xi.h File Reference

Completed Riemann xi evaluation helpers. More...

#include <stddef.h>
#include "zeta.h"

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Classes
struct	SimXiContext
	Evaluation controls for completed Riemann xi calculations. More...
struct	SimXiResult
	Value, error estimates, branch metadata, and status from xi evaluation. More...
struct	SimXiDerivativeResult
	Xi value and first derivative with associated error and branch metadata. More...
struct	SimXiZeroResult
	Critical-line zero-search result for the completed xi function. More...

Macros
#define	SIM_XI_FLAG_USED_LOG_ASSEMBLY   0x1u
#define	SIM_XI_FLAG_USED_REFLECTION   0x2u
#define	SIM_XI_FLAG_EXACT_LIMIT   0x4u
#define	SIM_XI_FLAG_USED_NEAR_ONE_EXPANSION   0x8u
#define	SIM_XI_FLAG_USED_LOCAL_EXPANSION   0x10u
#define	SIM_XI_FLAG_USED_ZERO_REFINEMENT   0x20u
#define	SIM_XI_FLAG_ZERO_BRACKET_VALIDATED   0x40u
#define	SIM_XI_FLAG_ZERO_BRACKET_FORMAL   0x80u

Typedefs
typedef struct SimXiContext	SimXiContext
	Evaluation controls for completed Riemann xi calculations.
typedef struct SimXiResult	SimXiResult
	Value, error estimates, branch metadata, and status from xi evaluation.
typedef struct SimXiDerivativeResult	SimXiDerivativeResult
	Xi value and first derivative with associated error and branch metadata.
typedef struct SimXiZeroResult	SimXiZeroResult
	Critical-line zero-search result for the completed xi function.

Functions
SimXiContext	sim_xi_context_default (void)
	Default tolerances and limit-handling thresholds for the Phase 3 xi evaluator.
SimXiContext	sim_xi_context_interactive (void)
	Low-latency xi context for interactive visual exploration.
const char *	sim_xi_status_string (SimZetaStatus status)
	Human-readable description of an xi status value.
SimXiResult	sim_xi_eval (SimComplexDouble s, const SimXiContext *context)
	Evaluate the completed xi function.
SimXiResult	sim_xi_eval_critical_line (double t, const SimXiContext *context)
	Evaluate xi on the critical line.
SimXiDerivativeResult	sim_xi_eval_with_derivative (SimComplexDouble s, const SimXiContext *context)
	Evaluate xi(s) and its first complex derivative.
SimComplexBall	sim_xi_eval_ball (SimComplexDouble s, const SimXiContext *context)
	Ball-style enclosure derived from the current xi error model.
SimXiZeroResult	sim_xi_find_critical_zero (double t_lo, double t_hi, const SimXiContext *context)
	Heuristically locate a critical-line zero of Xi(t) inside a bracket with a sign change.

Detailed Description

Completed Riemann xi evaluation helpers.

Function Documentation

sim_xi_context_default()

SimXiContext sim_xi_context_default

(

void

)

Default tolerances and limit-handling thresholds for the Phase 3 xi evaluator.

Returns

Default xi evaluation context.

sim_xi_context_interactive()

SimXiContext sim_xi_context_interactive

(

void

)

Low-latency xi context for interactive visual exploration.

This pairs xi with sim_zeta_context_interactive() and relaxed tolerances so renderers can use sim_xi_eval() for responsive previews while keeping the exact/certified path separate.

Returns

Relaxed low-latency xi evaluation context.

sim_xi_eval()

SimXiResult sim_xi_eval	(	SimComplexDouble	s,
		const SimXiContext *	context
	)

Evaluate the completed xi function.

Parameters

s	Complex xi argument.
context	Optional xi context.

Returns

Structured xi value, errors, zeta branch, status, and flags.

sim_xi_eval_ball()

SimComplexBall sim_xi_eval_ball	(	SimComplexDouble	s,
		const SimXiContext *	context
	)

Ball-style enclosure derived from the current xi error model.

The returned rigor metadata distinguishes heuristic, validated, and future formal interval results. The current implementation provides heuristic and cross-checked validated balls, plus formal enclosures for exact xi limits and real integer arguments whose reflected zeta ball is formal. Finite NO_CONVERGENCE evaluations are exposed as heuristic balls so callers can still use decisive sign information without treating it as certification.

Parameters

s	Complex xi argument.
context	Optional xi context.

Returns

Complex ball enclosure with status and rigor metadata.

sim_xi_eval_critical_line()

SimXiResult sim_xi_eval_critical_line	(	double	t,
		const SimXiContext *	context
	)

Evaluate xi on the critical line.

Parameters

t	Critical-line height in s = 1/2 + i t.
context	Optional xi context.

Returns

Structured xi result on the critical line.

sim_xi_eval_with_derivative()

SimXiDerivativeResult sim_xi_eval_with_derivative	(	SimComplexDouble	s,
		const SimXiContext *	context
	)

Evaluate xi(s) and its first complex derivative.

Parameters

s	Complex xi argument.
context	Optional xi context.

Returns

Structured value and derivative result.

sim_xi_find_critical_zero()

SimXiZeroResult sim_xi_find_critical_zero	(	double	t_lo,
		double	t_hi,
		const SimXiContext *	context
	)

Heuristically locate a critical-line zero of Xi(t) inside a bracket with a sign change.

SIM_ZETA_STATUS_OK indicates a refined zero candidate supported by a validated-or-better sign-changing bracket. bracket_rigor reports the strength of that bracket explicitly, and SIM_XI_FLAG_USED_ZERO_REFINEMENT only means a derivative-based refinement step ran; it is not itself a certification flag. SIM_ZETA_STATUS_NO_CONVERGENCE may still carry a finite best-effort candidate obtained from heuristic balls.

Parameters

t_lo	Lower critical-line height.
t_hi	Upper critical-line height; must exceed t_lo.
context	Optional xi context.

Returns

Zero-search result with candidate, ball, derivative, and rigor.

sim_xi_status_string()

const char * sim_xi_status_string

(

SimZetaStatus

status

)

Human-readable description of an xi status value.

Parameters

status

Status returned by xi evaluators.

Returns

Static diagnostic string.