Let $a_1$, $a_2$, and $a_3$ be distinct reduced residues modulo $q$ satisfying the congruences $a_1^2 \equiv a_2^2 \equiv a_3^2 \pmod q$. We conditionally derive an asymptotic formula, with an error term that has a power savings in $q$, for the logarithmic density of the set of real numbers $x$ for which $π(x;q,a_1) > π(x;q,a_2) > π(x;q,a_3)$. The relationship among the $a_i$ allows us to normalize the error terms for the $π(x;q,a_i)$ in an atypical way that creates mutual independence among their distributions, and also allows for a proof technique that uses only elementary tools from probability.
We investigate the distribution of the function $ω(n)$, the number of distinct prime divisors of $n$, in residue classes modulo $q$ for natural numbers $q$ greater than 2. In particular we ask `prime number races' style questions, as suggested by Coons and Dahmen in their paper `On the residue class distribution of the number of prime divisors of an integer'.
We survey results about prime number races, that is, results about the relative sizes of prime counting functions $π_{q,a}(x)$, with $q$ fixed and $a$ varying. In particular, we describe recent work by the authors on these problems.
Rubinstein and Sarnak have shown, conditional on the Riemann hypothesis (RH) and the linear independence hypothesis (LI) on the non-real zeros of $ζ(s)$, that the set of real numbers $x\ge2$ for which $π(x)>$ li$(x)$ has a logarithmic density, which they computed to be about $2.6\times10^{-7}$. A natural problem is to examine the actual primes in this race. We prove, assuming RH and LI, that the logarithmic density of the set of primes $p$ for which $π(p)>$ li$(p)$ relative to the prime numbers exists and is the same as the Rubinstein-Sarnak density. We also extend such results to a broad clas…
We prove explicit versions of the Kronecker-Weyl theorems, both in a discrete and a continuous settings, without any linear independence hypothesis. As an application, we propose an alternative approach to problems concerning asymptotic densities in prime number races, over number fields and over function fields in one variable over finite fields, in the language of random variables. Our approach allows us to prove new results on the existence and positivity of some of those densities, which, in the case of races over function fields, do not require any linear independence hypothesis.
Chebyshev observed in a letter to Fuss that there tends to be more primes of the form $4n+3$ than of the form $4n+1$. The general phenomenon, which is referred to as Chebyshev's bias, is that primes tend to be biased in their distribution among the different residue classes $\bmod q$. It is known that this phenomenon has a strong relation with the low-lying zeros of the associated $L$-functions, that is if these $L$-functions have zeros close to the real line, then it will result in a lower bias. According to this principle one might believe that the most biased prime number race we will ever …
We show, for any $q\ge 3$ and distinct reduced residues $a,b \pmod q$, the existence of certain hypothetical sets of zeros of Dirichlet $L$-functions lying off the critical line implies that $π(x;q,a)<π(x;q,b)$ for a set of real $x$ of asymptotic density 1.
This is a survey article on prime number races. Chebyshev noticed in the first half of the nineteenth century that for any given value of x, there always seem to be more primes of the form 4n+3 less than x then there are of the form 4n+1. Similar observations have been made with primes of the form 3n+2 and 3n+1, with primes of the form 10n+3/10n+7 and 10n+1/10n+9, and many others besides. More generally, one can consider primes of the form qn+a, qn+b, qn+c, >... for our favorite constants q, a, b, c, ... and try to figure out which forms are "preferred" over the others. In this paper, we descr…
Let $χ$ be a real and non-principal Dirichlet character, $L(s,χ)$ its Dirichlet $L$-function and let $p$ be a generic prime number. We prove the following result: If for some $0\leq σ<1$ the partial sums $\sum_{p\leq x}χ(p)p^{-σ}$ change sign only for a finite number of $x$, then there exists $ε>0$ such that $L(s,χ)$ has no zeros in the half plane $Re(s)>1-ε$.
