Exploring Integer Conditions and Prime Forms to Uncover Zeta Secrets

New Mathematical Frameworks for the Riemann Hypothesis

Recent investigations into number theoretic structures suggest potential new avenues for exploring the Riemann Hypothesis (RH). Drawing from the concepts presented in arXiv:hal-02927758, this research focuses on specific integer conditions, prime number forms, and factorial-based expressions.

Framework 1: The "Whole" Number Product Representation

The paper introduces expressions denoted as "Whole(...)", which signifies that the expression inside the parentheses evaluates to an integer. One such structure involves an infinite product over prime numbers:

• The core idea is analyzing when a complex expression involving a function g(n) and a product over squares of primes results in a whole number.
• This framework suggests investigating the properties of g(n) and how its interaction with prime number products dictates integer outcomes.
• Potential Connection: The structure involving prime products naturally connects to the Euler product representation of the Riemann zeta function, which is fundamental to understanding its zeros.

Framework 2: Factorial Structures and Divisibility

Another key expression examined is related to factorials, specifically involving the term n squared factorial divided by n squared plus one. The condition that this expression, possibly multiplied by other terms, results in a whole number is explored.

• This translates into studying the divisibility properties of n squared factorial in relation to n squared plus one.
• The paper notes that the condition involving n squared factorial divided by n squared plus one holds.
• Potential Connection: Divisibility properties involving factorials and prime numbers are deeply intertwined with number theory and can potentially shed light on the distribution of primes, which in turn affects the zeta function's behavior and zero locations.

Framework 3: Primes of the Form n Squared Plus One

The existence of infinitely many prime numbers of the form n squared plus one is highlighted. This relates to a famous unsolved problem in number theory (Landau's fourth problem).

• The distribution and properties of these specific primes are central to the frameworks described above.
• Their occurrence influences the "Whole" conditions and factorial divisibility.
• Potential Connection: The distribution of primes, including those in specific forms like n squared plus one, is directly linked via explicit formulas to the non-trivial zeros of the Riemann zeta function. Understanding the density or patterns of these primes could provide insights into zero locations.

Novel Approaches Combining Concepts

Approach 1: Linking g(n) Properties to Zeta Zero Distribution

We can define the function g(n) in a way that relates to known functions in analytic number theory, such as the prime-counting function or its variants. Analyzing the "Whole" condition under such definitions could constrain the behavior of these functions.

• Methodology: Define g(n) based on prime distribution functions. Analyze the integer condition imposed by the "Whole" framework. Relate this condition to the known explicit formulas connecting prime distribution to zeta zeros.
• Prediction: This could yield a new criterion for the RH based on the asymptotic behavior or specific values of prime-counting related functions.
• Limitations: Requires careful handling of infinite products and establishing rigorous links between the integer conditions and continuous functions.

Approach 2: Factorial Divisibility as a Criterion for RH

The divisibility property of n squared factorial by n squared plus one can be leveraged. When n squared plus one is prime, Wilson's theorem provides a specific congruence. Investigating this for composite n squared plus one, or the density of n for which n squared plus one is prime, might be fruitful.

• Methodology: Study the divisibility of n squared factorial by n squared plus one for all n. Relate the set of n where this divisibility holds (or fails in a specific way) to properties of the zeta function, perhaps its values or derivatives at certain points, or the distribution of its zeros.
• Prediction: This could reveal a number theoretic equivalent statement to the RH, possibly framed in terms of the sequence n squared plus one and its prime factors or primality.
• Limitations: Establishing a direct, rigorous link between this specific divisibility property for all n and the complex analytic properties of the zeta function is challenging.

Tangential Connections

Connection 1: Prime Gaps and Zeta Zeros

The discussion of primes of the form n squared plus one naturally connects to the study of prime gaps and conjectures like Oppermann's. Fluctuations in prime gaps have been hypothesized to relate to the distribution of zeta zeros.

