š Wessen Getachew Gap Decomposition Theorem (2025)
Ī¶(s) = āgā{1,2,4,6,...} ( āp: gap(p)=g (1 - p-s)-1 )
š„ Revolutionary Discovery:
1. Unique Prime Usage:
Each prime p appears exactly once in the product - classified by gap(p) = next_prime(p) - p
For example: 2 has gap(2)=1, 3 has gap(3)=2, 5 has gap(5)=2, 7 has gap(7)=4, etc.
2. Direct Factorization Method:
Instead of traditional analytic continuation approaches, this theorem provides
direct factorization of Ī¶(s) through prime gap structure!
3. Gap Classification Principle:
Prime gaps encode fundamental arithmetic information that directly reconstructs zeta functions
through discrete spacing structure rather than continuous analysis.
Gap Classes & Their Role:
ā¢ Gap 1: Only prime 2 (since gap(2) = 3-2 = 1)
ā¢ Gap 2: Primes 3, 5 (since gap(3) = 5-3 = 2, gap(5) = 7-5 = 2)
ā¢ Gap 4: Primes 7, 13 (since gap(7) = 11-7 = 4, gap(13) = 17-13 = 4)
ā¢ Gap 6: Primes 11, 23 (since gap(11) = 17-11 = 6, gap(23) = 29-23 = 6)
ā¢ Larger gaps: Each prime p classified by gap(p) = next_prime(p) - p
Ļ Reconstruction: Ļ ā ā(6 Ć Ī¶(2)) where Ī¶(2) is built from gap products!
This shows how prime gaps directly encode geometric constants!
ā” Key Insight: Each prime contributes exactly once to the product, ensuring proper factorization!
Special Cases:
Ī¶(2) = ĻĀ²/6 | Ī¶(4) = Ļā“/90 | Ī¶(6) = Ļā¶/945
Innovation:
Direct gap-based computation of these exact values!
š§® Gap Inversion Sieve Theory
Original Research by Wessen Getachew
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"Bridging discrete prime gaps with continuous zeta functions through modular arithmetic"