Recent mathematical explorations suggest that analyzing the properties of certain polynomials with coefficients derived from fundamental arithmetic functions could offer novel insights into the Riemann Hypothesis. Specifically, polynomials constructed using the Liouville and Möbius functions, which are deeply connected to the Riemann zeta function, are being investigated.
The core idea involves studying analytic trigonometric polynomials defined as:
P_n(z) = Σ_{j=1}^n λ(j) z^j, |z| = 1Q_n(z) = Σ_{j=1}^n μ(j) z^j, |z| = 1where λ(n) is the Liouville function and μ(n) is the Möbius function. These functions have direct links to the zeta function via Dirichlet series: Σ_{n=1}^∞ λ(n)/n^s = ζ(2s)/ζ(s) and Σ_{n=1}^∞ μ(n)/n^s = 1/ζ(s) for Re(s) > 1. Understanding the behavior of these polynomials, particularly their zero distribution on the unit circle, may shed light on the vertical distribution of zeta zeros on the critical line Re(s) = 1/2.
Concepts like L^α-semi-flatness and ultraflatness, traditionally applied to trigonometric polynomials, are relevant. A sequence of polynomials (P_n) is L^α-semi-flat if its normalized L^α norm is bounded. The paper mentions the non-L^α-flatness of certain polynomials with specific coefficients. Applying these ideas to P_n(z) and Q_n(z) could constrain the possible growth or oscillation behavior of related sums, potentially impacting our understanding of ζ(s) near the critical strip.
The analysis involves discrete averages, essentially evaluating the polynomials at roots of unity and considering their moments:
( rac{1}{N^{1+ rac{α}{2}}} Σ_{k=1}^{N-1}|Σ_{n=1}^{N-1} λ(n) ξ_{N, k}^{n}|^{α})_{N ≥ 1} where ξ_{N,k} = exp(2πik/N).
Bounds on these discrete averages could translate into bounds on moments of the zeta function on the critical line, a key area of research for the Riemann Hypothesis.
A novel approach is to formally connect the L^α-semi-flatness of the Liouville/Möbius polynomials (perhaps localized or weighted versions) to zero-density estimates for the zeta function away from the critical line. If these polynomials exhibit a certain degree of flatness or bounded growth on the unit circle, it might imply that the zeta function does not have too many zeros off the Re(s) = 1/2 line. This requires proving non-trivial bounds on the polynomial norms and then using complex analytic tools (like Jensen's formula or contour integration) to translate these bounds into statements about zeta zeros.
Another avenue is to rigorously investigate the distribution of zeros of P_n(z) and Q_n(z). If it could be shown that as n → ∞, the zeros of these polynomials cluster exclusively on the unit circle in a specific manner, this might provide indirect evidence or even a formal link to the Riemann Hypothesis, which states that all non-trivial zeros of ζ(s) lie on the line Re(s) = 1/2. This would involve techniques from the theory of polynomials and their asymptotic root distribution.
The concepts explored also touch upon other mathematical areas:
A research agenda could focus on proving conjectures derived from the frameworks above:
P_n(z) is L^α-semi-flat for some specific α > 0.P_n(z) and Q_n(z) converge in some sense to the unit circle and their distribution is linked to the distribution of prime numbers.P_n(z) evaluated at roots of unity imply specific bounds on the moments of ζ(s) on the critical line.Success requires tools from analytic number theory, complex analysis, real analysis (especially L^p theory), and potentially probability or ergodic theory. Intermediate goals include obtaining improved estimates for sums involving λ(n) and μ(n) and establishing precise relationships between polynomial norms and the behavior of their coefficients.
This analysis draws upon the mathematical structures presented in hal-02326464.