zeta.h File Reference

Riemann zeta evaluation helpers with branch metadata and error estimates. More...

#include <stddef.h>
#include "oakfield/field.h"

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Classes
struct	SimComplexBall
	Complex ball enclosure with evaluator status, rigor, and refinement metadata. More...
struct	SimZetaContext
	Evaluation controls for the Riemann zeta dispatcher. More...
struct	SimZetaResult
	Value, error estimates, branch metadata, and status from zeta evaluation. More...
struct	SimZetaDerivativeResult
	Zeta value and first derivative with associated error and branch metadata. More...

Macros
#define	SIM_ZETA_FLAG_USED_REFLECTION   0x1u
#define	SIM_ZETA_FLAG_NEAR_POLE   0x2u
#define	SIM_ZETA_FLAG_TRIVIAL_ZERO   0x4u
#define	SIM_ZETA_FLAG_USED_ETA   0x8u
#define	SIM_ZETA_FLAG_USED_NEAR_ONE_LAURENT   0x10u
#define	SIM_ZETA_FLAG_USED_AFE   0x20u
#define	SIM_ZETA_FLAG_USED_ADAPTIVE_REFINEMENT   0x40u
#define	SIM_ZETA_FLAG_USED_RIEMANN_SIEGEL   0x80u
#define	SIM_ZETA_FLAG_USED_LOCAL_EXPANSION   0x100u
#define	SIM_ZETA_FLAG_ZERO_PROXIMITY   0x200u

Typedefs
typedef enum SimZetaStatus	SimZetaStatus
	Status values returned by zeta evaluators.
typedef enum SimBallRigor	SimBallRigor
	Rigor level attached to complex-ball enclosures.
typedef enum SimZetaBranch	SimZetaBranch
	Dispatcher branch used for a zeta evaluation.
typedef struct SimComplexBall	SimComplexBall
	Complex ball enclosure with evaluator status, rigor, and refinement metadata.
typedef struct SimZetaContext	SimZetaContext
	Evaluation controls for the Riemann zeta dispatcher.
typedef struct SimZetaResult	SimZetaResult
	Value, error estimates, branch metadata, and status from zeta evaluation.
typedef struct SimZetaDerivativeResult	SimZetaDerivativeResult
	Zeta value and first derivative with associated error and branch metadata.

Enumerations
enum	SimZetaStatus {   SIM_ZETA_STATUS_OK = 0 , SIM_ZETA_STATUS_SINGULAR , SIM_ZETA_STATUS_INVALID_ARGUMENT , SIM_ZETA_STATUS_NO_CONVERGENCE ,   SIM_ZETA_STATUS_NUMERIC_FAILURE }
	Status values returned by zeta evaluators. More...
enum	SimBallRigor { SIM_BALL_RIGOR_HEURISTIC = 0 , SIM_BALL_RIGOR_VALIDATED , SIM_BALL_RIGOR_FORMAL }
	Rigor level attached to complex-ball enclosures. More...
enum	SimZetaBranch {   SIM_ZETA_BRANCH_DIRECT_EULER_MACLAURIN = 0 , SIM_ZETA_BRANCH_ETA_ACCELERATED , SIM_ZETA_BRANCH_APPROXIMATE_FUNCTIONAL_EQUATION , SIM_ZETA_BRANCH_RIEMANN_SIEGEL ,   SIM_ZETA_BRANCH_NEAR_ONE_LAURENT , SIM_ZETA_BRANCH_LOCAL_EXPANSION , SIM_ZETA_BRANCH_REFLECTION }
	Dispatcher branch used for a zeta evaluation. More...

Functions
const char *	sim_zeta_status_string (SimZetaStatus status)
	Human-readable description of a zeta status value.
const char *	sim_zeta_branch_string (SimZetaBranch branch)
	Human-readable description of a zeta branch value.
SimZetaContext	sim_zeta_context_default (void)
	Default tolerances and truncation limits for the current zeta dispatcher.
SimZetaContext	sim_zeta_context_interactive (void)
	Low-latency zeta context for interactive visual exploration.
SimZetaResult	sim_zeta_eval_direct_euler_maclaurin (SimComplexDouble s, const SimZetaContext *context)
	Evaluate zeta(s) with the direct Euler-Maclaurin branch.
SimZetaResult	sim_zeta_eval_eta_accelerated (SimComplexDouble s, const SimZetaContext *context)
	Evaluate zeta(s) with the Phase 2 accelerated eta/Hasse branch.
SimZetaResult	sim_zeta_eval_approximate_fe (SimComplexDouble s, const SimZetaContext *context)
	Evaluate zeta(s) with the Phase 3 approximate functional equation branch.
SimZetaResult	sim_zeta_eval_riemann_siegel (double t, const SimZetaContext *context)
	Evaluate zeta(1/2 + i t) with the Phase 4 Riemann-Siegel branch.
SimZetaDerivativeResult	sim_zeta_eval_with_derivative (SimComplexDouble s, const SimZetaContext *context)
	Evaluate zeta(s) and its first complex derivative.
SimComplexBall	sim_zeta_eval_ball (SimComplexDouble s, const SimZetaContext *context)
	Ball-style enclosure derived from the current zeta error model.
SimZetaResult	sim_zeta_eval (SimComplexDouble s, const SimZetaContext *context)
	Evaluate zeta(s) using the Phase 3 dispatcher and adaptive refinement layer.

Detailed Description

Riemann zeta evaluation helpers with branch metadata and error estimates.

Enumeration Type Documentation

SimBallRigor

enum SimBallRigor

Rigor level attached to complex-ball enclosures.

