mathhook_core/functions/special/

zeta.rs

//! Riemann zeta function
//!
//! Implements the Riemann zeta function ζ(s) with analytic continuation to
//! the entire complex plane except for a simple pole at s=1.
//!
//! # Mathematical Background
//!
//! The Riemann zeta function is defined for Re(s) > 1 as:
//! ζ(s) = Σ(n=1 to ∞) 1/n^s
//!
//! It extends to the entire complex plane via analytic continuation, with
//! a simple pole at s=1 with residue 1. The functional equation relates
//! ζ(s) to ζ(1-s), enabling evaluation across all complex values.
//!
//! The zeta function is central to number theory, appearing in the prime
//! number theorem and the Riemann hypothesis.
//!
//! # Performance
//!
//! Uses Euler-Maclaurin acceleration (50 terms) for 200x speedup vs
//! direct summation (10,000 terms). Achieves 14-digit accuracy.

use crate::core::{Expression, Number};
use crate::functions::special::gamma::lanczos_gamma;
use std::f64::consts::PI;

/// Riemann zeta function ζ(s)
///
/// The Riemann zeta function extends the series Σ 1/n^s to the entire
/// complex plane via analytic continuation.
///
/// # Mathematical Properties
///
/// - ζ(2) = π²/6 (Basel problem)
/// - ζ(4) = π⁴/90
/// - ζ(6) = π⁶/945
/// - ζ(8) = π⁸/9450
/// - ζ(10) = π¹⁰/93555
/// - ζ(0) = -1/2
/// - ζ(-1) = -1/12 (famous result used in string theory)
/// - ζ(-2n) = 0 for positive integers n (trivial zeros)
/// - ζ(-3) = 1/120
/// - ζ(-5) = -1/252
/// - ζ(-7) = 1/240
/// - Pole at s=1 with residue 1
/// - Functional equation: ζ(s) = 2^s π^(s-1) sin(πs/2) Γ(1-s) ζ(1-s)
///
/// # Arguments
///
/// * `s` - Expression argument to evaluate zeta function at
///
/// # Examples
///
/// ```rust
/// use mathhook_core::functions::special::zeta;
/// use mathhook_core::{Expression, Number};
///
/// let zeta_2 = zeta(&Expression::Number(Number::Integer(2)));
/// ```
pub fn zeta(s: &Expression) -> Expression {
    match s {
        Expression::Number(Number::Integer(n)) => zeta_integer(*n),
        Expression::Number(Number::Float(val)) => {
            if (*val - val.round()).abs() < 1e-10 {
                zeta_integer(val.round() as i64)
            } else {
                let result = zeta_numerical(*val);
                Expression::Number(Number::Float(result))
            }
        }
        _ => Expression::function("zeta", vec![s.clone()]),
    }
}

/// Evaluate zeta function at integer arguments
///
/// Returns exact symbolic values for known special cases, or numerical
/// evaluation for general integers.
fn zeta_integer(n: i64) -> Expression {
    match n {
        // ζ(0) = -1/2
        0 => Expression::rational(-1, 2),

        // ζ(1) is undefined (pole with residue 1)
        1 => Expression::function("zeta", vec![Expression::integer(1)]),

        // ζ(2) = π²/6 (Basel problem)
        2 => Expression::div(
            Expression::pow(Expression::pi(), Expression::integer(2)),
            Expression::integer(6),
        ),

        // ζ(4) = π⁴/90
        4 => Expression::div(
            Expression::pow(Expression::pi(), Expression::integer(4)),
            Expression::integer(90),
        ),

        // ζ(6) = π⁶/945
        6 => Expression::div(
            Expression::pow(Expression::pi(), Expression::integer(6)),
            Expression::integer(945),
        ),

        // ζ(8) = π⁸/9450
        8 => Expression::div(
            Expression::pow(Expression::pi(), Expression::integer(8)),
            Expression::integer(9450),
        ),

        // ζ(10) = π¹⁰/93555
        10 => Expression::div(
            Expression::pow(Expression::pi(), Expression::integer(10)),
            Expression::integer(93555),
        ),

        // ζ(-1) = -1/12
        -1 => Expression::rational(-1, 12),

        // ζ(-2n) = 0 for positive integers n (trivial zeros)
        n if n < 0 && n % 2 == 0 => Expression::integer(0),

        // ζ(-3) = 1/120
        -3 => Expression::rational(1, 120),

        // ζ(-5) = -1/252
        -5 => Expression::rational(-1, 252),

        // ζ(-7) = 1/240
        -7 => Expression::rational(1, 240),

        // For other positive even integers, use numerical evaluation
        n if n > 0 && n % 2 == 0 => {
            let result = zeta_numerical(n as f64);
            Expression::Number(Number::Float(result))
        }

        // For other negative odd integers, use numerical evaluation
        n if n < 0 && n % 2 != 0 => {
            let result = zeta_numerical(n as f64);
            Expression::Number(Number::Float(result))
        }

        // For all other cases, use numerical evaluation
        n => {
            let result = zeta_numerical(n as f64);
            Expression::Number(Number::Float(result))
        }
    }
}

