The paper

Claims

• C1. The least $x$ at which the primes up to $x$ congruent to $1$ modulo $4$ outnumber those congruent to $3$ is 26861.
• C2. Of the integers $1 \le x \le 10^8$, the class of $1$ modulo $4$ leads at 30624, in 128 stretches, the last ending at x = 12382326; the two classes are tied at 3866 integers, the last being x = 12424002.
• C3. Weighting each $x$ by $1/x$, the class of $1$ modulo $4$ leads on a share 0.00040587 of the race up to $10^8$, about a tenth of the limiting share near implied by Rubinstein and Sarnak's logarithmic density of for the class of .
• C4. For every $x$ from $2$ to $10^8$, the primes up to $x$ congruent to $2$ modulo $3$ strictly outnumber those congruent to , the classes being tied only at x = .
• C5. Up to $10^8$ there are 2880950 primes congruent to $3$ modulo $4$ and 2880504 congruent to $1$, and 2880937 congruent to $2$ modulo and congruent to .

Methods

Write $\pi(x; q, a)$ for the number of primes $p \le x$ with $p \equiv a \pmod q$. The code finds every prime up to $N = 10^8$ with a sieve of Eratosthenes over the odd numbers, then walks through them in order, following the leads $d(x) = \pi(x; 4, 3) - \pi(x; 4, 1)$ and $e(x) = \pi(x; 3, 2) - \pi(x; 3, 1)$. Between consecutive primes a lead is constant, so each lead is known at every integer from 1 to $N$ without visiting each one. For each race the code counts the integers at which each side leads and at which the two are tied, and records each maximal stretch of integers at which the usual loser leads (the class of 1 in both races), and each stretch of ties.

The weighted share in C3 is

$$\frac{\sum_{x \le N,\ d(x) < 0} 1/x}{\sum_{x \le N} 1/x},$$

where each sum over a stretch $a \le x \le b$ is $H(b) - H(a - 1)$, the harmonic numbers taken directly below 1000 and otherwise from the asymptotic series $\ln m + \gamma + \frac{1}{2m} - \frac{1}{12m^2} + \frac{1}{120m^4} - \frac{1}{252m^6}$, whose error is below $10^{-24}$ there. As $N \to \infty$, the share of $x$ at which the class of 3 leads, weighted this way, tends to the logarithmic density $\delta(4; 3, 1)$, which Rubinstein and Sarnak computed as about $0.9959$ assuming the Generalized Riemann Hypothesis and that the imaginary parts of the zeros of the relevant L-functions are linearly independent over the rationals. Ties have logarithmic density zero, so the share on which the class of 1 leads tends to about $1 - 0.9959 = 0.0041$. Rubinstein and Sarnak computed the density, not this finite share; the comparison in C3 is ours.

Everything except the weighted share is an exact integer count. The code needs only Python's standard library, takes about five seconds and about 130 MB of memory, and code/run runs it and writes results/R1.json, which also lists where the first 20 stretches of the class of 1's lead begin, and the longest stretch.

As checks, the counts add up, with the prime 2, to $\pi(10^8) = 5{,}761{,}455$, the known number of primes below $10^8$, and the same code with $N = 30{,}000$ gives the same first lead, leading integers, and ties as a separate count by trial division.

Results

The race modulo $4$ up to $10^8$:

The class of $1$ leads only near three places: at 26861 for a moment, through a cluster of short stretches beginning at 616841, and through a longer cluster that holds the longest stretch, from 12361933 to 12377290. From the last tie at 12424002 to $10^8$, the class of $3$ leads at every integer.

Weighted by $1/x$, the share on which the class of $1$ leads, 0.00040587, is about a tenth of the limiting share. That isn't evidence against the limit: the race's behavior is driven by the zeros of an L-function, and the stretches in which the class of $1$ leads come in rare, large episodes, so a range that ends before the next large episode will undercount them, and the weighting by $1/x$ makes the convergence slow.

Modulo $3$, the class of $2$ leads at every integer from $2$ to $10^8$, with 0 integers at which the class of $1$ leads and no tie after x = 1; its lead at $10^8$ is 420.

Limitations

The study stops at $10^8$. Much longer computations of both races exist, to well beyond $10^{11}$, and nothing here extends them; its value is an exact, checkable account of the races up to $10^8$, with every count declared, from code anyone can run in seconds. The comparison with the limiting share relies on Rubinstein and Sarnak's computation, which assumes two unproven hypotheses. The weighted share depends on how ties and the first few integers are counted, which affects its value at $10^8$ but not its limit. Only two races are followed, both between quadratic residues and non-residues, where the bias is strongest.

Provenance

Written and run by Claude, a model from Anthropic. The code was written for this study, and the results are the code's output, unedited. No data were collected. See provenance.json and references.json.