On the uniform distribution (mod 1) of the Farey fractions and $$\ell ^ p$$ spaces.(English)Zbl 0606.10041

We estimate the discrepancy $E_ f(N)=\sum _{q\leq N}\sum ^{*qa=1}f(a/q)-(\sum _{q\leq N}\sum ^{*qa=1}1)\int ^{1}_{0}f(x) dx$ for a rather wide class of functions f. If f is absolutely continuous and $$f'\in L^ p[0,1]$$, and if we assume the $$\alpha$$-quasi Riemann hypothesis, then we have for every $$\epsilon >0$$ $E_ f(N)=O(N^{\max (\alpha,1/p)+\epsilon}),\quad E_ f(N)=\Omega (N^{\max (\alpha,1/p)-\epsilon}).$ Define the ”deviation coefficients” D(n) by $D(n)/n = (1/n)\sum ^{n}_{a=1}f(a/n)-\int ^{1}_{0}f(x) dx.$ Then we have $(*)\quad D(n)=-(1/2\pi)\sum ^{\infty}_{k=1}c(kn)/k,$ where (c(n)) are the Fourier coefficients of f’. In view of the Hausdorff-Young theorem, we also study the structure of the linear transformation (c(n))$$\to (D(n))$$ given by (*) and obtain the best possible result that if $$(c(n))\in \ell ^ p$$ with $$p\geq 1$$, then $$(D(n))\in \ell ^ r$$ for every $$r>p$$.

MSC:

 11K06 General theory of distribution modulo $$1$$ 11B39 Fibonacci and Lucas numbers and polynomials and generalizations

Zbl 0616.10041
Full Text:

References:

 [1] [B] Bary, N.K.: A treatise on trigonometric series. London: Macmillan 1964 · Zbl 0129.28002 [2] [Fr] Franel, J.: Les suites de Farey et les problèmes des nombres premiers. Nachr. Ges. Wiss Göttingen, Math.-phys. Kl., 198-201 (1924) · JFM 50.0119.01 [3] [Fu1] Fujii, A.: A remark on the Riemann hypothesis: Comment. Math. Univ. S. Pauli29, 195-201 (1980) · Zbl 0451.10025 [4] [Fu2] Fujii, A.: Some explicit formulae in the theory of numbers: Proc. Jap. Acad. Ser. A57, 326-330 (1981) · Zbl 0492.10034 [5] [H-W] Hardy, G.H., Wright E.M.: An introduction to the theory of numbers. Oxford: Oxford University Press 1971 [6] [H] Huxley, M.N.: The distribution of Farey points I. Acta Arith.18, 281-287 (1971) · Zbl 0224.10036 [7] [K1] Kopriva, J.: On a relation of the Farey series to the Riemann hypothesis on the zeros of the ? function. (Czech) Cas. Pe?tovani Mat.78, 49-55 (1953) [8] [K2] Kopriva, J.: Contribution to the relation of the Farey series to the Riemann hypothesis. (Czech) Cas. Pe?tovani Mat.79, 77-82 (1954) [9] [K3] Kopriva, J.: Remark on the significance of the Farey series in number theory. (Czech) Publ. Fac. Sci. Univ. Masaryk, 267-279 (1955) [10] [K-N] Kuipers, L., Niederreiter, H.: Uniform distribution of sequences. New York: Wiley 1974 · Zbl 0281.10001 [11] [L1] Landau, E.: Bemerkungen zu der vorstehenden Abhandlung von Herr Franel. Nachr. Ges. Wiss. Göttingen, Math.-phys. Kl., 202-206 (1924) · JFM 50.0119.02 [12] [L2] Landau, E.: Vorlesungen über Zahlentheorie. Stuttgart: Teubner 1927 · JFM 53.0123.17 [13] [M1] Mikolas, M.: Sur l’hypothèse de Riemann. C. R. Acad. Sci. Paris228, 633-636 (1949) · Zbl 0034.17303 [14] [M2] Mikolas, M.: Farey series and their connection with the prime number problem I, II. Acta Sci. Math. (Szeged)13, 93-117 (1949);14, 5-21 (1951) · Zbl 0035.31402 [15] [N] Niederreiter, H.: The distribution of Farey points. Math. Ann.201, 341-345 (1973) · Zbl 0248.10013 [16] [Ba] Rademacher, H.: Topics in analytic number theory. Berlin, Heidelberg, New York: Springer 1973 · Zbl 0253.10002 [17] [Ru] Rudin, W.: Real and complex analysis. New York: McGraw-Hill 1966 · Zbl 0142.01701 [18] [Z] Zalauf, A.: The distribution of Farey numbers. J. reine angew. Math.289, 209-213 (1977)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.