/// Numerical evaluation of Riemann zeta function
///
/// Uses different algorithms depending on the value of s:
/// - Euler-Maclaurin acceleration for Re(s) > 1.5 (50 terms for 14-digit accuracy)
/// - Functional equation for Re(s) < 0
/// - Dirichlet eta relation for 0 < Re(s) < 1.5
///
/// # Mathematical Algorithm
///
/// For Re(s) > 1.5, uses Euler-Maclaurin acceleration:
/// ζ(s) = Σ(n=1 to N) 1/n^s + integral correction + Bernoulli corrections
///
/// For Re(s) < 0, uses the functional equation:
/// ζ(s) = 2^s π^(s-1) sin(πs/2) Γ(1-s) ζ(1-s)
pub fn zeta_numerical(s: f64) -> f64 {
    // Handle invalid inputs
    if s.is_nan() || s.is_infinite() {
        return f64::NAN;
    }

    // Handle special cases
    if (s - 1.0).abs() < 1e-10 {
        return f64::INFINITY;
    }

    if s.abs() < 1e-10 {
        return -0.5;
    }

    // For s > 1.5, use Euler-Maclaurin acceleration
    if s > 1.5 {
        zeta_euler_maclaurin(s)
    } else if s < 0.0 {
        zeta_functional_equation(s)
    } else {
        zeta_series_eta(s)
    }
}

/// Euler-Maclaurin acceleration for Re(s) > 1.5
///
/// Uses only 50 terms instead of 10,000 for 200x speedup while maintaining
/// 14-digit accuracy. The acceleration formula includes:
/// 1. Direct sum of first N terms
/// 2. Integral approximation for tail
/// 3. Bernoulli number corrections
///
/// Mathematical formula:
/// ζ(s) ≈ Σ(n=1 to N) 1/n^s + N^(1-s)/(s-1) + 1/(2N^s) + s/(12N^(s+1)) - s(s+1)(s+2)/(720N^(s+3))
fn zeta_euler_maclaurin(s: f64) -> f64 {
    const N: usize = 50; // Euler-Maclaurin needs only 50 terms for 14-digit accuracy

    // 1. Direct sum for first N terms
    let mut sum = 0.0;
    for n in 1..=N {
        sum += 1.0 / (n as f64).powf(s);
    }

    let n = N as f64;

    // 2. Integral approximation: ∫[N to ∞] x^(-s) dx = N^(1-s)/(s-1)
    let integral = n.powf(1.0 - s) / (s - 1.0);

    // 3. Bernoulli corrections (first 3 terms for accuracy)
    let correction1 = 1.0 / (2.0 * n.powf(s));
    let correction2 = s / (12.0 * n.powf(s + 1.0));
    let correction3 = -s * (s + 1.0) * (s + 2.0) / (720.0 * n.powf(s + 3.0));

    sum + integral + correction1 + correction2 + correction3
}

/// Dirichlet eta function relation for 0 < Re(s) < 1.5
///
/// Uses the relation: ζ(s) = η(s) / (1 - 2^(1-s))
/// where η(s) = Σ(n=1 to ∞) (-1)^(n-1) / n^s (alternating zeta)
///
/// Includes convergence check to stop when additional terms contribute
/// less than epsilon.
fn zeta_series_eta(s: f64) -> f64 {
    const N_TERMS: usize = 10000; // Maximum terms
    const EPSILON: f64 = 1e-14; // Convergence threshold (double precision limit)

    let mut eta = 0.0;
    let mut prev_eta = 0.0;

    for n in 1..=N_TERMS {
        let sign = if n % 2 == 1 { 1.0 } else { -1.0 };
        eta += sign / (n as f64).powf(s);

        // Check convergence after some iterations
        if n > 100 && (eta - prev_eta).abs() < EPSILON * eta.abs() {
            break;
        }

        prev_eta = eta;
    }

    let factor = 1.0 - 2.0_f64.powf(1.0 - s);
    eta / factor
}

/// Functional equation for Re(s) < 0
///
/// Uses the functional equation:
/// ζ(s) = 2^s π^(s-1) sin(πs/2) Γ(1-s) ζ(1-s)
///
/// This relates negative values to positive values where the series converges.
/// Uses high-accuracy Lanczos gamma function instead of Stirling approximation.
fn zeta_functional_equation(s: f64) -> f64 {
    let one_minus_s = 1.0 - s;
    let zeta_reflected = zeta_numerical(one_minus_s);

    let factor1 = 2.0_f64.powf(s);
    let factor2 = PI.powf(s - 1.0);
    let factor3 = (PI * s / 2.0).sin();
    let factor4 = lanczos_gamma(one_minus_s);

    factor1 * factor2 * factor3 * factor4 * zeta_reflected
}

#[cfg(test)]
mod tests {
    use super::*;

    #[test]
    fn test_zeta_2_basel_problem() {
        let result = zeta(&Expression::integer(2));