Enumerator
SIM_BALL_RIGOR_HEURISTIC	Heuristic error bounds only.
SIM_BALL_RIGOR_VALIDATED	Bounds validated by runtime checks.
SIM_BALL_RIGOR_FORMAL	Formally justified enclosure.

SimZetaBranch

enum SimZetaBranch

Dispatcher branch used for a zeta evaluation.

Enumerator
SIM_ZETA_BRANCH_DIRECT_EULER_MACLAURIN	Direct Euler-Maclaurin summation branch.
SIM_ZETA_BRANCH_ETA_ACCELERATED	Eta-accelerated series branch.
SIM_ZETA_BRANCH_APPROXIMATE_FUNCTIONAL_EQUATION	Approximate functional equation.
SIM_ZETA_BRANCH_RIEMANN_SIEGEL	Riemann-Siegel branch.
SIM_ZETA_BRANCH_NEAR_ONE_LAURENT	Laurent expansion near s=1.
SIM_ZETA_BRANCH_LOCAL_EXPANSION	Local expansion around a cached center.
SIM_ZETA_BRANCH_REFLECTION	Reflection/formula-transformed branch.

SimZetaStatus

enum SimZetaStatus

Status values returned by zeta evaluators.

Enumerator
SIM_ZETA_STATUS_OK	Evaluation completed successfully.
SIM_ZETA_STATUS_SINGULAR	Argument lies on a singularity or pole.
SIM_ZETA_STATUS_INVALID_ARGUMENT	Input argument or output pointer was invalid.
SIM_ZETA_STATUS_NO_CONVERGENCE	Series/refinement did not converge within budget.
SIM_ZETA_STATUS_NUMERIC_FAILURE	Evaluation produced an unstable numeric result.

Function Documentation

sim_zeta_branch_string()

const char * sim_zeta_branch_string

(

SimZetaBranch

branch

)

Human-readable description of a zeta branch value.

Parameters

branch

Branch identifier from a zeta result.

Returns

Static diagnostic string.

sim_zeta_context_default()

SimZetaContext sim_zeta_context_default

(

void

)

Default tolerances and truncation limits for the current zeta dispatcher.

Returns

Default zeta evaluation context.

sim_zeta_context_interactive()

SimZetaContext sim_zeta_context_interactive

(

void

)

Low-latency zeta context for interactive visual exploration.

This keeps the existing dispatcher but relaxes tolerances, caps work budgets, and disables validation so renderers can use sim_zeta_eval() for responsive previews without changing the exact path.

Returns

Relaxed low-latency zeta evaluation context.

sim_zeta_eval()

SimZetaResult sim_zeta_eval	(	SimComplexDouble	s,
		const SimZetaContext *	context
	)

Evaluate zeta(s) using the Phase 3 dispatcher and adaptive refinement layer.

Parameters

s	Complex zeta argument.
context	Optional evaluation context.

Returns

Structured zeta result from the adaptive dispatcher.

sim_zeta_eval_approximate_fe()

SimZetaResult sim_zeta_eval_approximate_fe	(	SimComplexDouble	s,
		const SimZetaContext *	context
	)

Evaluate zeta(s) with the Phase 3 approximate functional equation branch.

Parameters

s	Complex zeta argument.
context	Optional evaluation context.

Returns

Structured zeta result for the approximate functional equation branch.

sim_zeta_eval_ball()

SimComplexBall sim_zeta_eval_ball	(	SimComplexDouble	s,
		const SimZetaContext *	context
	)

Ball-style enclosure derived from the current zeta 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 special values and direct-Euler-Maclaurin right-half-plane paths, including a genuinely complex family with Re(s) > 1.

Parameters

s	Complex zeta argument.
context	Optional evaluation context.

Returns

Complex ball enclosure with status and rigor metadata.

sim_zeta_eval_direct_euler_maclaurin()

SimZetaResult sim_zeta_eval_direct_euler_maclaurin	(	SimComplexDouble	s,
		const SimZetaContext *	context
	)

Evaluate zeta(s) with the direct Euler-Maclaurin branch.

Parameters

s	Complex zeta argument.
context	Optional evaluation context.

Returns

Structured zeta result for the direct branch.

sim_zeta_eval_eta_accelerated()

SimZetaResult sim_zeta_eval_eta_accelerated	(	SimComplexDouble	s,
		const SimZetaContext *	context
	)

Evaluate zeta(s) with the Phase 2 accelerated eta/Hasse branch.

Parameters

s	Complex zeta argument.
context	Optional evaluation context.

Returns

Structured zeta result for the eta/Hasse branch.

sim_zeta_eval_riemann_siegel()

SimZetaResult sim_zeta_eval_riemann_siegel	(	double	t,
		const SimZetaContext *	context
	)

Evaluate zeta(1/2 + i t) with the Phase 4 Riemann-Siegel branch.

Parameters

t	Critical-line height.
context	Optional evaluation context.

Returns

Structured zeta result on the critical line.

sim_zeta_eval_with_derivative()

SimZetaDerivativeResult sim_zeta_eval_with_derivative	(	SimComplexDouble	s,
		const SimZetaContext *	context
	)

Evaluate zeta(s) and its first complex derivative.

Parameters

s	Complex zeta argument.
context	Optional evaluation context.

Returns

Structured value and derivative result.

sim_zeta_status_string()

const char * sim_zeta_status_string

(

SimZetaStatus

status

)

Human-readable description of a zeta status value.

Parameters

status

Status returned by zeta/ball evaluators.

Returns

Static diagnostic string.