mathxlab.experiments.number_theory_suite

Number theory experiment suite (E094–E123).

Quickstart

from mathxlab.experiments.number_theory_suite import run_e094, run_e095, ParamsE094

Public API

Name	Kind	Summary
ParamsE094	class	Parameters for E094.
run_e094	function	E094: ω(n) vs Ω(n): Erdős–Kac side-by-side.
ParamsE095	class	Parameters for E095.
run_e095	function	E095: Squarefree filter: ω(n)=Ω(n) when μ(n)≠0.
ParamsE096	class	Parameters for E096.
run_e096	function	E096: Record-holders for τ(n).
ParamsE097	class	Parameters for E097.
run_e097	function	E097: σ(n)/n landscape: deficient, perfect, abundant.
ParamsE098	class	Parameters for E098.
run_e098	function	E098: Extremals of σ(n)/n^α across α.
ParamsE099	class	Parameters for E099.
run_e099	function	E099: Jordan totients J_k: atlas and identities.
ParamsE100	class	Parameters for E100.
run_e100	function	E100: Carmichael λ(n) vs Euler φ(n).
ParamsE101	class	Parameters for E101.
run_e101	function	E101: Reduced residues: (Z/qZ)^× as a concrete set.
ParamsE102	class	Parameters for E102.
run_e102	function	E102: Dirichlet convolution identity zoo.
ParamsE103	class	Parameters for E103.
run_e103	function	E103: Chebyshev ψ(x): prime powers drive the jumps.
ParamsE104	class	Parameters for E104.
run_e104	function	E104: Von Mangoldt Λ: support and statistics.
ParamsE105	class	Parameters for E105.
run_e105	function	E105: Mertens function M(x): scaling views.
ParamsE106	class	Parameters for E106.
run_e106	function	E106: Real vs complex Dirichlet characters (small moduli).
ParamsE107	class	Parameters for E107.
run_e107	function	E107: Conductor distribution for characters.
ParamsE108	class	Parameters for E108.
run_e108	function	E108: Character orthogonality matrix heatmap.
ParamsE109	class	Parameters for E109.
run_e109	function	E109: Gauss sums τ(χ) for prime modulus.
ParamsE110	class	Parameters for E110.
run_e110	function	E110: Dirichlet L-series partial sums (toy).
ParamsE111	class	Parameters for E111.
run_e111	function	E111: Euler product vs Dirichlet series for L(s,χ).
ParamsE112	class	Parameters for E112.
run_e112	function	E112: Prime race π(x;q,a) − π(x;q,b).
ParamsE113	class	Parameters for E113.
run_e113	function	E113: First prime in each reduced residue class.
ParamsE114	class	Parameters for E114.
run_e114	function	E114: ζ(1/2+it) via η-series: truncation error curves.
ParamsE115	class	Parameters for E115.
run_e115	function	E115: Hardy Z(t): sign-change scan near low zeros.
ParamsE116	class	Parameters for E116.
run_e116	function	E116: Zero-count proxy vs Riemann–von Mangoldt main term.
ParamsE117	class	Parameters for E117.
run_e117	function	E117: Functional equation check: ζ(s) ≈ χ(s) ζ(1−s).
ParamsE118	class	Parameters for E118.
run_e118	function	E118: Euler product approximation vs ζ(s): where it degrades.
ParamsE119	class	Parameters for E119.
run_e119	function	E119: ψ(x)−x with simple smoothing.
ParamsE120	class	Parameters for E120.
run_e120	function	E120: Pretentious distance toy: sum_{p<=x} (1−Re χ(p))/p.
ParamsE121	class	Parameters for E121.
run_e121	function	E121: Multiplicativity stress tests (random coprime pairs).
ParamsE122	class	Parameters for E122.
run_e122	function	E122: Arithmetic function atlas: μ(n), ω(n), φ(n)/n heatmaps.
ParamsE123	class	Parameters for E123.
run_e123	function	E123: Correlation matrix: arithmetic functions on 1..N.

Reference

Classes

class mathxlab.experiments.number_theory_suite.ParamsE094(n_max=200000, max_k=12)

Bases: object

Parameters for E094.

Parameters:

• n_max – Upper bound N (inclusive).

• max_k – Maximum factor-count bin to display.

class mathxlab.experiments.number_theory_suite.ParamsE095(n_max=200000, max_k=12)

Bases: object

Parameters for E095.

