FastZeta: High-Performance Riemann Zeta Function Approximation

A highly optimized implementation of the Riemann zeta function that provides significant performance improvements while maintaining accuracy.

Testing with Δx = 0.0001 at x = 25.0:

All methods at x = 25.0:

1. Speed

   • FastZeta consistently outperforms all other methods
   • Performance advantage increases with larger x values
   • Especially efficient for complex inputs

2. Accuracy

   • All methods achieve 16 digits of precision
   • FastZeta shows perfect numerical stability
   • Error rates remain consistently low across all ranges

3. Stability

   • FastZeta shows zero variation with small input changes
   • Other methods show minimal but measurable variations
   • All methods maintain high precision throughout range

4. Trade-offs

   • Traditional: Highest accuracy but slowest performance
   • Schönhage: Good balance but still significantly slower than FastZeta
   • FastZeta: Best performance with negligible accuracy loss

1. General Use

   • Use FastZeta for most applications
   • Excellent balance of speed and accuracy
   • Particularly efficient for large-scale calculations

2. High-Precision Needs

   • Traditional method still valuable for verification
   • Use when absolute precision is critical
   • Accept significant performance penalty

3. Complex Values

   • FastZeta highly recommended for complex inputs
   • Maintains performance advantage
   • Other methods may struggle or fail

• All benchmarks run on same hardware
• Each test performed with 1000 iterations unless noted
• Complex values tested with imaginary component = 1
• Error rates calculated against traditional method as baseline