        // The result is a symbolic expression π²/6
        // Verify it's not a simple number
        match result {
            Expression::Number(_) => panic!("Expected symbolic expression for ζ(2)"),
            _ => {
                // Good - it's symbolic
            }
        }

        // Numerical verification
        let numerical = zeta_numerical(2.0);
        let pi_squared_over_6 = PI * PI / 6.0;
        assert!(
            (numerical - pi_squared_over_6).abs() < 1e-3,
            "ζ(2) numerically = {}, expected π²/6 ≈ {}",
            numerical,
            pi_squared_over_6
        );
    }

    #[test]
    fn test_zeta_0_exact() {
        let result = zeta(&Expression::integer(0));
        let expected = Expression::rational(-1, 2);
        assert_eq!(result, expected, "ζ(0) should equal -1/2");
    }

    #[test]
    fn test_zeta_negative_1() {
        let result = zeta(&Expression::integer(-1));
        let expected = Expression::rational(-1, 12);
        assert_eq!(result, expected, "ζ(-1) should equal -1/12");
    }

    #[test]
    fn test_zeta_4_exact() {
        let result = zeta(&Expression::integer(4));

        // Should be a symbolic expression π⁴/90
        match result {
            Expression::Number(_) => panic!("Expected symbolic expression for ζ(4)"),
            _ => {
                // Good - it's symbolic
            }
        }

        // Numerical verification with Euler-Maclaurin method
        let numerical = zeta_numerical(4.0);
        let pi_fourth_over_90 = PI.powi(4) / 90.0;
        assert!(
            (numerical - pi_fourth_over_90).abs() < 1e-6,
            "ζ(4) numerically = {}, expected π⁴/90 ≈ {}",
            numerical,
            pi_fourth_over_90
        );
    }

    #[test]
    fn test_zeta_numerical_convergence() {
        let val = zeta_numerical(3.0);
        let expected = 1.202064903159592;
        assert!(
            (val - expected).abs() < 1e-6,
            "ζ(3) = {}, expected ≈ {}",
            val,
            expected
        );
    }

    #[test]
    fn test_zeta_pole_at_1() {
        let result = zeta(&Expression::integer(1));

        match result {
            Expression::Function { name, args } => {
                assert_eq!(name, "zeta");
                assert_eq!(args.len(), 1);
            }
            _ => panic!("ζ(1) should remain symbolic (pole)"),
        }
    }

    #[test]
    fn test_zeta_negative_odd_integers() {
        let result = zeta(&Expression::integer(-3));
        let expected = Expression::rational(1, 120);
        assert_eq!(result, expected, "ζ(-3) should equal 1/120");
    }

    #[test]
    fn test_zeta_trivial_zeros() {
        // ζ(-2n) = 0 for positive integers n
        for n in 1..=10 {
            let result = zeta(&Expression::integer(-2 * n));
            assert_eq!(
                result,
                Expression::integer(0),
                "ζ({}) should be 0 (trivial zero)",
                -2 * n
            );
        }
    }

    #[test]
    fn test_zeta_symbolic_fallback() {
        let s = Expression::symbol(crate::core::Symbol::scalar("s"));
        let result = zeta(&s);

        match result {
            Expression::Function { name, args } => {
                assert_eq!(name, "zeta");
                assert_eq!(args.len(), 1);
            }
            _ => panic!("Expected symbolic function for variable input"),
        }
    }

    #[test]
    fn test_zeta_float_rounding() {
        let result = zeta(&Expression::Number(Number::Float(2.0)));

        // Should recognize 2.0 as integer and return symbolic form
        match result {
            Expression::Number(_) => {
                panic!("ζ(2.0) should be treated as ζ(2) and return symbolic")
            }
            _ => {
                // Good - treated as symbolic
            }
        }
    }

    #[test]
    fn test_zeta_large_argument() {
        let val = zeta_numerical(10.0);
        let expected = 1.000994575127818;
        assert!(
            (val - expected).abs() < 1e-9,
            "ζ(10) = {}, expected ≈ {}",
            val,
            expected
        );
    }

    #[test]
    fn test_zeta_8_exact() {
        let result = zeta(&Expression::integer(8));

        // Should be a symbolic expression π⁸/9450
        match result {
            Expression::Number(_) => panic!("Expected symbolic expression for ζ(8)"),
            _ => {
                // Good - it's symbolic
            }
        }
    }

    #[test]
    fn test_zeta_10_exact() {
        let result = zeta(&Expression::integer(10));

        // Should be a symbolic expression π¹⁰/93555
        match result {
            Expression::Number(_) => panic!("Expected symbolic expression for ζ(10)"),
            _ => {
                // Good - it's symbolic
            }
        }
    }

    #[test]
    fn test_zeta_negative_5() {
        let result = zeta(&Expression::integer(-5));
        let expected = Expression::rational(-1, 252);
        assert_eq!(result, expected, "ζ(-5) should equal -1/252");
    }

    #[test]
    fn test_zeta_negative_7() {
        let result = zeta(&Expression::integer(-7));
        let expected = Expression::rational(1, 240);
        assert_eq!(result, expected, "ζ(-7) should equal 1/240");
    }
}