Fix an integer $r\geq 3$. Let $q$ be a large positive integer and $a_1,...,a_r$ be distinct residue classes modulo $q$ that are relatively prime to $q$. In this paper, we establish an asymptotic formula for the logarithmic density $δ_{q;a_1,...,a_r}$ of the set of real numbers $x$ such that $π(x;q,a_1)>π(x;q,a_2)>...>π(x;q,a_r),$ as $q\to\infty$; conditionally on the assumption of the Generalized Riemann Hypothesis GRH and the Grand Simplicity Hypothesis GSH. Several applications concerning these prime number races are then deduced. Indeed, comparing with a recent work of D. Fiorilli and G. Ma…
Let $π_{q,a}(x)$ denote the number of primes $\le x$ in the progression $a$ modulo $q$. We study subtle inequities in these functions, with $q$ fixed and variable $a$ (sometimes called 'prime race problems'). It is known unconditionally for many triples $(q,a,b)$ that the difference $π_{q,a}(x) - π_{q,b}(x)$ changes sign infinitely often, although there may be a pronounced bias toward one sign (first observed by Chebyshev in 1853). Similar results for the comparison of three or more prime counting functions all require the assumption of ERH (extended Riemann Hypothesis for the Dirichlet L-func…
We continue to investigate the race between prime numbers in many residue classes modulo $q$, assuming the standard conjectures GRH and LI. We show that provided $n/\log q \rightarrow \infty$ as $q \rightarrow \infty$, we can find $n$ competitor classes modulo $q$ so that the corresponding $n$-way prime number race is extremely biased. This improves on the previous range $n \geq φ(q)^ε$, and (together with an existing result of Harper and Lamzouri) establishes that the transition from all $n$-way races being asymptotically unbiased, to biased races existing, occurs when $n = \log^{1+o(1)}q$. T…
We investigate the race between prime numbers in many residue classes modulo $q$, assuming the standard conjectures GRH and LI. Among our results we exhibit, for the first time, prime races modulo $q$ with $n$ competitor classes where the biases do not dissolve when $n, q\to \infty$. We also study the leaders in the prime number race, obtaining asymptotic formulae for logarithmic densities when the number of competitors can be as large as a power of $q$, whereas previous methods could only allow a power of $\log q$. The proofs use harmonic analysis related to the Hardy--Littlewood circle metho…
We study the prime number race for elliptic curves over the function field of a proper, smooth and geometrically connected curve over a finite field. This constitutes a function field analogue of prior work by Mazur, Sarnak and the second author. In this geometric setting we can prove unconditional results whose counterparts in the number field case are conditional on a Riemann Hypothesis and a linear independence hypothesis on the zeros of the implied L-functions. Notably we show that in certain natural families of elliptic curves, the bias generically dissipates as the conductor grows. This …
Fix a prime $p >2$ and a finite field $\mathbb{F}_{q}$ with $q$ elements, where $q$ is a power of $p$. Let $m$ be a monic polynomial in the polynomial ring $\mathbb{F}_{q}[T]$ such that $deg(m)$ is large. Fix an integer $r\geq 2$, and let $a_1,\dots,a_r$ be distinct residue classes modulo $m$ that are relatively prime to $m$. In this paper, we derive an asymptotic formula for the natural density $δ_{m;a_1,\dots,a_r}$ of the set of all positive integers $X$ such that $\sum\limits_{N=1}^{X} π_{q}(a_1,m,N) > \sum\limits_{N=1}^{X} π_{q}(a_2,m,N) > \dots > \sum\limits_{N=1}^{X} π_{q}(a_r,m,N)$, whe…
