A variant of the Bombieri-Vinogradov theorem with explicit constants and applications

Abstract: We give an effective version with explicit constants of a mean value theorem of Vaughan related to the values of ψ(y, χ), the twisted summatory function associated to the von Mangoldt function Λand a Dirichlet character χ. As a consequence of this result we prove an effective variant of the Bombieri-Vinogradov theorem with explicit constants. This effective variant has the potential to provide explicit results in many problems. We give examples of such results in several number theoretical problems related to shifted primes.

• The paper’s main contribution is an effective Bombieri-Vinogradov theorem variant that computes explicit constants using Vaughan’s method.
• It refines the theorem by constraining moduli and specifying numerical bounds, thereby enhancing its computational applicability.
• The work offers improved estimates for applications like shifted prime analysis and Turaán-type inequalities in number theory.

A Variant of the Bombieri-Vinogradov Theorem with Explicit Constants

Introduction

The paper "A variant of the Bombieri-Vinogradov theorem with explicit constants and applications" (1309.2730) by Amir Akbary and Kyle Hambrook addresses significant improvements to the Bombieri-Vinogradov theorem by providing explicit constants for its effective variant. This development has potential applications in number theory, particularly in problems concerning shifted primes and prime number distribution in arithmetic progressions. By integrating explicit constants into the mean value theorem related to the twisted summatory function and the von Mangoldt function, this research contributes to a more tangible understanding and application of analytic number theory results.

Methodology and Results

The authors' primary achievement is an effective variant of the Bombieri-Vinogradov theorem using an inequality from Vaughan's method with explicit constants. According to their findings, the improved form maintains similar asymptotic behaviors but makes previously abstract constants computable. The effective nature and explicit computation of results offer substantial improvements for practical applications.

The main technical contribution involves constraining moduli $q$ to integers, enhancing the theorem's applicability. Their results set quantitative improvements over previous results, specifically stating numerical values for the constants δ(θ) and X(a, θ) in Harman’s theorem for specific values of $a$ and $\theta$. Notably, their contributions refine the parameters where the Bombieri-Vinogradov theorem's implications are robustly applicable, offering tighter bounds and facilitating more precise analytical tools.

Implications

The explicit constants provided in this variant extend the utility of the Bombieri-Vinogradov theorem across various applications concerning shifted primes. These applications include determining the density of primes whose shifts have large prime factors, effective Turaán-type inequalities, and improved estimates for ω(p−1), the number of prime divisors of p−1. Such results underscore the relevance of explicit numerical constants, enabling a direct computation and implementation into broader analytic problems.

As number theory relies heavily on abstract housekeeping like establishing the efficiency of mean theorems, explicit constants imply computational feasibility, thus fostering advanced research potential. For concretely computing the shifted prime distribution or mean values of prime-related functions within intelligible bounds, this theorem variant acts as a cornerstone.

Future Prospects

Looking forward, expanding the effectiveness of this variant to encapsulate larger moduli or improvement upon the restrictions set by the additional factors influencing (log x) power might further enhance its applicability. The constraints and freedom inherent in the manner constants are computed and refined remain a vital area of exploration. Moreover, applying these explicit constraints, aiding computational complexity reduction, and furthering primality-related algorithms are probable domains for future investigations.

Conclusion

The refinement of the Bombieri-Vinogradov theorem via explicit constants signifies a meaningful stride toward pragmatic arithmetic applications. By determining explicit expressions and improving mean theorem applicability, this variant embodies a potential growth area in computational and analytic number theory. The work's comprehensive approach aligns theoretical analyses with practical computations, poised to influence efficiently solving complex problems in number theory and cryptography moving forward.