Prime Patterns: Unlocking the Riemann Hypothesis with Modular Arithmetic?

Unveiling Prime Number Secrets: A Path to the Riemann Hypothesis?

The Riemann Hypothesis, a cornerstone of number theory, remains one of mathematics' greatest unsolved problems. Recent research, as explored in arXiv:1903.08570, delves into the fascinating world of prime number patterns and their potential link to this elusive hypothesis.

Mathematical Frameworks and Formulations

One promising approach involves analyzing the geometric distribution of prime numbers. The idea is that primes aren't scattered randomly but follow specific patterns when viewed through the lens of modular arithmetic. For example, the paper highlights the relationship p² = k × 24 + 1, where p is a prime. This suggests a deeper structure governing prime distribution.

Another framework explores the properties of digital roots in prime number sequences. Digital roots are found by repeatedly summing the digits of a number until a single digit remains. While the paper's initial statements on digital roots requires correction, a more rigorous investigation into the distribution of digital roots of primes could reveal subtle connections to the zeta function.

Novel Approaches: Combining Ideas

One novel approach combines modular arithmetic with Dirichlet L-functions, generalizations of the Riemann zeta function. By expressing the prime counting function in terms of arithmetic progressions modulo 24 and using the explicit formula related to L-functions, we can search for correlations between prime distribution and the zeros of these functions.

Another avenue involves investigating prime gaps within specific arithmetic progressions. If the Riemann Hypothesis is false, we might expect to find unusually large prime gaps in certain progressions modulo 24.

Tangential Connections and Research Agenda

Even seemingly unrelated concepts like Lucas sequences (number sequences where each term is the sum of the two preceding terms) might hold clues. The distribution of Lucas sequence terms modulo a prime could be related to the prime's properties and, indirectly, to the zeta function.

A detailed research agenda would involve:

• Proving conjectures about the distribution of values generated by the equation p² = k × 24 + 1.
• Expressing the prime counting function in terms of Dirichlet L-functions.
• Searching for deviations from uniformity in the distribution of these values if the Riemann Hypothesis is false.

By rigorously pursuing these research pathways, we may uncover innovative insights into the distribution of prime numbers and potentially unlock the secrets of the Riemann Hypothesis.