number-theory

Let ξ(s) = ½s(s−1)π^(−s/2)Γ(s/2)ζ(s) be the completed Riemann zeta function and let {γ_ρ} denote the positive imaginary parts of its nontrivial zeros. We first prove a universal product formula: for all s∈ℂ (under the Riemann Hypothesis), ξ(s)/ξ(½) = ∏_{γ_ρ>0} (1 + (s−½)²/γ_ρ²), expressing the ratio ξ(s)/ξ(½) as a product over the Riemann zeros with coupling a(s) = |s−½|. The formula has two regi…

The definition of infinity is that it is how many natural numbers there are. You can take those infinite natural numbers and slice them into an infinite number of infinite sets, each of which can then be sliced the same way ad infinitum. What does this mean/imply?

We formalize a structural perspective on the Riemann zeta function, highlighting a natural balance between the "inside" (integer counting) and "outside" (density/reflection) perspectives of the number system. Using divisor pairings, information measures, and the functional equation, we show that s = 1/2 is a unique symmetry axis where information is balanced. While this framework does not prove t…

Abstract This paper presents a complete proof of Goldbach's Conjecture by establishing an equivalence with the positivity of a density function D(n) over symmetric prime parametrizations, analyzed through Wilson's Theorem. We parametrize all possible prime pairs (p,q) with p+q=n as p=(n-m)/2 and q=(n+m)/2, where m is the symmetric distance parameter. Using Wilson's quotients k_p = ((p-1)!+1)/p an…

This paper proves a strengthened form of the strong Goldbach conjecture by showing that its negation is unprovable in ZFC, assuming this theory is sound. We reformulate the conjecture using an infinite set and show, based on properties of this set, that the assumption that there is a proof of its negation leads to a contradiction. Whereas the traditional approaches focus on the control over the d…

This paper proves that both the strong Goldbach conjecture and a strengthened form of it are independent of ZFC and Peano arithmetic (PA), with the latter result being a corollary of the former, assuming these theories are sound. For each of the conjectures, we define an infinite set with which we reformulate the conjecture and show that this set always remains the same, regardless of whether the…

It seems to me that the formulation of Schinzel's hypothesis H is related to Pólya's results on integer-valued polynomials (1) and my question is if this conjecture can be generalized as below? Given ...

In a series of works [1-6] an algebraic construction was developed that describes the dynamical change of the dimension of configuration space. In the present work these constructions are systematized in the form of a unified family of Khodakovsky superalgebras X_q, where the parameter q = 0,1,2,3,4,5 corresponds to the successive extension of the original algebra X (dimension n, rank m) through …

The Collatz process is traditionally approached as an unsolved numerical conjecture concerning iterative parity operations. This paper proposes a different interpretation. Rather than treating Collatz solely as a destination-oriented puzzle asking whether all trajectories converge to 1, this paper examines the process as a minimal parity-gated recursive architecture composed of alternating expans…

This note presents the Andrews–Gordon qqq-series as a higher-rank test case for the local-to-global reconstruction principle developed in the companion paper From Ramanujan Summation to Modular Monodromy. The Rogers–Ramanujan pair corresponds to the first non-trivial vector-valued case of rank 222. The Andrews–Gordon hierarchy extends the same mechanism to arbitrary rank k≥2k \geq 2k≥2, producing…

Markov numbers are integer solutions to x² + y² + z² = 3xyz. The Wikipedia article on Markov numbers mentions that Don Zagier studied Markov numbers by looking the approximating equation x² + y² + z² = 3xyz + 4/9 which is equivalent to f(x) + f(y) = f(z) where f(t) is defined as arccosh(3t/2). It wasn’t clear to me why the […] The post Approximating Markov’s equation first appeared on John D. Coo…

In October 2024 I attended a workshop at Harvard University where mathematicians talked through the uses of artificial intelligence in their field. Most were less worried about the future of math than excited about a new tool they might use. During one coffee break, I found myself in a group of participants who all agreed that it made no difference whether a human or a computer solved their favor…

For the mixed CM point (a, b, c) = (1/6, 1/3, 1), define A_n^mix := 108^n [z^n] ₂F₁(1/6, 1/3; 1; z)^3. For every split prime p ≥ 7, p ≡ 1 mod 3, and every m ≥ 1, A_{mp}^mix ≡ A_m^mix (mod p^4). The exponent 4 exceeds the generic weight-3 Hodge-gap prediction of 3; the extra factor of p is a CM enhancement attached to j = 0. The matching unconditional inert-prime obstruction (p ≡ 2 mod 3) is also …

We introduce a divisor-defined sequence M(x) counting integers n in [2, x] with g(n)³ < n, where g(n) is the largest divisor of n at or below √n (OEIS A033676). The prime-counting function π(x) satisfies the exact triangular identity π(x) = M(x) − Σ ( π(⌊x/c⌋) − π(c²) ), summed over c = 2 to ⌊x^(1/3)⌋. Every π-argument on the right is strictly less than x, so the identity determines π(x) bottom-u…

I am trying to understand a bound appearing in Section 4 of the paper of Iwaniec--Sarnak (1999) on Dirichlet L-functions at the central point. They use an estimate of the shape $$ \sum_{mn\neq 1} ...

We investigate the Diophantine condition d(\sigma(\varphi(abc)))=k\mathrm{\thinsp} d(abc),\bigm where d(n)denotes the divisor-counting function, \sigma(n)denotes the sum-of-divisors function, and \varphi(n)denotes Euler’s totient function. The variables a,b,care positive integers satisfying a+b=c,\gcd\funcapply(a,b,c)=1.\bigm Computational evidence suggests that for each fixed integer k\in{1,2,3,…

The a2b2 Conjecture (VC5S3) proposes that for every integer a>1, there exists an integer b such that: 0

We investigate the Diophantine condition$$a+b=c, \quad \gcd(a,b,c)=1,$$together with the arithmetic constraint$$\Omega(\sigma(\varphi(abc)))=\Omega(abc),$$where $\varphi$ denotes Euler’s totient function, $\sigma$ the sum-of-divisors function, and $\Omega(n)$ the total number of prime factors of $n$, counted with multiplicity.

In this paper, we consider the ab2 conjecture, a proposal concerning the relationship between natural numbers and prime numbers through perfect squares. This conjecture states that for every natural number a>1, there exists a natural number b such that b

We present a novel conjecture concerning sums of fourth powers and primes, named the a4b4 Conjecture. The conjecture asserts that for every integer a>1, there exists a natural number b such that a^4+b^4 is prime, b is bounded by a logarithmic function of a, and parity conditions are satisfied. This paper formulates the conjecture, discusses its plausibility, and outlines initial computational app…