[Paper Review] Fourier-Stieltjes coefficients of the Minkowski question mark function

Giedrius Alkauskas|arXiv (Cornell University)|Aug 24, 2010

Point processes and geometric inequalities17 references3 citations

TL;DR

This paper investigates the Fourier-Stieltjes coefficients $ d_n $ of the Minkowski question mark function, proposing a refined conjecture that implies Salem's 1943 problem—whether these coefficients vanish at infinity—is true. It establishes infinite linear identities among the coefficients using zeta functions and provides a high-precision method to compute special values of the associated zeta function, achieving over 30-digit accuracy via series involving moments and special functions.

ABSTRACT

In this paper we investigate the Fourier-Stieltjes coefficients of the Minkowski question mark function. In 1943, R. Salem asked whether these coefficients vanish at infinity. We propose the conjecture which implies the affirmative answer to Salem's problem. Further, we prove infinite linear identities among these Fourier-Stieltjes coefficients. We also provide a method to numerically calculate special values at integers of the associated zeta function with a high precision (more than 30 digits).

Motivation & Objective

• To resolve Salem's 1943 problem on whether the Fourier-Stieltjes coefficients of the Minkowski question mark function vanish at infinity.
• To establish infinite linear relations among the coefficients $ d_n $ using zeta functions and special values.
• To develop a high-precision numerical method for computing special values of the zeta function associated with $ d_n $.
• To characterize the coefficients $ d_n $ uniquely via canonical linear identities under symmetry and continuity constraints.

Proposed method

• Derives a canonical identity for each odd $ r \in \mathbb{N} $, expressing $ \sum_{n=1}^\infty \xi_{r,n} d_n = A_r $, where coefficients involve $ \Gamma(s) $, $ \zeta(s) $, $ \pi $, $ \log n $, and definite integrals.
• Uses the functional equation $ F(x) + F(1/x) = 1 $, with $ F(x) = ?(x/(x+1)) $, to derive symmetry-based identities for $ d_n $.
• Applies the Laplace-Fourier transform $ \mathfrak{m}(t) = \int_0^1 e^{xt} d?(x) $, with $ d_n = \mathfrak{m}(2\pi i n) $, to connect coefficients to zeta functions.
• Employs the Taylor series $ \pi \cot(\pi x) = \frac{1}{x} - 2\sum_{n=1}^\infty \zeta(2n) x^{2n-1} $ to expand the cotangent kernel in the integral representation of $ \mathfrak{M}(1) $.
• Uses the identity $ \int_0^1 (?(x) - x) x^{2n-1} dx = \frac{1}{2n(2n+1)} - \frac{m_{2n}}{2n} $, where $ m_n $ are moments of the measure, to express $ \mathfrak{M}(1) $ as a rapidly convergent series.
• Leverages known results on moments $ m_n $, which decay as $ n^{1/4} C^{-\sqrt{n}} $ with $ C = e^{-2\sqrt{\log 2}} $, to ensure fast convergence in numerical evaluation.

Experimental results

Research questions

• RQ1 Do the Fourier-Stieltjes coefficients $ d_n $ of the Minkowski question mark function tend to zero as $ n \to \infty $, resolving Salem's problem?
• RQ2 Can infinite linear identities be established among the coefficients $ d_n $, and what is their canonical structure?
• RQ3 What is the precise value of $ \mathfrak{M}(1) = \sum_{n=1}^\infty \frac{d_n}{n} $, and how can it be computed with high precision?
• RQ4 How do the special values of the zeta function associated with $ d_n $ behave, and what is their analytical structure?

Key findings

• The paper proves that for each odd $ r \in \mathbb{N} $, there exists a canonical linear identity $ \sum_{n=1}^\infty \xi_{r,n} d_n = A_r $, with absolute convergence and explicit algebraic constants involving $ \Gamma(s) $, $ \zeta(s) $, $ \pi $, $ \log n $, and definite integrals.
• The simplest such identity for $ r = 1 $ is $ \frac{3}{4} \sum_{n=1}^\infty 2^{-n} \log n - \frac{1}{4} \log 2\pi - \sum_{n=1}^\infty \frac{d_n}{4n} - \sum_{n=1}^\infty \frac{1 - d_n}{\pi n} \int_n^\infty \frac{\sin(2\pi x)}{5 - 4\cos(2\pi x)} \frac{dx}{x} = 0 $.
• The constant $ \mathfrak{M}(1) = \sum_{n=1}^\infty \frac{d_n}{n} $ is computed to over 30-digit precision as $ -0.455959203740245619075047841829\ldots $.
• The value $ \mathfrak{M}(1) $ is expressed via a rapidly convergent series: $ \mathfrak{M}(1) = \sum_{n=1}^\infty \frac{(\zeta(2n)-1) m_{2n}}{n} - 3 \sum_{n=1}^\infty \frac{m_n}{n \cdot 2^n} - \log \pi + \log 4 $, where $ m_n $ are moments of the measure.
• The moments $ m_n $ decay as $ n^{1/4} C^{-\sqrt{n}} $ with $ C = e^{-2\sqrt{\log 2}} \approx 0.18916 $, ensuring fast convergence of the series.
• The method enables numerical computation of other special values of the zeta function associated with $ d_n $ at odd positive integers with high accuracy.