We continue our examination the effects of certain hypothetical configurations of zeros of Dirichlet $L$-functions lying off the critical line ("barriers") on the relative magnitude of the functions $π_{q,a}(x)$. Here $π_{q,a}(x)$ is the number of primes $\le x$ in the progression $a \mod q$. In particular, we construct barriers so that $π_{q,1}(x)$ is simultaneously greater than, or simultaneously less than, each of $k$ functions $π_{q,a_i}(x)$ ($1\le i\le k$). We also construct barriers so that only a small number of the $r!$ possible orderings of functions $π_{q,a_i}(x)$ ($1\le i\le r$) occ…
Under certain plausible assumptions, M. Rubinstein and P. Sarnak solved the Shanks--Rényi race problem, by showing that the set of real numbers $x\geq 2$ such that $π(x;q,a_1)>π(x;q,a_2)>...>π(x;q,a_r)$ has a positive logarithmic density $δ_{q;a_1,...,a_r}$. Furthermore, they established that if $r$ is fixed, $δ_{q;a_1,...,a_r}\to 1/r!$ as $q\to \infty$. In this paper, we investigate the size of these densities when the number of contestants $r$ tends to infinity with $q$. In particular, we deduce a strong form of a recent conjecture of A. Feuerverger and G. Martin which states that $δ_{q;a_1,…
Let $π(x;q,a)$ denote the number of primes up to $x$ that are congruent to $a$ modulo $q$. A prime number race, for fixed modulus $q$ and residue classes $a_1, \ldots, a_r$, investigates the system of inequalities $π(x;q,a_1) > π(x;q,a_2) > \cdots > π(x;q,a_r)$. The study of prime number races was initiated by Chebyshev and further studied by many others, including Littlewood, Shanks-Rényi, Knapowski-Turan, and Kaczorowski. We expect that this system of inequalities should have arbitrarily large solutions $x$, and moreover we expect the same to be true no matter how we permute the residue clas…
It has been well-observed that an inequality of the type $π(x;q,a) > π(x;q,b)$ is more likely to hold if $a$ is a non-square modulo $q$ and $b$ is a square modulo $q$ (the so-called ``Chebyshev Bias''). For instance, each of $π(x;8,3)$, $π(x;8,5)$, and $π(x;8,7)$ tends to be somewhat larger than $π(x;8,1)$. However, it has come to light that the tendencies of these three $π(x;8,a)$ to dominate $π(x;8,1)$ have different strengths. A related phenomenon is that the six possible inequalities of the form $π(x;8,a_1) > π(x;8,a_2) > π(x;8,a_3)$ with $\{a_1,a_2,a_3\}=\{3,5,7\}$ are not all equally lik…
Let $q\geq 3$, $2\leq r\leq ϕ(q)$ and $a_1,...,a_r$ be distinct residue classes modulo $q$ that are relatively prime to $q$. Assuming the Generalized Riemann Hypothesis and the Grand Simplicity Hypothesis, M. Rubinstein and P. Sarnak showed that the vector-valued function $E_{q;a_1,...,a_r}(x)=(E(x;q,a_1),..., E(x;q,a_r)),$ where $E(x;q,a)= \frac{\log x}{\sqrt{x}}(ϕ(q)π(x;q,a)-π(x))$, has a limiting distribution $μ_{q;a_1,...,a_r}$ which is absolutely continuous on $\mathbb{R}^r$. Under the same assumptions, we determine the asymptotic behavior of the large deviations $μ_{q;a_1,...,a_r}(||\vx|…
We establish a conjectured result of Dummit, Granville and Kisilevsky for the maximum bias in P2 races, which compare the number of P2's $\leq x$ whose prime factors, evaluated at a given quadratic character, take specifc values.
