Dirichlet’s theorem and PNT(AP) in the form used by the experiments

This page is the Phase 2 background for experiments about primes in residue classes and prime races. It states Dirichlet’s theorem and the prime number theorem in arithmetic progressions (PNT(AP)) exactly in the normalization used by the experiments:

  • baseline: \(\mathrm{li}(x)/\varphi(q)\),

  • error plots: \(\pi(x;q,a)-\mathrm{li}(x)/\varphi(q)\) (and derived race differences).

The counting function \(\pi(x;q,a)\)

Fix integers \(q\ge 1\) and \(a\). Define

\[ \pi(x;q,a) := \#\{p\le x : p\ \text{prime and}\ p\equiv a\pmod q\}. \]

Only reduced residue classes participate in the classical equidistribution theorems:

  • If \(\gcd(a,q)=1\), then the congruence class \(a\bmod q\) contains infinitely many primes.

  • If \(\gcd(a,q)>1\), then the class contains at most one prime (the common divisor itself), so it is not part of the PNT(AP) story.

Let

\[ (\mathbb{Z}/q\mathbb{Z})^\times = \{a\bmod q : \gcd(a,q)=1\} \]
denote the reduced residue system, and let \(\varphi(q)=\#(\mathbb{Z}/q\mathbb{Z})^\times\) be Euler’s totient.

The baseline \(\mathrm{li}(x)/\varphi(q)\)

The logarithmic integral is

\[ \mathrm{li}(x) := \operatorname{PV}\int_0^x \frac{dt}{\log t}. \]

In numerical work one often uses the “offset” version

\[ \mathrm{Li}(x) := \int_2^x \frac{dt}{\log t}, \]
and treats the two as essentially the same baseline for large \(x\) (they differ by a constant). Phase 2 uses the baseline in the form
\[ \text{baseline}(x;q) := \frac{\mathrm{li}(x)}{\varphi(q)}. \]

Interpretation: if primes are “evenly spread” across the \(\varphi(q)\) reduced residue classes, then each class should get about a \(1/\varphi(q)\) share of the total prime mass predicted by \(\mathrm{li}(x)\).

Dirichlet’s theorem (existence of primes in each reduced residue class)

Theorem (Dirichlet, 1837). If \(\gcd(a,q)=1\), then there are infinitely many primes \(p\) such that \(p\equiv a\pmod q\).

This is the minimal statement the experiments rely on: every reduced residue class shows up forever, so “race leaders” can change infinitely often in principle.

Why characters and \(L\)-functions enter (one paragraph)

The proof introduces Dirichlet characters \(\chi\bmod q\) and their \(L\)-functions

\[ L(s,\chi) = \sum_{n\ge 1}\frac{\chi(n)}{n^s}\qquad (\Re(s)>1), \]
and shows that for nonprincipal characters \(\chi\), the value \(L(1,\chi)\) is nonzero. Using character orthogonality, one can express the indicator of a residue class \(a\bmod q\) as an average over characters, which lets one “filter primes by congruence class” and prove that every reduced class contains infinitely many primes.

Prime number theorem in arithmetic progressions (equidistribution)

Dirichlet’s theorem says primes exist in each reduced class; PNT(AP) says they are asymptotically equidistributed.

Theorem (PNT(AP), fixed modulus form). Fix \(q\ge 1\) and let \(\gcd(a,q)=1\). Then, as \(x\to\infty\),

\[ \pi(x;q,a) \sim \frac{\mathrm{li}(x)}{\varphi(q)}. \]

Equivalently, the difference

\[ E(x;q,a) := \pi(x;q,a) - \frac{\mathrm{li}(x)}{\varphi(q)} \]
is “small compared to” \(\mathrm{li}(x)/\varphi(q)\) in the limit \(x\to\infty\).

The experiment’s error-term plots

Most Phase 2 plots are based on \(E(x;q,a)\). Typical visualizations include:

  • Raw error: plot \(E(x;q,a)\) vs \(x\).

  • Multiple residues: plot \(E(x;q,a)\) for several \(a\in(\mathbb{Z}/q\mathbb{Z})^\times\) on the same axes.

  • Race differences (baseline cancels): for two residues \(a,b\) define

    \[ \Delta(x;q,a,b) := \pi(x;q,a)-\pi(x;q,b) = E(x;q,a)-E(x;q,b). \]

So, race experiments are directly about comparing error terms across residues.

What size should \(E(x;q,a)\) have? (qualitative)

For the ranges used in computational experiments, it is normal that \(E(x;q,a)\) oscillates and does not look “small” pointwise. The key phenomenon is not monotone convergence but oscillation around the baseline.

A useful benchmark statement (not required for running the experiments, but helpful for interpretation) is: under GRH one expects roughly

\[ E(x;q,a) = O\!\left(\sqrt{x}\,\log x\right) \]
for fixed \(q\) (and more refined uniform statements are known with restrictions on \(q\)). This heuristic explains why, even when \(\pi(x;q,a)\) is close to the baseline in relative terms, the raw difference can still have visible swings.

How \(L\)-function zeros explain oscillations (high level)

A guiding principle is: zeros of Dirichlet \(L\)-functions drive oscillations in residue-class prime counts. Very roughly, character orthogonality lets you write “error terms in a class” as sums over contributions from nonprincipal characters, and analytic number theory relates those contributions to the zeros of \(L(s,\chi)\). You do not need to compute zeros for Phase 2, but this explains why race plots can show long-lasting biases and sign changes.

Practical numerical notes for experiments

  • Always restrict to \(a\) with \(\gcd(a,q)=1\) when discussing equidistribution or races.

  • Sampling matters: using a linear grid in \(x\) vs a log grid in \(x\) weights different ranges differently and can change “leader fractions”.

  • If you approximate \(\mathrm{li}(x)\) numerically, document the method and keep it consistent across experiments (baseline consistency matters more than ultimate accuracy for qualitative plots).

References

See References.

[Davenport, 2000, Iwaniec and Kowalski, 2004, Lejeune Dirichlet, 1837, Montgomery and Vaughan, 2006]

Experiments in this repository

  • E070 — \(\pi(x;q,a)\) for several reduced residue classes (baseline comparison).

  • E071 — \(E(x;q,a)=\pi(x;q,a)-\mathrm{li}(x)/\varphi(q)\) error-term plots.

  • E072–E075 — Prime races and race distributions (differences of \(\pi(x;q,a)\)).

  • E081 — Effect of modulus on PNT(AP) error behavior (comparative error plots).