New large value estimates for Dirichlet polynomials (2405.20552v1)

Abstract: We prove new bounds for how often Dirichlet polynomials can take large values. This gives improved estimates for a Dirichlet polynomial of length $N$ taking values of size close to $N^{3/4}$, which is the critical situation for several estimates in analytic number theory connected to prime numbers and the Riemann zeta function. As a consequence, we deduce a zero density estimate $N(\sigma,T)\le T^{30(1-\sigma)/13+o(1)}$ and asymptotics for primes in short intervals of length $x^{17/30+o(1)}$.

• The paper establishes new bounds on the frequency of large values in Dirichlet polynomials, outperforming classical estimates for V near N^(3/4).
• It employs refined techniques that enhance zero-density estimates for the Riemann zeta function, offering more precise analytical tools.
• The results offer improved insights into prime distribution in short intervals, paving the way for future research in computational number theory.

Analysis of New Large Value Estimates for Dirichlet Polynomials

Guth and Maynard's paper offers substantial advancements in the domain of Dirichlet polynomials through the derivation of novel bounds that delineate the frequency with which these polynomials can attain large values. This investigation is pivotal for the critical scenario concerning Dirichlet polynomials of significant length and values, with implications for related inquiries such as the zero-density estimates associated with the Riemann zeta function.

Key Contributions and Numerical Findings

The primary contribution of the paper is the establishment of new bounds for Dirichlet polynomials, specifically targeting the frequency of occurrences when these functions reach large magnitudes. This advancement is encapsulated in their main theorem, which contrasts favourably with earlier results derived from the classical Mean Value Theorem and Montgomery-Halasz-Huxley's large value estimate. The new bounds demonstrate superiority for certain ranges of V (value sizes) relative to the length N, providing improved estimations when V is close to the critical threshold where V ≈ N^3/4.

Moreover, their results directly impact our understanding of the zero density of the Riemann zeta function, a crucial area in analytic number theory. The paper leverages these new bounds to propose an alternative zero density estimate for the Riemann zeta function that effectively supersedes previous bounds when σ, the real part of the complex zeros considered, falls within specified intervals.

Implications for Analytic Number Theory

These findings bolster the theoretical framework for handling large values of Dirichlet polynomials and offer refined tools for ongoing investigations into zeta functions and L-functions. Theoretical implications include more precise zero-density estimates, which lay a groundwork for further explorations into the distribution of zeros, contributing to an enriched understanding of the Riemann zeta function's behaviour.

Practically, the results provide an enhanced basis for assessing the distribution of primes within short intervals. The improved bounds on large values translate into more robust estimates for such distributions, with potential implications for cryptographic algorithms and the broader field of computational number theory.

Potential for Future Research and Developments

The future avenues opened by this research are abundant, with the primary focus likely to be on extending these results to other zero-density related functions and more generalized Dirichlet polynomial cases. Additionally, the methodological innovations presented could stimulate further investigations into additive number theory and the application of zero-density techniques to new classes of mathematical problems.

Ultimately, Guth and Maynard's paper contributes significant depth to the current understanding of Dirichlet polynomials, reinforces connections to critical open problems in analytic number theory, and sets a new standard for future research within this essential facet of mathematical inquiry.