New Combinatorial and Factorization Perspectives on the Riemann Hypothesis

Exploring Combinatorial Structures

Recent work introduces intriguing combinatorial quantities derived from binomial coefficients, such as sums and differences denoted as s_k^N and D_k^N. These are defined as:

• s_k^N = C_N+2k-1,2k + C_{N+(2k-1)-1,2k-1}
• D_k^N = C_N+2k-1,2k - C_{N+(2k-1)-1,2k-1}

The paper establishes inequalities and bounds for these sums, including a key finding that a sum involving s_k^N is bounded by a series related to a hyperbolic cosine function. This suggests a potential link between combinatorial growth patterns and analytic functions relevant to number theory.

Potential Research Directions:

• Analyze Asymptotics: Rigorously determine the asymptotic behavior of s_k^N and D_k^N as N and k tend to infinity using tools like Stirling's approximation.
• Relate to Number Theory: Explore if these combinatorial quantities can be related to coefficients of Dirichlet series or other functions important in analytic number theory, potentially via generating functions.
• Establish Convergence: Study the convergence properties of infinite sums involving these terms and their bounds, particularly in the context of complex variables.

Factorization Quantities and Prime Distributions

Another concept introduced is Q_q^ℤ, defined as a limit of binomial coefficients, which is interpreted as the "quantity of unique factorizations having q". This quantity relates to the distribution of integers based on the number of their prime factors.

Potential Research Directions:

• Formal Definition: Provide a precise number-theoretic definition for Q_q^ℤ and prove its equivalence to the combinatorial limit.
• Connect to Omega Function: Relate Q_q^ℤ to the number of integers n with exactly q prime factors, counted with multiplicity (Ω(n)=q). This could involve connecting to results like the Erdos-Kac theorem on the distribution of additive functions.
• Analyze Distribution: Study the distribution of Q_q^ℤ across different q values and relate it to known distributions of arithmetic functions.

Novel Approaches Linking Combinatorics to Zeta Function

Approach 1: Combinatorial Approximation of Zeta

Can we express the fundamental building blocks of the Riemann zeta function, such as 1/n^s, using the combinatorial quantities s_k^N or D_k^N? If successful, this could lead to a novel combinatorial series representation or approximation of ζ(s).

• Methodology: Find functions A(N, k, s) and B(N, k, s) such that 1/n^s can be approximated by a limit involving A, B, and s_k^N. Then, attempt to sum this approximation over all n to represent ζ(s).
• Challenges: Justifying the interchange of limits and infinite sums, and ensuring the approximation is sufficiently accurate to preserve the zero structure of the zeta function.
• Prediction: A successful approximation might reveal combinatorial conditions that force the zeros onto the critical line, potentially linking properties of s_k^N to the location of zeros.

Approach 2: Factorization Quantities and Zeta Poles

The zeta function has a simple pole at s=1. Its behavior near this pole is well-understood and related to fundamental constants like the Euler-Mascheroni constant. Can the factorization quantity Q_q^ℤ provide insight into this behavior?

• Methodology: Relate Q_q^ℤ to the distribution of prime factors, and then attempt to express the coefficients of the Laurent series expansion of ζ(s) around s=1 in terms of Q_q^ℤ.
• Challenges: Establishing a precise quantitative link between the asymptotic distribution of integers with q prime factors and the coefficients of the zeta function's Laurent series.
• Prediction: If such a link is found, the Riemann Hypothesis might be equivalent to Q_q^ℤ satisfying certain specific asymptotic properties related to the structure of the zeta function near its pole.

Tangential Connections and Conjectures

Erdos-Kac Theorem and Combinatorial Sums

The Erdos-Kac theorem describes the probabilistic distribution of the number of prime factors of a typical integer. If the combinatorial sums s_k^N or D_k^N could be used to express quantities central to this theorem (like log log n), it would create a bridge between these combinatorial structures and probabilistic number theory, which in turn is linked to the zeta function.

• Conjecture: Specific linear combinations or limits involving s_k^N or D_k^N can approximate or represent terms like log log n.
• Experiment: Computational analysis comparing values of s_k^N/D_k^N for large N, k with log log n for various n to identify potential relationships.

Detailed Research Agenda

The research pathway could proceed as follows:

1. Phase 1: Foundations
   • Precisely define Q_q^ℤ and prove its combinatorial and number-theoretic interpretations.
   • Establish rigorous asymptotic formulas for s_k^N and D_k^N.
   • Formulate conjectures linking these quantities to known arithmetic functions or distributions.
2. Phase 2: Developing Connections
   • Attempt to express 1/n^s or related terms using s_k^N/D_k^N, aiming for a combinatorial series representation of ζ(s).
   • Explore the relationship between Q_q^ℤ and the coefficients of the Laurent series of ζ(s) at s=1.
   • Investigate connections to the Erdos-Kac theorem by attempting to represent log log n or related quantities using s_k^N/D_k^N.
3. Phase 3: Proving the Hypothesis
   • If a combinatorial representation of ζ(s) is found, analyze its analytic properties and zeros. Prove that its non-trivial zeros lie on the critical line.
   • If a link between Q_q^ℤ and the zeta pole is established, prove that the properties of Q_q^ℤ imply the necessary structure of the zeta function near s=1, which in turn supports the Riemann Hypothesis.
   • If a strong connection to the Erdos-Kac theorem is made, explore whether the implied distribution of prime factors forces the zeros of the zeta function onto the critical line.

This research requires tools from combinatorics, analytic number theory, complex analysis, and potentially probabilistic number theory. Intermediate results would include proving asymptotic formulas, establishing specific identities, and finding explicit relationships between the combinatorial quantities and known number-theoretic functions or constants. The source material for these ideas is found in arXiv:hal-01220071.