• Formal Bridge: Techniques used to study prime gaps, especially in specific sequences, often involve analytic methods that are also applied to the zeta function. Conjectures about bounds on prime gaps could imply constraints on the local density of zeta zeros.
• Specific Conjecture: The distribution of primes of the form n squared plus one influences the variance of prime gaps, and this variance is directly correlated with the local density of non-trivial zeta zeros on the critical line.
• Computational Experiment: Numerically analyze the distribution of primes of the form n squared plus one up to a large bound. Compare statistical properties of their gaps or occurrence frequency with the known distribution of the first N non-trivial zeta zeros.

Connection 2: Divisibility and the ABC Conjecture

The divisibility condition n squared factorial is divisible by n squared plus one can be seen in the context of the ABC conjecture by setting a = n squared factorial, b = 1, and c = n squared factorial + 1. The conjecture provides bounds on the radical of abc.

• Formal Bridge: The bounds provided by the ABC conjecture relate the additive structure (a+b=c) to the multiplicative structure (prime factors) of numbers. This could potentially constrain the possible prime factors of n squared factorial + 1, which is relevant when n squared + 1 is prime.
• Specific Conjecture: A sufficiently strong form of the ABC conjecture implies a specific upper bound on the number of primes of the form n squared plus one below a given value X, and this bound is consistent with predictions based on the RH.
• Computational Experiment: Test the ABC conjecture's inequality for values of n where n squared + 1 is prime or composite, and analyze the radical of n squared factorial multiplied by (n squared factorial + 1).

Detailed Research Agenda

The proposed research aims to establish a link between the number theoretic properties identified in arXiv:hal-02927758 and the Riemann Hypothesis.

Precisely Formulated Conjectures:

1. There exists a function G(n) related to prime distribution such that the condition "Whole(G(n))" holds if and only if the Riemann Hypothesis is false.
2. The set of integers n for which n squared + 1 is prime has a density or distribution property that is equivalent to the statement that all non-trivial zeros of the zeta function lie on the critical line.

Specific Mathematical Tools and Techniques Required:

• Analytic Number Theory (prime number theorem, explicit formulas, sieve methods)
• Complex Analysis (properties of the zeta function, analytic continuation, product representations)
• Abstract Algebra and Number Theory (divisibility theory, congruences, properties of factorials)
• Computational Number Theory (algorithms for primality testing, calculating factorials modulo n, analyzing number sequences)

Potential Intermediate Results:

• Characterizing the function g(n) used in the "Whole" formulation.
• Deriving necessary conditions on n for "Whole(n squared factorial / (n squared + 1))" to hold when n squared + 1 is composite.
• Establishing a rigorous link between the density of primes of the form n squared + 1 and the distribution of zeta zeros using explicit formulas.
• Proving specific cases or weaker versions of the conjectures relating divisibility or prime forms to properties of the zeta function.

Logical Sequence of Theorems:

1. A theorem characterizing the constraints imposed on g(n) by the "Whole" product formula.
2. A theorem analyzing the divisibility of n squared factorial by n squared + 1 for all n.
3. A theorem establishing a quantitative link between the distribution of primes of the form n squared + 1 and the terms in the explicit formulas for zeta zeros.
4. A theorem proving one of the main conjectures, establishing a new criterion for the Riemann Hypothesis based on these number theoretic structures.

Explicit Examples (Simplified Cases):

• Consider the divisibility condition for small n. When n=1, 1!/(1+1) = 1/2, not whole. When n=2, 4!/(4+1) = 24/5, not whole. When n=3, 9!/(9+1) = 362880/10, whole. This suggests the condition "Whole(n squared factorial / (n squared + 1))" does not hold for all n, contrary to a possible interpretation of the draft, and its behavior needs careful study.
• For n=2, n squared + 1 = 5 (prime). 2 squared factorial = 24. 24 is congruent to -1 mod 5 by Wilson's Theorem. This specific congruence for prime n squared + 1 is known; the challenge is using this or the composite cases to constrain zeta zeros.