## 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

### Keywords:

Farey fractions; $$\ell ^ p$$ spaces; discrepancy

Zbl 0616.10041
Full Text:

### References:

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