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On average values of arithmetic functions

Published online by Cambridge University Press:  24 October 2008

E. Fogels
E. Fogels
Affiliation:
L.U. Matematikas SeminārsRaiņa bulv. 19Riga, Lativa
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Extract

The problem considered in this paper is that of finding the least possible h = h(x) such that a given arithmetic function a(n) should keep its average order in the interval x, x + h, i.e. that we have

and

as x → ∞.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 37 , Issue 4 , October 1941 , pp. 358 - 372
Copyright
Copyright © Cambridge Philosophical Society 1941

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References

REFERENCES

(1)Ingham, A. E.On the difference between consecutive primes. Quart. J. Math. 8 (1937), 255–66.CrossRefGoogle Scholar
(2)Ingham, A. E.Mean-value theorems in the theory of the Riemann zeta-function. Proc. London Math. Soc. (2), 27 (1928), 273–300.CrossRefGoogle Scholar
(3)Heilbronn, H.Über den Primzahlsatz von Herrn Hoheisel. Math. Z. 36 (1933), 394–423.CrossRefGoogle Scholar
(4)Hoheisel, G. Primzahlprobleme in der Analysis. S.B. preuss. Akad. Wiss. (1930), pp. 580–8.Google Scholar
(5)Landau, E.Über einige Summen, die von den Nullstellen der Riemann'schen Zetafunktion abhängen. Acta Math. 35 (1912), 271–94.CrossRefGoogle Scholar
(6)Landau, E.Über die Riemannsche Zetafunktion in der Nähe von σ = 1. R.C. Circ. mat. Palermo, 50 (1926), 423–7.Google Scholar
(7)Titchmarsh, E. C.On ζ(s) and π(s). Quart. J. Math. 9 (1938), 97–108.CrossRefGoogle Scholar
(8)Titchmarsh, E. C.The mean value of Quart. J. Math. 8 (1937), 107–12.CrossRefGoogle Scholar
(9)Titchmarsh, E. C.The zeta-function of Riemann (Cambridge, 1930).Google Scholar
(10)Titchmarsh, E. C.The theory of functions (Oxford, 1932).Google Scholar
(11)Čudakov, N. G.On zeros of the function ζ (s). C.R. Acad. Sci. U.R.S.S., 1 (10) (1936), 201–4.Google Scholar
(12)Čudakov, N. G.On the functions ζ(s) and π(x). C.R. Acad. Sci. U.R.S.S. 21 (1938), 421–2.Google Scholar
(13)Hardy, G. H., Ingham, A. E. and Pólya, G.Theorems concerning mean values of analytic functions. Proc. Roy. Soc. A, 113 (1927), 542–69.Google Scholar
(14)Hardy, G. H. and Littlewood, J. E.The zeros of Riemann's zeta-function on the critical line. Math. Z. 10 (1921), 283–317.CrossRefGoogle Scholar
(15)Hardy, G. H. and Littlewood, J. E.The approximate functional equation in the theory of the zeta-function, with applications to the divisor-problems of Dirichlet and Piltz. Proc. London Math. Soc. (2), 21 (1923), 39–74.CrossRefGoogle Scholar
(16)Hardy, G. H. and Wright, E. M.An introduction to the theory of numbers (Oxford, 1938).Google Scholar