Parameters:

• n_max – Upper bound N (inclusive).

• max_k – Maximum factor-count bin to display.

class mathxlab.experiments.number_theory_suite.ParamsE096(n_max=300000, max_points=150)

Bases: object

Parameters for E096.

Parameters:

• n_max – Upper bound N (inclusive).

• max_points – Maximum number of record points to include in the plot.

class mathxlab.experiments.number_theory_suite.ParamsE097(n_max=200000, stride=20)

Bases: object

Parameters for E097.

Parameters:

• n_max – Upper bound N (inclusive).

• stride – Downsampling stride for scatter plotting.

class mathxlab.experiments.number_theory_suite.ParamsE098(n_max=60000, alpha_min=0.5, alpha_max=2.0, alpha_steps=40)

Bases: object

Parameters for E098.

Parameters:

• n_max – Upper bound N (inclusive).

• alpha_min – Minimum alpha value.

• alpha_max – Maximum alpha value.

• alpha_steps – Number of alpha grid points.

class mathxlab.experiments.number_theory_suite.ParamsE099(n_max=30000, k_max=4, stride=10)

Bases: object

Parameters for E099.

Parameters:

• n_max – Upper bound N (inclusive).

• k_max – Maximum k for J_k.

• stride – Downsampling stride for scatter plotting.

class mathxlab.experiments.number_theory_suite.ParamsE100(n_max=25000, max_ratio=1.0, bins=40)

Bases: object

Parameters for E100.

Parameters:

• n_max – Upper bound N (inclusive).

• max_ratio – Max ratio bin for histogram (values clipped).

• bins – Number of histogram bins.

class mathxlab.experiments.number_theory_suite.ParamsE101(q_values=(10, 12, 15, 21, 30))

Bases: object

Parameters for E101.

Parameters:

q_values – Moduli to inspect.

class mathxlab.experiments.number_theory_suite.ParamsE102(n_max=20000)

Bases: object

Parameters for E102.

Parameters:

n_max – Maximum n for convolution checks.

class mathxlab.experiments.number_theory_suite.ParamsE103(n_max=200000, stride=5)

Bases: object

Parameters for E103.

Parameters:

• n_max – Upper bound N (inclusive).

• stride – Downsampling stride for plotting.

class mathxlab.experiments.number_theory_suite.ParamsE104(n_max=200000, bins=30)

Bases: object

Parameters for E104.

Parameters:

• n_max – Upper bound N (inclusive).

• bins – Histogram bins.

class mathxlab.experiments.number_theory_suite.ParamsE105(n_max=200000, stride=5)

Bases: object

Parameters for E105.

Parameters:

• n_max – Upper bound N (inclusive).

• stride – Downsampling stride for plotting.

class mathxlab.experiments.number_theory_suite.ParamsE106(q_values=(5, 8, 10, 12, 15))

Bases: object

Parameters for E106.

Parameters:

q_values – Moduli to inspect.

class mathxlab.experiments.number_theory_suite.ParamsE107(q=60)

Bases: object

Parameters for E107.

Parameters:

q – Composite modulus to inspect.

class mathxlab.experiments.number_theory_suite.ParamsE108(q=15)

Bases: object

Parameters for E108.

Parameters:

q – Modulus (small, to keep the matrix readable).

class mathxlab.experiments.number_theory_suite.ParamsE109(q=11)

Bases: object

Parameters for E109.

Parameters:

q – Prime modulus.

class mathxlab.experiments.number_theory_suite.ParamsE110(q=5, n_max=5000, s_re=1.0)

Bases: object

Parameters for E110.

Parameters:

• q – Modulus.

• n_max – Truncation for the Dirichlet series.

• s_re – Real part of s for convergent evaluation.

class mathxlab.experiments.number_theory_suite.ParamsE111(q=5, s_re=1.5, n_series=8000, p_max=2000, p_steps=30)

Bases: object

Parameters for E111.

Parameters:

• q – Modulus.

• s_re – Real part of s (>1).

• n_series – Truncation for Dirichlet series.

• p_max – Max prime for Euler product.

• p_steps – Number of prime cutoffs to test.

class mathxlab.experiments.number_theory_suite.ParamsE112(q=8, a=1, b=3, p_max=800000)

Bases: object

Parameters for E112.

Parameters:

• q – Modulus.

• a – First residue (must be coprime to q).

