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Explicit Formula Construction Using Chebyshev Functions and a Proof of the Twin Prime Conjecture Kohei Okawa (Seifu High School) [email protected] June 5, 2025 1 / 26

Abstract First, I created an explicit formula using the von Mangoldt function. I then proved that the zeta function deviates at (s − 1) when squared with the von Mangoldt function or when adding similar functions. I then invalidated the twin primes constant, which is the leading term of the explicit formula, using the sieve method, and invalidated the GRH dominance of the other terms using the BV method, proving the twin primes conjecture. 2 / 26

Table of Contents 1 Introduction 2 Construction of the Explicit Formula 3 Quasi-explicit Formula for Twin Primes 4 Numerical Experiments 5 Eliminating the Twin Prime Constant 6 Addressing the Error Term via GRH 3 / 26

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Introduction In prime number distribution theory, following the zeta function, the Chebyshev functions are among the most studied objects. The second Chebyshev function is a sum involving the non-trivial zeros of the Riemann zeta function, known as the von Mangoldt function, which has achieved the remarkable feat of providing a rigorous explicit formula for ψ(x). This can be expressed as follows: X X ψ(x) = Λ(n) = ln p n≤x p k ≤x In this work, the goal is to derive and verify an asymptotic explicit formula for twin primes using the Chebyshev functions, and to explore the infinitude of such a formula, thereby aiming to resolve the twin prime conjecture. 4 / 26

Explicit Formula: General Concept Understanding the explicit formula is crucial. In mathematics, an explicit formula expresses the relationship between variables directly, without recursive or iterative definitions. In number theory, the explicit formula related to the Riemann zeta function is most relevant: X π(x) = R(x) − R(x ρ ) + correction terms ρ Here, R(x) is ”Riemann’s explicit function”: R(x) = ∞ X µ(n) n=1 n Li(x 1/n ) where µ(n) is the Möbius function, and Li(x) is the logarithmic integral: Z x dt Li(x) = 2 ln t 5 / 26

Explicit Formula: Prime Number Theorem This term provides a ”smooth” approximation for the distribution of primes. For example, the prime number theorem approximates: π(x) ≈ Li(x) − 1 √ Li( x) + (zero contributions) + · · · 2 This structure underpins the current study’s formulas. 6 / 26

Dirichlet Series for Twin von Mangoldt Functions Define the Dirichlet series for the twin von Mangoldt functions: F (s) := ∞ X Λ(n)Λ(n + 2) ns n=1 Perform Möbius inversion:  Λ(n)Λ(n + 2) = − (ℜ(s) > 2)  X d|n µ(d) ln d  −  X µ(e) ln e  e|n+2 7 / 26

Dirichlet Convolution and Rearrangement From the properties of Dirichlet convolution: X µ(d) ln d = Λ(n) d|n and rewriting F (s): F (s) = ∞ X 1 n=1 ns    X X  µ(d) ln d   µ(e) ln e  d|n e|n+2 8 / 26

Chinese Remainder Theorem and Congruences The inner sum runs over n satisfying: d|n, e|n + 2 which can be expressed as: ( n ≡ 0 (mod d) n ≡ −2 (mod e) By the Chinese Remainder Theorem, when gcd(d, e) = 1, the system has a unique solution modulo de. 9 / 26

Density and Euler Product The sum is approximated by ∞ X n=1 d|n, e|n+2 1 1 ≈ ns (de)s X n≡0 (mod d) n≡−2 (mod e) 1 ns Thus, F (s) = X d,e µ(d)µ(e) log d log e X n≡0 (mod d) n≡−2 (mod e) 1 ns 10 / 26

Local Factors and Euler Product For each prime p, X Λ(p k ) k≥1 p ks = log p 1 s p 1 − p −s Squaring gives Fp (s) = (log p)2 X (m − 1) p −ms m≥2 and Fp (s) = (log p)2 · p −s X j p −js j≥1 11 / 26

Geometric Series and Zeta Function From the geometric series, X j xj = j≥1 x (1 − x)2 Insert x = p −s : X j≥1 j p −js = (|x| < 1) p −s (1 − p −s )2 Thus, Fp (s) = (log p)2 p −s (1 − p −s )2 12 / 26

Euler Product for F (s) The local factors (Euler factors) for F (s) are then derived,  1− −1   χ2 (p) −1 · 1 − p s−1 p s−1 1 Take the logarithmic derivative:   log p χ2 (p) log p F ′ (s) X = + F (s) p s−1 − 1 p s−1 − χ2 (p) p 13 / 26

Multiplicity and Local Factors For odd primes p, χ2 (p) = 1, and for p = 2, χ2 (2) = 0. Thus,  −1 1  (p = 2) χ2 (p) −1 =  1 − s−1 1  1 − s−1 p (p odd) p 14 / 26