Rubinstein and Sarnak investigated systems of inequalities of the form pi(x;q,a_1) > ... > pi(x;q,a_r), where pi(x;q,a) denotes the number of primes up to x that are congruent to a 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…
Chebyshev was the first to observe a bias in the distribution of primes in residue classes. The general phenomenon is that if $a$ is a nonsquare\mod q and $b$ is a square\mod q, then there tend to be more primes congruent to $a\mod q$ than $b\mod q$ in initial intervals of the positive integers; more succinctly, there is a tendency for $π(x;q,a)$ to exceed $π(x;q,b)$. Rubinstein and Sarnak defined $δ(q;a,b)$ to be the logarithmic density of the set of positive real numbers $x$ for which this inequality holds; intuitively, $δ(q;a,b)$ is the "probability" that $π(x;q,a) > π(x;q,b)$ when $x$ is "…
We give formulas for local densities of diagonal integral ternary quadratic forms at odd primes. Exponential sums and quadratic Gauss sums are used to obtain these formulas. These formulas (along with 2-adic densities and Siegel's mass formula) can be used to compute the representation numbers of certain ternary quadratic forms.
We generalize results of Alladi, Dawsey, and Sweeting and Woo for Chebotarev densities to general densities of sets of primes. We show that if $K$ is a number field and $S$ is any set of prime ideals with natural density $δ(S)$ within the primes, then \[ -\lim_{X \to \infty}\sum_{\substack{2 \le \operatorname{N}(\mathfrak{a})\le X\\ \mathfrak{a} \in D(K,S)}}\frac{μ(\mathfrak{a})}{\operatorname{N}(\mathfrak{a})} = δ(S), \] where $μ(\mathfrak{a})$ is the generalized Möbius function and $D(K,S)$ is the set of integral ideals $ \mathfrak{a} \subseteq \mathcal{O}_K$ with unique prime divisor of min…
A subset of the integers larger than 1 is $primitive$ if no member divides another. Erdos proved in 1935 that the sum of $1/(a\log a)$ for $a$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained for the set of prime numbers. In this paper we make some progress on several fronts, and show a connection to certain prime number "races" such as the race between $π(x)$ and li$(x)$.
The set of primes where a hypergeomeric series with rational parameters is $p$-adically bounded is known by [10] to have a Dirichlet density. We establish a formula for this Dirichlet density and conjecture that it is rare for the density to be large. We prove this conjecture for hypergeometric series whose parameters have denominators equal to a prime of the form $p = 2q^r + 1$, where $q$ is an odd prime, by establishing an upper bound on the density of bounded primes in this case. The general case remains open. This paper is the output of an undergraduate research course taught by the first …
We generalize current known distribution results on Shanks--Rényi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical case that goes back to Chebyshev and function field analogues developed in the recent years. More precisely, let $π(x;q,a)$ be the number of primes up to $x$ that are congruent to $a$ modulo $q$. For a fixed integer $q$ and distinct invertible congruence classes $a_0,a_1,\ldots,a_D$, assuming the generalized Riemann Hypothesis and a weak version of the linear independence hypothesis, we show that the set of $…
In this paper we introduce new densities on the set of primes of a number field. If $K/K_0$ is a Galois extension of number fields, we associate to any element $x \in {\rm Gal}_{K/K_0}$ a density $δ_{K/K_0,x}$ on primes of $K$. In particular, the density associated to $x = 1$ is the usual Dirichlet density on $K$. After establishing some properties of these densities, we use them to show that the maximal solvable extension of a number field unramified outside an almost Chebotarev set realize the maximal local extension at each prime lying outside this set.
