Skip to main content
Log in

Abelian Theorems, Farey Series and the Riemann Hypothesis

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

The aim of this paper is to provide unconditional estimates for the error terms associated with Farey series that are comparable to error terms in the prime number theorem, and also to provide error terms for Farey series based on implications of the RH(α).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. P. Codecà, “Alcune proprietà della discepanza locale delle sequenze di Farey,” Atti della Accademia delle Scienze dell'Instituto di Bologna, 13 (1981), 163–173.

    Google Scholar 

  2. P. Codecà and A. Perelli, “On the uniform distribution (mod 1) of the Farey fraction and l p spaces,” Math. Ann. 279 (1988), 413–422.

    Google Scholar 

  3. M. Ishibashi and S. Kanemitsu, “Fractional part sums and divisor functions I,” in Number Theory and Com-binatorics (J. Akiyama et al., eds.), World Sci., 1985, pp. 119–183.

  4. A. Ivić, The Riemann Zeta-Function,Wiley-Interscience, 1985.

  5. S. Kanemitsu, T. Kuzumaki and M. Yoshimoto, “Some sums involving Farey fractions II,” J. Math. Soc. Japan 52 (2000), 915–947.

    Google Scholar 

  6. S. Kanemitsu and M. Yoshimoto, “Farey series and the Riemann hypothesis,” Acta Arith. 75 (1996), 351–374.

    Google Scholar 

  7. S. Kanemitsu and M. Yoshimoto, “Farey series and the Riemann hypothesis III,” The Ramanujan J. 1 (1997), 363–378.

  8. G. Kolesnik, “On the order of Dirichlet L-functions,” Pacific J. Math. 82 (1979), 479–484.

    Google Scholar 

  9. J. van de Lune, H.J.J. te Riele and D.T. Winter, “On the zeros of the Riemann zeta function in the critical strip IV,” Math. Comp. 46 (1986), 667–681.

    Google Scholar 

  10. M. Mikolás, “On a theorem of J. Franel,” Kgl. Norske Videnskabers Selskabs Forhandlinger 21 (1948), 98–101.

  11. M. Mikolás, “Farey series and their connection with the prime number problem I,” Acta Sci. Math. (Szeged) 13 (1949), 93–117.

    Google Scholar 

  12. M. Mikolás, “Farey series and their connection with the prime number problem II,” Acta Sci. Math. (Szeged) 14 (1951), 5–21.

    Google Scholar 

  13. J. Milnor, “On polylogarithms, Hurwitz zeta functions, and the Kubert identities,” Enseign. Math. 29 (1983), 281–322.

    Google Scholar 

  14. H. Niederreiter, “The distribution of Farey points,” Math. Ann. 201 (1973), 341–345.

    Google Scholar 

  15. H. Niederreiter, “Application of Diophantine approximations to numerical integration,” in Diophantine ap-proximation and its applications (C.F. Osgood, ed.), Academic Press (1973), 129–199.

  16. M. Szalkowski, A remark on the Farey fractions, Discussiones Math. 8 (1986), 59–60.

    Google Scholar 

  17. E.C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford UP, 1948; 2nd ed. (revised by D.R. Heath-Brown), Oxford UP, 1988.

  18. M. Yoshimoto, “Farey series and the Riemann hypothesis II,” Acta Math. Hung. 78 (1998), 287–304.

    Google Scholar 

  19. M. Yoshimoto, “Farey series and the Riemann hypothesis IV,” Acta Math. Hung. 87 (2000), 109–119.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yoshimoto, M. Abelian Theorems, Farey Series and the Riemann Hypothesis. The Ramanujan Journal 8, 131–145 (2004). https://doi.org/10.1023/B:RAMA.0000040478.59518.9b

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:RAMA.0000040478.59518.9b

Navigation