• b – Second residue (must be coprime to q).

• p_max – Max prime to include.

class mathxlab.experiments.number_theory_suite.ParamsE113(q=30, p_max=200000)

Bases: object

Parameters for E113.

Parameters:

• q – Modulus.

• p_max – Search primes up to this bound.

class mathxlab.experiments.number_theory_suite.ParamsE114(t_max=30.0, t_steps=61, n_terms_values=(100, 200, 400), ref_dps=70, eval_dps=50)

Bases: object

Parameters for E114.

Parameters:

• t_max – Max height t.

• t_steps – Number of t samples.

• n_terms_values – Eta truncation sizes.

• ref_dps – mpmath precision for reference zeta.

• eval_dps – mpmath precision for eta evaluation.

class mathxlab.experiments.number_theory_suite.ParamsE115(t_max=40.0, dt=0.1, dps=40)

Bases: object

Parameters for E115.

Parameters:

• t_max – Max t to scan.

• dt – Step size in t.

• dps – mpmath precision used by hardy_Z.

class mathxlab.experiments.number_theory_suite.ParamsE116(t_max=120.0, dt=0.2, dps=40)

Bases: object

Parameters for E116.

Parameters:

• t_max – Max t to scan.

• dt – Step size in t.

• dps – mpmath precision used by hardy_Z.

class mathxlab.experiments.number_theory_suite.ParamsE117(t_max=25.0, t_steps=60, sigma=0.2, dps=70)

Bases: object

Parameters for E117.

Parameters:

• t_max – Max imaginary part t.

• t_steps – Number of t samples.

• sigma – Real part σ for s=σ+it.

• dps – mpmath precision for evaluation.

class mathxlab.experiments.number_theory_suite.ParamsE118(p_max=50000, p_steps=40, dps=60, s_values=(1.2 + 0j, 1.05 + 0j, 0.9 + 0j))

Bases: object

Parameters for E118.

Parameters:

• p_max – Max prime for Euler product.

• p_steps – Number of prime cutoffs.

• dps – mpmath precision.

• s_values – s values to compare.

class mathxlab.experiments.number_theory_suite.ParamsE119(n_max=300000, stride=10, window=500)

Bases: object

Parameters for E119.

Parameters:

• n_max – Upper bound N (inclusive).

• stride – Downsampling stride for plotting.

• window – Smoothing window (moving average) for ψ(x)−x.

class mathxlab.experiments.number_theory_suite.ParamsE120(q=8, p_max=400000)

Bases: object

Parameters for E120.

Parameters:

• q – Modulus.

• p_max – Max prime to include.

class mathxlab.experiments.number_theory_suite.ParamsE121(n_max=100000, pairs=3000)

Bases: object

Parameters for E121.

Parameters:

• n_max – Max n for function tables.

• pairs – Number of random coprime pairs to test.

class mathxlab.experiments.number_theory_suite.ParamsE122(side=256)

Bases: object

Parameters for E122.

Parameters:

side – Side length for the atlas grid (side^2 numbers).

class mathxlab.experiments.number_theory_suite.ParamsE123(n_max=80000, stride=1)

Bases: object

Parameters for E123.

Parameters:

• n_max – Upper bound N (inclusive).

• stride – Stride for sampling (increase to speed up).

Functions

mathxlab.experiments.number_theory_suite.run_e094(*, out_dir, seed, figures_dir, report_path, params_path)

E094: ω(n) vs Ω(n): Erdős–Kac side-by-side.

mathxlab.experiments.number_theory_suite.run_e095(*, out_dir, seed, figures_dir, report_path, params_path)

E095: Squarefree filter: ω(n)=Ω(n) when μ(n)≠0.

mathxlab.experiments.number_theory_suite.run_e096(*, out_dir, seed, figures_dir, report_path, params_path)

E096: Record-holders for τ(n).

mathxlab.experiments.number_theory_suite.run_e097(*, out_dir, seed, figures_dir, report_path, params_path)

E097: σ(n)/n landscape: deficient, perfect, abundant.

mathxlab.experiments.number_theory_suite.run_e098(*, out_dir, seed, figures_dir, report_path, params_path)

E098: Extremals of σ(n)/n^α across α.

mathxlab.experiments.number_theory_suite.run_e099(*, out_dir, seed, figures_dir, report_path, params_path)

E099: Jordan totients J_k: atlas and identities.

