Experimental Mathematics

Biases in the Shanks-Rényi prime number race

Andrey Feuerverger and Greg Martin


Rubinstein and Sarnak investigated systems of inequalities of the form $\pi(x;q,a_1)>\cdots>\pi(x;q,a_r)$, where $\pi(x;q,b)$ denotes the number of primes up to x that are congruent to b mod q. They showed, under standard hypotheses on the zeros of Dirichlet L-functions mod q, that the set of positive real numbers x for which these inequalities hold has positive (logarithmic) density $\delta_{q;a_1,\dots,a_r}>0$. They also discovered the surprising fact that a certain distribution associated with these densities is not symmetric under permutations of the residue classes $a_j$ in general, even if the $a_j$ are all squares or all nonsquares mod q (a condition necessary to avoid obvious biases of the type first observed by Chebyshev). This asymmetry suggests, contrary to prior expectations, that the densities $\delta_{q;a_1,\dots,a_r}$ themselves vary under permutations of the $a_j$.

Here we derive (under the hypotheses used by Rubinstein and Sarnak) a general formula for the densities $\delta_{q;a_1,\dots,a_r}$, and we use this formula to calculate many of these densities when $q\le12$ and $r\le4$. For the special moduli q=8 and q=12, and for $\{a_1,a_2,a_3\}$ a permutation of the nonsquares {3,5,7} mod 8 and {5,7,11} mod12, respectively, we rigorously bound the error in our calculations, thus verifying that these densities are indeed asymmetric under permutation of the $a_j$. We also determine several situations in which the densities $\delta_{q;a_1,\dots,a_r}$ remain unchanged under certain permutations of the $a_j$, and some situations in which they are provably different.

Article information

Experiment. Math., Volume 9, Issue 4 (2000), 535-570.

First available in Project Euclid: 20 February 2003

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11N13: Primes in progressions [See also 11B25]
Secondary: 11Y35: Analytic computations

Chebyshev's bias comparative prime number theory primes in arithmetic progression Shanks-Rényi race


Feuerverger, Andrey; Martin, Greg. Biases in the Shanks-Rényi prime number race. Experiment. Math. 9 (2000), no. 4, 535--570. https://projecteuclid.org/euclid.em/1045759521

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