Local Moments of Möbius Fourier Polynomials and the Riemann Hypothesis

2026/07/27 by Alberto Verjovsky
#math.PR #math.NT

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Abstract

Let P_N(t)=\frac1√ N∑_{n≤ N}μ(n)\e^{2πi nt}, t∈\T=\mathbb R/\mathbb Z. We give a local probabilistic reformulation of the Riemann hypothesis by evaluating the normalized Möbius Fourier polynomial \(P_N\) at a uniform random point in an arc of radius \(c/N\). We prove that RH is equivalent to subpolynomial growth of arbitrarily high finite local moments of these random variables. The principal quantitative tool is a local moment-to-point-value inequality which recovers the value \(P_N(0)=M(N)/√ N\) from local \(L^q\)-data (where M denotes the Mertens function). This provides a critical-scale local criterion for RH and complements Denjoy's random-walk heuristic, Kahane's theory of random Fourier series, and global \(L^p\)-semiflatness criteria for Möbius polynomials.

Citations

• Exponential sums of the Möbius function and flat polynomials
• Flat Littlewood Polynomials Exist

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