The methodology for analyzing polynomials on roots of unity, as presented in the source paper, can be adapted to study sums of arithmetic functions central to the Riemann Hypothesis (RH). Let's define a "zeta-approximating polynomial" $Q_q(z; f)$ as:
$Q_q(z; f) = \frac{1}{\sqrt{q}} \sum_{j=1}^{q} f(j) z^{j-1}$
where $f(n)$ is an arithmetic function (e.g., the Möbius function, von Mangoldt function, or coefficients of $1/\zeta(s)$). The source paper analyzes expressions like $\frac{1}{q} \sum_{k=0}^{q-1} |Q_q(\xi_{q,k})|^4$, which are discrete $L_4$ norms. Adapting this to the RH involves exploring the behavior of these norms as $q \to \infty$ and relating them to properties of $\zeta(s)$. For instance, a potential theorem could link the growth rate of the $L_4$ norm to the distribution of zeros.
The paper's use of discrete Fourier analysis on roots of unity offers another avenue. The Riemann zeta function has connections to various transforms, including the Mellin transform. Exploring analogous discrete Fourier transforms of arithmetic functions, guided by the methods in the source paper, might reveal new relationships between the zeros and the underlying arithmetic structure.
A potential theorem could connect the spacing of zeros to the decay rate of the discrete Fourier transform of the coefficients of the zeta function's Laurent series expansion.
The source paper's careful analysis of error terms in polynomial approximations offers valuable insights. The prime number theorem and its error terms are closely tied to the Riemann Hypothesis. By adapting the source paper's techniques, we can investigate how the error terms in approximating the zeta function relate to the location of its zeros. This may involve developing new bounds or inequalities for error terms in prime number counting functions, analogous to those in the source paper.
We can generate a sequence of polynomials $Q_q(z; f)$ for various arithmetic functions $f(n)$ and study their statistical properties. This could involve analyzing the distribution of their roots, their $L_p$ norms, and their correlations. The behavior of these statistical properties as $q \to \infty$ might reveal information about the distribution of zeros of $\zeta(s)$.
The source paper provides precise bounds for error terms in certain polynomial approximations. We can adapt these techniques to improve the bounds on error terms in the prime number theorem. This can lead to stronger constraints on the distribution of zeros of $\zeta(s)$.
The statistical analysis of the polynomials $Q_q(z; f)$ could be compared to results from random matrix theory, which has shown connections to the distribution of zeros of the Riemann zeta function. The source paper's analysis could provide a new framework for refining or extending these connections. This could lead to new conjectures regarding the statistical behavior of zeta zeros and possibly new ways to constrain their distribution.
1. Establish preliminary results: Prove convergence of polynomial approximations to $\zeta(s)$. Develop explicit error bounds using the techniques from the source paper.
2. Key conjectures to prove: Formulate precise conjectures about the relationship between the statistical properties of $Q_q(z; f)$ and the distribution of zeros of $\zeta(s)$. For example, connect the $L_4$ norm's growth to the density of zeros.
3. Computational experiments: Generate a large number of polynomials $Q_q(z; f)$ for various $q$ and $f(n)$. Analyze the distribution of their roots and norms. Compare the results to known data about $\zeta(s)$ zeros.
4. Refinement of Error Bounds: Improve the error bounds in the prime number theorem using the techniques of the source paper. Show how these improved bounds directly affect the distribution of zeros of $\zeta(s)$.
This approach leverages the insights from the source paper to explore new connections between polynomial analysis and the Riemann Hypothesis, offering a promising avenue for further research.