Residues and Laurent Expansion If we express F (s) as the ζ function, F (s) = 1 d2 ζ ′ (s) · log ζ(s) − ds 2 ζ(s) s − 1 The Laurent expansion in the ρ neighborhood is ζ(s) = (s − ρ)ζ ′ (ρ) + (s − ρ)2 ′′ ζ (ρ) + · · · 2 Ress=ρ F (s) = Cρ = 1 ζ ′′ (ρ) − ′ 2 2ζ (ρ) ρ(ρ − 1) Thus, 15 / 26

Perron’s Formula and Residue Theorem For any c > 2: X n≤x 1 Λ(n)Λ(n + 2) = 2πi Z c+iT F (s) c−iT xs ds s From the poles around the origin, expand F (s) around s = 0: F (s) = a−2 a−1 + + a0 + · · · s2 s Apply the residue theorem: Ress=0 F (s)x s a0 = a−2 log x + a−1 + log2 x s 2 16 / 26

Zero Terms and Oscillatory Sums The formula for the zero terms can be expressed as a sum of complex exponentials (assuming GRH): ! ∞ X Cρ∗k x −iγk Cρ∗k x iγk + iγk −iγk k=1 Using Euler’s formula, this becomes: K X ak sin(γk log x + ϕk ) k=1 17 / 26

Explicit Formula for Twin Primes The explicit formula then takes the form: X n≤x K X xρ X ∗ 2 Λ(n)Λ(n+2) = S(2)x− C +c1 log x+c2 log x+c3 + ak sin(γk log ρ ρ ρ k=1 where the constants ci are explicitly determined, and the oscillatory sum captures fluctuations due to zeros. This is called the ”preF-twin” formula. 18 / 26

Numerical Experiments (Python) Python code snippet: import numpy as np from mpmath import zetazero S2 = 1.320323632 def theoret ical_C _rho ( rho ): return (1 - rho . conjugate ()) / (2 * abs ( rho )**2 + 1e -8) def get_zeros_batch ( n_zeros ): return [ zetazero ( n ) for n in range (1 , n_zeros + 1)] zeros_100 = get_zeros_batch (100) # Use only 100 zeros for memo def i m p r o v e d _ e x p l i c i t _f o r m u l a (x , zeros ): main_term = S2 * x rho_terms = sum (( x ** rho / rho * theoretical_C_rho ( rho )). rea log_correction = 1.837 * np . log ( x )**2 - 25.63 * np . log ( x ) + return main_term - rho_terms + log_correction 19 / 26

Numerical Results (Excerpt) Output: x Actual value Theoretical value Error Relative error (%) 1000 1135.46 1411.41 -275.95 24.30% 112000 146652.48 148006.21 -1353.73 0.92% 223000 293905.24 294575.81 -670.57 0.23% 20 / 26

How to Eliminate the Twin Prime Constant? 1 Selberg Sieve: Apply the Selberg sieve and Cauchy-Schwarz inequality: 2 P Λ(n) X n≤x Λ(n)Λ(n + 2) ≈ x n≤x P Since n≤x Λ(n) ∼ x, this ratio approximates 1, and error terms can absorb S(2). 21 / 26

Bombieri-Vinogradov Theorem and Error Terms The Bombieri-Vinogradov theorem states that, without assuming GRH, primes are equidistributed on average among arithmetic progressions: X q≤Q max π(x; q, a) − (a,q)=1 Li(x) x ≪ φ(q) (log x)A for Q = x 1/2 /(log x)B . Applying this, the zero contributions (oscillations) can be controlled, and √ the error term reduces from GRH-dependent O( x) to an unconditional O(x 1/2+ϵ ). 22 / 26

Final Explicit Formula and Infinitude Consequently, the explicit formula becomes:   X x + oscillations + O(x 1/2 (log x)5 ) Λ(n)Λ(n + 2) = x + O 1/10 (log x) n≤x Oscillations tend to cancel out infinitely often, implying the sum grows linearly, thus confirming the infinitude of twin primes. 23 / 26

Additional Note Replacing S(2) with an expression involving π2 (x): S(2) ≈ π2 (x)(log x)2 x and using the asymptotic π2 (x) ∼ (log2xx)2 , we recover the classical heuristic that twin primes are infinite. 24 / 26

Author Information Kohei Okawa (Seifu High School) 12-16 Ishigatsuchicho, Tennoji-ku, Osaka, Japan, 543-0031 Email: [email protected] 25 / 26

References and Software In this research, the Python programs for numerical experiments heavily relied on xAI (Grok 3). I gratefully acknowledge this support. References: 1 Chen, J. R.: On the representation of a large even integer as the sum of a prime and the product of at most two primes. Kexue Tongbao, 17, 385–386 (1966). 2 Kazuo Matsuzaka, ”Introduction to Algebraic Systems” (2018). 3 Takashi Nakamura, “The generalized strong recurrence for the Riemann zeta function” (Tokyo University of Science). 4 Gérald Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Graduate Studies in Mathematics, Vol. 163, AMS, 2015. 26 / 26