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).
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References
P. Codecà, “Alcune proprietà della discepanza locale delle sequenze di Farey,” Atti della Accademia delle Scienze dell'Instituto di Bologna, 13 (1981), 163–173.
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.
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.
A. Ivić, The Riemann Zeta-Function,Wiley-Interscience, 1985.
S. Kanemitsu, T. Kuzumaki and M. Yoshimoto, “Some sums involving Farey fractions II,” J. Math. Soc. Japan 52 (2000), 915–947.
S. Kanemitsu and M. Yoshimoto, “Farey series and the Riemann hypothesis,” Acta Arith. 75 (1996), 351–374.
S. Kanemitsu and M. Yoshimoto, “Farey series and the Riemann hypothesis III,” The Ramanujan J. 1 (1997), 363–378.
G. Kolesnik, “On the order of Dirichlet L-functions,” Pacific J. Math. 82 (1979), 479–484.
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.
M. Mikolás, “On a theorem of J. Franel,” Kgl. Norske Videnskabers Selskabs Forhandlinger 21 (1948), 98–101.
M. Mikolás, “Farey series and their connection with the prime number problem I,” Acta Sci. Math. (Szeged) 13 (1949), 93–117.
M. Mikolás, “Farey series and their connection with the prime number problem II,” Acta Sci. Math. (Szeged) 14 (1951), 5–21.
J. Milnor, “On polylogarithms, Hurwitz zeta functions, and the Kubert identities,” Enseign. Math. 29 (1983), 281–322.
H. Niederreiter, “The distribution of Farey points,” Math. Ann. 201 (1973), 341–345.
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.
M. Szalkowski, A remark on the Farey fractions, Discussiones Math. 8 (1986), 59–60.
E.C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford UP, 1948; 2nd ed. (revised by D.R. Heath-Brown), Oxford UP, 1988.
M. Yoshimoto, “Farey series and the Riemann hypothesis II,” Acta Math. Hung. 78 (1998), 287–304.
M. Yoshimoto, “Farey series and the Riemann hypothesis IV,” Acta Math. Hung. 87 (2000), 109–119.
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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
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DOI: https://doi.org/10.1023/B:RAMA.0000040478.59518.9b