In this paper we develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. As an application we show that the logarithmic densities of any automatic sequence along squares $(n^2)_{n\geq 0}$ and primes $(p_n)_{n\geq 1}$ exist and are computable. Furthermore, we give for these subsequences a criterion to decide whether the densities exist, in which case they are also computable. In particular in the prime case these densities are all rational. We also deduce from a recent result of the third author and Lemańczyk…
If $E$ is an elliptic curve over $\mathbb{Q}$, then it follows from work of Serre and Hooley that, under the assumption of the Generalized Riemann Hypothesis, the density of primes $p$ such that the group of $\mathbb{F}_p$-rational points of the reduced curve $\tilde{E}(\mathbb{F}_p)$ is cyclic can be written as an infinite product $\prod δ_\ell$ of local factors $δ_\ell$ reflecting the degree of the $\ell$-torsion fields, multiplied by a factor that corrects for the entanglements between the various torsion fields. We show that this correction factor can be interpreted as a character sum, and…
We give a new formula for the Chebotarev densities of Frobenius elements in Galois groups. This formula is given in terms of smallest prime factors $p_{\mathrm{min}}(n)$ of integers $n\geq2$. More precisely, let $C$ be a conjugacy class of the Galois group of some finite Galois extension $K$ of $\mathbb{Q}$. Then we prove that $$-\lim_{X\rightarrow\infty}\sum_{\substack{2\leq n\leq X\\[1pt]\left[\frac{K/\mathbb{Q}}{p_{\mathrm{min}}(n)}\right]=C}}\frac{μ(n)}{n}=\frac{\#C}{\#G}.$$ This theorem is a generalization of a result of Alladi from 1977 that asserts that largest prime divisors $p_{\mathr…
Let P denote the set of all primes. Suppose that P_1, P_2, P_3 are three subsets of P with the sum of their lower densities relative to P is greater than 2. We prove that for sufficiently large odd integer n, there exist p_i\in P_i such that n=p_1+p_2+p_3.
For any prime p we consider the density of elements in the multiplicative group of the finite field F_p having order, respectively index, congruent to a(mod d). We compute these densities on average, where the average is taken over all finite fields of prime order. Some connections between the two densities are established. It is also shown how to compute these densities with high numerical accuracy.
This paper uses concepts introduced by Fiorilli and Martin to develop a more accurate remainder term in calculations of the distribution of primes in residue classes. Using it in Rubinstein and Sarnak's method estimates the limiting logarithmic frequency of pi(x)>Li(x) as 0.00000026300 with only 5 zeta zeros used explicitly, and a reliable value between 0.000000262996732 and 0.000000262996733 is obtained with fewer than 100 zeros. Accurate results for "prime number races" can usually be obtained with explicit use of only the first zero of each L-function involved, thus bringing out their depen…
We study the prime numbers that lie in Beatty sequences of the form $\lfloor αn + β\rfloor$ and have prescribed algebraic splitting conditions. We prove that the density of primes in both a fixed Beatty sequence and a Chebotarev class of some Galois extension is precisely the product of the densities $α^{-1}\cdot\frac{|C|}{|G|}$. Moreover, we show that the primes in the intersection of these sets satisfy a Bombieri--Vinogradov type theorem. This allows us to prove the existence of bounded gaps for such primes. As a final application, we prove a common generalization of the aforementioned bound…
Partitions of the set of primes are introduced based on the Chebyshev polynomials at rationals. The prime densities of all such partitions are established. Euler's Criterion for $SL(2,\mathbb Q)$ is formulated, which is the bridge between the algebra of Chebyshev polynomials and number-theoretic properties of the partitions. It is shown how to obtain in this way some of the classical theory of Lucas sequences. A hidden symmetry of the problem is revealed by the new language. As an application number-theoretic properties of simple dynamical systems (rotations and certain interval maps) are disc…
It follows from the work of Artin and Hooley that, under assumption of the generalized Riemann hypothesis, the density of the set of primes $q$ for which a given non-zero rational number $r$ is a primitive root modulo $q$ can be written as an infinite product $\prod_p δ_p$ of local factors $δ_p$ reflecting the degree of the splitting field of $X^p-r$ at the primes $p$, multiplied by a somewhat complicated factor that corrects for the `entanglement' of these splitting fields. We show how the correction factors arising in Artin's original primitive root problem and some of its generalizations ca…
We generalize the Chebotarev density formulas of Dawsey (2017) and Alladi (1977) to the setting of arbitrary finite Galois extensions of number fields $L/K$. In particular, if $C \subset G = \textrm{Gal}(L/K)$ is a conjugacy class, then we establish that the Chebotarev density is the following limit of partial sums of ideals of $K$: \[ -\lim_{X\rightarrow\infty} \sum_{\substack{2\leq N(I)\leq X \\ I \in S(L/K; C)}} \frac{μ_K(I)}{N(I)} = \frac{|C|}{|G|}, \] where $μ_K(I)$ denotes the generalized Möbius function and $S(L/K;C)$ is the set of ideals $I\subset \mathcal{O}_K$ such that $I$ has a uni…
Let $\mathbf{G}$ be the set of all finite or infinite increasing sequences of positive integers beginning with 1. For a sequence $S=\{s(n)\}, n\geq1,$ from $\mathbf{G},$ a positive number $N$ is called an exponentially $S$-number $(N\in E(S)),$ if all exponents in its prime power factorization are in $S.$ The author \cite{2} proved that, for every sequence $S\in \mathbf{G},$ the sequence of exponentially $S$-numbers has a density $h=h(E(S))\in [\frac{6}{π^2}, 1].$ In this paper we study the set $\{h(E(S)\}$ of all such densities.
