Motivated by a famous Franel’s theorem dealing with distributions of Farey series, we consider an analogous problem related to Stern-Brocot sequences. We prove that the remainder <Equation ID="Equ37"> <EquationSource Format="TEX">$ R_n=\sum _{j=1}^{2^n}\left( \xi _{j,n}-\frac{j}{2^n}\right) ^2-2^n\int _0^1(?(x)-x))^2\text {d}x $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>R</mi> <mi>n</mi> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </munderover> <msup> <mfenced close=")" open="("> <msub> <mi>ξ</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>-</mo> <mfrac> <mi>j</mi> <msup> <mn>2</mn> <mi>n</mi> </msup> </mfrac> </mfenced> <mn>2</mn> </msup> <mo>-</mo> <msup> <mn>2</mn> <mi>n</mi> </msup> <msubsup> <mo>∫</mo> <mn>0</mn> <mn>1</mn> </msubsup> <msup> <mrow> <mrow> <mo stretchy="false">(</mo> <mo>?</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mtext>d</mtext> <mi>x</mi> </mrow> </math></EquationSource> </Equation>tends to 0 when n tends to infinity, where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\xi _{j,n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ξ</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> are elements of the Stern-Brocot sequence and ?(x) denotes the Minkowski question-mark Function. We present some extended results and give a correct proof of a theorem on the Fourier-Stieltjes coefficient of the inverse function of ?(x).

Sums with Stern-Brocot sequences and the Minkowski question-mark function

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发表日期：

2025年12月17日

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Haomin Liu,
Jiadong Lü,
Yonghao Xie

摘要

Motivated by a famous Franel’s theorem dealing with distributions of Farey series, we consider an analogous problem related to Stern-Brocot sequences. We prove that the remainder $ R_n=\sum _{j=1}^{2^n}\left( \xi _{j,n}-\frac{j}{2^n}\right) ^2-2^n\int _0^1(?(x)-x))^2\text {d}x $ $R_{n} = \sum_{j = 1}^{2^{n}} {(ξ_{j, n} - \frac{j}{2^{n}})}^{2} - 2^{n} \int_{0}^{1} {(? (x) - x))}^{2} d x$ tends to 0 when n tends to infinity, where $\xi _{j,n}$ $ξ_{j, n}$ are elements of the Stern-Brocot sequence and ?(x) denotes the Minkowski question-mark Function. We present some extended results and give a correct proof of a theorem on the Fourier-Stieltjes coefficient of the inverse function of ?(x).