\(k\)-moments of distances between centers of Ford circles (Q465185)

The subject matter of the paper is the investigation of the distribution of Ford circles. A Ford circle is a circle with center at \((p/q,1/(2q^2))\) and radius \(1/(2q^2),\) where \(p,q\) are coprime positive integers. The authors consider the set \(\mathcal{F}_{I,Q}\) of the Ford circles being tangent to the real axis at points \((a/q,0)\) where \(\frac{a}{q}\in I=[\alpha,\beta]\subseteq[0,1], \; q\leq Q.\) Let \(N_I(Q)=\#\mathcal{F}_{I,Q},\) and let NEWLINE\[NEWLINEO_{Q,j},\quad j=1,\ldots,N_I(Q)NEWLINE\]NEWLINE be centers of the circles. Define the following moments NEWLINE\[NEWLINE \mathcal{M}_{k,I}(Q)=\frac{1}{|I|}\sum_{j=1}^{N_I(Q)-1}\left(\text{dist}(O_{Q,j},O_{Q,j+1})\right)^k,NEWLINE\]NEWLINE NEWLINE\[NEWLINE \mathcal{A}_{k,I}(X)=\frac{1}{X}\int_X^{2X}\mathcal{M}_{k,I}([Y])\,dY. NEWLINE\]NEWLINE In the paper for any positive integer \(k\) very precise asymptotic formulas for \(\mathcal{M}_{k,I}(Q), \mathcal{A}_{k,I}(X)\) are obtained. The proof is based on the bijection of \(\mathcal{F}_{I,Q}\) with the set of Farey fractions \(\mathcal{F}_{Q}.\) This allows to rewrite the moments in terms of Farey fractions and different moments of Farey fractions. Using the properties of Farey fractions and Perron's formula the problem is reduced to evaluation of integrals involving the Riemann zeta function. To estimate the integrals the estimates on zeta function (also in the Vinogradov-Korobov zero free region) are used. To obtain asymptotic formulas for desired moment of Farey fractions Weil bounds for Kloosterman sums were used see \textit{F. P. Boca} et al. [J. Reine Angew. Math. 535, 207--236 (2001; Zbl 1006.11053)].