. Turán”,ibid., Acta Math. Acad. Sci. Hung.,9 (1958), pp. 378–380. The ingenious counterexample which shows that $$c< \frac{{2e}}{{\log 2}}$$ is false, is due toE. Makai; seeMakai l. c.
4. Λ (n) stands for the Mangoldt-symbol, equals logp ifn=prime powerp α and 0 otherwise.
5. See his paper “Some theorems concerning prime numbers”,Arkiv för Math. Astr. och Fys.,15 (1921).
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