Prime Numbers and Last Digits Analysis

• Prime sequence: 2, 3, 5, 7, 11, 13, 17, 19, 23...
• Analysis of last digits of primes up to various limits (100, 500, 1000, million)
• Pattern emerges: after initial primes 2 and 5, only digits 1, 3, 7, 9 appear as last digits
• Reason: numbers ending in even digits divisible by 2, numbers ending in 5 divisible by 5
• Base 10 = 2 × 5, eliminating digits with factors 2 or 5

Distribution Patterns Across Different Bases

• Base 10: ~25% chance each for digits 1, 3, 7, 9 as last digits
• Base 2 (binary): all primes after 2 have last digit 1 (all odd numbers)
• Base 3: primes have last digits 1 or 2 (50/50 split after eliminating multiples of 3)
• Base 101: roughly uniform distribution across all valid digits
• Pattern holds for any base B: equal distribution among digits with no shared factors with B

Prime Number Theorem for Arithmetic Progressions

• Mathematical notation: π(x) = number of primes up to x
• π(x,B,a) = number of primes up to x with last digit a in base B
• Result: π(x,B,a) ~ π(x)/φ(B) where φ(B) is Euler's totient function
• φ(B) counts digits with gcd(a,B) = 1
• Distribution evens out as x increases, but question remains: how soon?

Connection to Riemann Hypothesis

• Riemann Hypothesis predicts threshold: relationship holds when x > B²
• Generalized Riemann Hypothesis: works for bases B < √x as x → ∞
• Critical line of 1/2 in zeta function connects to √x threshold
• Connection between zeros of Riemann zeta function and digit distribution

Historical Progress Beyond Riemann Hypothesis

• 1965: Bombieri-Vinogradov theorem proved result holds for 99.9% of bases up to √x
• 1980s: First breakthrough going 15% beyond Riemann Hypothesis (exponent x^0.571)
• 2021: James Maynard extended to 20% beyond (exponent x^0.60)
• Current result: New world record of 23% beyond Riemann Hypothesis (exponent x^0.617)

Technical Details and Implications

• Statistical approach: studying families of numbers rather than individual cases
• Cannot pinpoint exact bases B where this holds, but can prove for large families
• For fixed individual bases, prime number theorem already guarantees convergence
• Challenge arises when bases allowed to grow moderately with x
• Siegel-Walfisz theorem (100+ years old): allows logarithmic growth of base with x

Mathematical Techniques and Connections

• Proof connects multiple mathematical areas:
  • Modular forms (crucial in Fermat's Last Theorem proof by Andrew Wiles)
  • Algebraic geometry
  • Weil bounds and Deligne bounds
  • Riemann Hypothesis for curves over finite fields
• Revival in field over last decade with new researchers contributing
• Acknowledgment: Sary Drappeau, Youness Lamzouri, Laszlo Merai, Alessandro Languasco, Alex Pascadi