mathxlab.experiments.number_theory_suite.run_e100(*, out_dir, seed, figures_dir, report_path, params_path)

E100: Carmichael λ(n) vs Euler φ(n).

mathxlab.experiments.number_theory_suite.run_e101(*, out_dir, seed, figures_dir, report_path, params_path)

E101: Reduced residues: (Z/qZ)^× as a concrete set.

mathxlab.experiments.number_theory_suite.run_e102(*, out_dir, seed, figures_dir, report_path, params_path)

E102: Dirichlet convolution identity zoo.

mathxlab.experiments.number_theory_suite.run_e103(*, out_dir, seed, figures_dir, report_path, params_path)

E103: Chebyshev ψ(x): prime powers drive the jumps.

mathxlab.experiments.number_theory_suite.run_e104(*, out_dir, seed, figures_dir, report_path, params_path)

E104: Von Mangoldt Λ: support and statistics.

mathxlab.experiments.number_theory_suite.run_e105(*, out_dir, seed, figures_dir, report_path, params_path)

E105: Mertens function M(x): scaling views.

mathxlab.experiments.number_theory_suite.run_e106(*, out_dir, seed, figures_dir, report_path, params_path)

E106: Real vs complex Dirichlet characters (small moduli).

mathxlab.experiments.number_theory_suite.run_e107(*, out_dir, seed, figures_dir, report_path, params_path)

E107: Conductor distribution for characters.

mathxlab.experiments.number_theory_suite.run_e108(*, out_dir, seed, figures_dir, report_path, params_path)

E108: Character orthogonality matrix heatmap.

mathxlab.experiments.number_theory_suite.run_e109(*, out_dir, seed, figures_dir, report_path, params_path)

E109: Gauss sums τ(χ) for prime modulus.

mathxlab.experiments.number_theory_suite.run_e110(*, out_dir, seed, figures_dir, report_path, params_path)

E110: Dirichlet L-series partial sums (toy).

mathxlab.experiments.number_theory_suite.run_e111(*, out_dir, seed, figures_dir, report_path, params_path)

E111: Euler product vs Dirichlet series for L(s,χ).

mathxlab.experiments.number_theory_suite.run_e112(*, out_dir, seed, figures_dir, report_path, params_path)

E112: Prime race π(x;q,a) − π(x;q,b).

mathxlab.experiments.number_theory_suite.run_e113(*, out_dir, seed, figures_dir, report_path, params_path)

E113: First prime in each reduced residue class.

mathxlab.experiments.number_theory_suite.run_e114(*, out_dir, seed, figures_dir, report_path, params_path)

E114: ζ(1/2+it) via η-series: truncation error curves.

mathxlab.experiments.number_theory_suite.run_e115(*, out_dir, seed, figures_dir, report_path, params_path)

E115: Hardy Z(t): sign-change scan near low zeros.

mathxlab.experiments.number_theory_suite.run_e116(*, out_dir, seed, figures_dir, report_path, params_path)

E116: Zero-count proxy vs Riemann–von Mangoldt main term.

mathxlab.experiments.number_theory_suite.run_e117(*, out_dir, seed, figures_dir, report_path, params_path)

E117: Functional equation check: ζ(s) ≈ χ(s) ζ(1−s).

mathxlab.experiments.number_theory_suite.run_e118(*, out_dir, seed, figures_dir, report_path, params_path)

E118: Euler product approximation vs ζ(s): where it degrades.

mathxlab.experiments.number_theory_suite.run_e119(*, out_dir, seed, figures_dir, report_path, params_path)

E119: ψ(x)−x with simple smoothing.

mathxlab.experiments.number_theory_suite.run_e120(*, out_dir, seed, figures_dir, report_path, params_path)

E120: Pretentious distance toy: sum_{p<=x} (1−Re χ(p))/p.

mathxlab.experiments.number_theory_suite.run_e121(*, out_dir, seed, figures_dir, report_path, params_path)

E121: Multiplicativity stress tests (random coprime pairs).

mathxlab.experiments.number_theory_suite.run_e122(*, out_dir, seed, figures_dir, report_path, params_path)

E122: Arithmetic function atlas: μ(n), ω(n), φ(n)/n heatmaps.

mathxlab.experiments.number_theory_suite.run_e123(*, out_dir, seed, figures_dir, report_path, params_path)

E123: Correlation matrix: arithmetic functions on 1..N.