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On the remainder-term of the prime-number formula, I

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References

  1. All the mentioned facts on ζ(s) can be found in every standard book of the subject, we quote onlyIngham's book “The distribution of prime-numbers”, Cambr. Tracts in Math. and Math. Phys., No. 30, which we shall quote often for reference.

  2. G. Pólya, Über das Vorzeichen des Restgliedes im Primzahlsatz,Gött. Nachrichten 1930, pp. 19–27.

  3. S. Skewes, On the difference π(x)−li(x),Journal of Lond. Math. Soc.,8 (1933), pp. 277–283. I learn from a letter of Prof.Littlewood that Mr.Skewes will publish a paper containing a complete solution of the problem of the least sign-change, giving without any supposition anexplicit numerical upper bound for it. (Letter dated from 29. Aug. 1949.)

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  4. A. E. Ingham, A note on the distribution of primes,Acta Arith.,1 (1936), pp. 201–211.

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  5. J. E. Littlewood, Mathematical notes (12). An inequality for a sum of cosines,Journ. of Lond. Math. Soc.,12 (1937), pp. 217–222.

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  6. For a discussion of the applications see my lecture at the Meeting of the Czechoslovakian and Polish Mathematical Associations in Prague delivered on 3. Sept. 1949 entitled “On a new method in the analysis with applications”.

  7. Forc 0 we have only to choose max (c 5,c 12,c 13,c 14); forc 1(ϱ0) we have only the requirements (15,2) and (16,5).

  8. See e. g.E. C. Titchmarsh, The Zeta-Function of Riemann, Cambr. Tracts in Math. and Math. Phys., No. 26, p. 45.

  9. P. Turán, On Riemann's hypothesis,Bull. de l'Acad. des Sciences de l'URSS,11 (1947), pp. 197–262.

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  10. P. Turán, On a theorem of Littlewood,Journ. of the Lond. Math. Soc.,21 (1946), pp. 268–275.

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  11. H. Cramér, Ein Mittelwertsatz in der Primzahltheorie,Math. Zeitschr.,12 (1946), pp. 147–153.

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  1. Budapest

    Paul Turán (corresponding member of the Academy)

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  1. Paul Turán
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Turán, P. On the remainder-term of the prime-number formula, I. Acta Mathematica Academiae Scientiarum Hungaricae 1, 48–63 (1950). https://doi.org/10.1007/BF02022552

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