Exploring Number Theoretic Functions for Clues to Prime Distribution

Investigating Fundamental Number Theoretic Functions

Recent research, building on classical number theory, highlights the potential of functions like the Möbius function (μ(n)) and its summatory counterpart, the Mertens function (M(x)), in shedding light on the distribution of prime numbers. The definition of μ(n) is central:

• μ(n) = 0 if n contains a square of a prime.
• μ(n) = +1 if n=1 or is square-free and has an even number of primes.
• μ(n) = -1 if n is square-free and has an odd number of primes.

The behavior of M(x) = Σ_{n ≤ x} μ(n) is closely tied to the Riemann Hypothesis (RH). Specifically, the conjecture that M(x) grows no faster than a power of x slightly larger than x^1/2 is equivalent to RH. Analyzing the asymptotic behavior, such as the limit of |M(x)|/x as x approaches infinity, offers a starting point.

Connections to Prime Distribution and Square-Free Integers

The distribution of prime numbers, quantified by the prime-counting function π(x), is deeply intertwined with the Möbius function. The paper details relationships involving square-free integers and π(x). For instance, the number of square-free integers formed by the product of two primes p_ip_j in a given range can be expressed using μ(p_i) and differences of π(x) values.

Proving tighter bounds on expressions linking M(x) and π(x) could offer new insights into the zero-free regions of the zeta function, a critical aspect of RH research.

Series Representations and Analytic Properties

The Dirichlet series Σ_n=1^∞ μ(n)/n^s represents 1/ζ(s) for Re(s) > 1. Exploring transformations and convergence properties of this series, as well as related alternating sums, is crucial. An equation presented in the source material connects alternating and non-alternating sums:

(1 - 1/2^s) sum((-1)^(n-1) mu(n)/n^s) = (1 + 1/2^s) sum(mu(n)/n^s)

Understanding the analytic properties and convergence criteria for these series could reveal properties of the zeta function's zeros on the critical strip.

Novel Research Directions

Combining these frameworks suggests promising research avenues:

• Refined Mertens Function Analysis: Integrate detailed knowledge of prime distribution, potentially from explicit formulas or bounds on π(x), into the analysis of M(x). The goal is to derive a sufficiently tight bound on |M(x)| that implies RH. This requires rigorous bounding of sums involving μ(n) and careful handling of error terms using techniques from analytic number theory.
• Alternating Sum Behavior and Zeta Zeros: Investigate the behavior of the alternating sum A(x) = Σ_n=1^x (-1)^n-1 μ(n)/n. A conjecture could be that the boundedness of A(x) is equivalent to RH. Connecting A(x) to the zeta function through its series representation and potentially using the Hadamard product formula could link the sum's properties directly to the location of zeta zeros.

Tangential Connections and Future Work

The statistical properties of the Möbius function sequence exhibit fascinating patterns. Connections can be explored with:

• Random Matrix Theory: Model the fluctuations of M(x) using statistical properties from random matrix ensembles, comparing observed distributions to theoretical predictions.
• Information Theory: Analyze the entropy or complexity of the μ(n) sequence to quantify its apparent randomness, which is related to the distribution of primes and potentially the zeros of ζ(s).

A detailed research agenda would involve formulating precise conjectures about the growth rate of M(x) and the boundedness of alternating sums, employing tools from analytic number theory, complex analysis, and potentially computational verification for initial cases. Proving intermediate theorems on square-free distributions and refined sum bounds would pave the way towards the main goal.

This analysis is based on the mathematical concepts presented in arXiv:0365.1451 and related work.