Centered Moments of Weighted One-Level Densities of $GL(2)$ $L$-Functions
University of Washington（华盛顿大学）；University of Michigan（密歇根大学）；Princeton University（普林斯顿大学）；Yale University（耶鲁大学）；Williams College（威廉姆斯学院）；Columbia University（哥伦比亚大学）；Carleton College（卡尔顿学院）；University of Wisconsin（威斯康星大学）
Let $a_0=b_0=0$ and $0<a_1\leq b_1<a_2\leq b_2<\ldots\leq b_{n}$ be integers. Let $Q\left(x;\bigcup_{j=1}^{n}[a_j,b_j]\right)$ be the number of integers between $1$ and $x$ such that all exponents in their prime factorization are in $\bigcup_{j=1}^{n}[a_j,b_j]$. The following formula holds: $$\lim_{x\to\infty}{\frac{Q\left(x;\bigcup_{j=1}^{n}[a_j,b_j]\right)}{x}}=\prod\limits_{p}\sum\limits_{i=0}^{n}\left(\frac{1}{p^{a_{i}}}-\frac{1}{p^{b_{i}+1}}\right).$$ In this paper, we prove this result and then generalize it.
The results of the computer investigation of the sign changes of the difference between the number of twin primes $π_2(x)$ and the Hardy--Littlewood conjecture $C_2\Li_2(x)$ are reported. It turns out that $d_2(x)=π_2(x) - C_2\Li_2(x)$ changes the sign at unexpectedly low values of $x$ and for $x<2^{48}=2.81\...\times10^{14}$ there are 477118 sign changes of this difference. It is conjectured that the number of sign changes of $d_2(x)$ for $x\in (1, T)$ is given by $\sqrt T/\log(T)$. The running logarithmic densities of the sets for which $d_2(x)>0$ and $d_2(x)<0$ are plotted for $x$ up to $2^…
We determine the asymptotic density $δ_k$ of the set of ordered $k$-tuples $(n_1,...,n_k)\in \N^k, k\ge 2$, such that there exists no prime power $p^a$, $a\ge 1$, appearing in the canonical factorization of each $n_i$, $1\le i\le k$, and deduce asymptotic formulae with error terms regarding this problem and analogous ones. We give numerical approximations of the constants $δ_k$ and improve the error term of formula (1.2) due to {\sc E. Cohen}. We point out that our treatment, based on certain inversion functions, is applicable also in case $k=1$ in order to establish asymptotic formulae with e…
Let a be a positive integer greater than 1, and Q_a(x;k,j) be the set of primes p less than x such that the residual order of a(mod p) is congruent to j modulo k. In this paper, the natural densities of Q_a(x;4,j) (j=0,1,2,3) are considered. We assume a is square-free and a is congruent to 1 (mod 4). Then, for j=0, 2, we can prove unconditionally that their natural densities are equal to 1/3. On the contrary, for j=1, 3, we assume Generalized Riemann Hypothesis, then we can prove that their densities are equal to 1/6.
