Marco Aymone
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 $Re(s)>1-ε$.
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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…
A Wasserstein metric approach to generalized Skewes' numbers. I. Prime number races
本文研究广义Skewes数，即两个可比素数计数函数间首次符号变化的位置。通过构造高度复合模数序列，证明了Fiorilli猜想的不成立，并在假设广义黎曼猜想和有效线性独立假设下，建立了广义Skewes数的条件上界。
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…
Prime Ideal Races With Several Competitors
本文推广素理想竞赛至多竞争共轭类，给出任意 r 路竞赛偏差的显式公式，并完全刻画阿贝尔情形下的中等偏差，发现 r≥3 时准则与 r 无关。
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,…
Ties in Function Field Prime Races
本文研究函数域中素数分布的平局现象，通过两种方法证明了在特定条件下存在无限多个平局情况，包括特征2的情形。
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…
On the role of higher roots in prime ideal races
研究伽罗瓦扩张中素理想竞赛问题，通过引入代数参数给出偏差对数密度位于特定区间的条件特征，构造出偏差源于\(2p\)次根数量差异的伽罗瓦扩张，给出最小群阶及广义构造，还使一估计无条件成立。
Densities in certain three-way prime number races
本文在特定同余条件下推导三元素数竞赛中满足特定不等式顺序的实数集合的对数密度渐近公式，通过非典型归一化实现误差项独立，并使用初等概率工具证明。
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…
A note on iterated sumsets races
本文构造同直径有限整数集 A、B，使迭代和集基数差序列的符号至少改变 n 次，回答 Nathanson 的迭代和集竞赛问题，并作为理解 Kravitz 更强结果的铺垫。
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 "…
Joint distribution of primes in multiple short intervals
在黎曼假设和线性独立猜想下，研究多个短区间中素数的联合分布，发现其遵循多元正态分布，并揭示了短区间素数竞赛的相变现象。
We survey the classical results on the prime number theorem
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$(x)$.
We implement the Maynard-Tao method of detecting primes in tuples to investigate small gaps between primes in arithmetic progressions, with bounds that are uniform over a range of moduli.
We investigate the distribution of $pθ$ modulo 1, where $θ$ is a complex number which is not contained in $\mathbb{Q}(i)$, and $p$ runs over the Gaussian primes.
Let $α, β\geq 0$ and $α+ β< 1$. In this short note, we show that $\liminf_{n \to \infty} p_n^β(p_{n+1}^α- p_n^α) = 0$, where $p_n$ is the $n$th prime. This notes an improvement over results of Sándor and gives additional evidence towards a conjecture of Andrica. This follows directly from recent results on prime pairs from Maynard, Tao, Zhang.
In this note, we revisit a result of Benli's and Pollack's on the number of small prime $k^{th}$ power residues. The proof is based on their idea of using reciprocity laws, but the argument is simplified and we prove a slightly stronger bound.
Let p be a prime number, and let K be a number field. For p=2, assume moreover K totally imaginary. In this note we prove the existence of asymptotically good extensions L{K of cohomological dimension 2 in which infinitely many primes split completely. Our result is inspired by a recent work of Hajir, Maire, and Ramakrishna [7].
Motivated by the connection to the pair correlation of the Riemann zeros, we investigate the second derivative of the logarithm of the Riemann zeta function, in particular the zeros of this function. Theorem 1 gives a zero-free region. Theorem 2 gives an asymptotic estimate for the number of nontrivial zeros to height T. Theorem 3 is a zero density estimate.
Instead of a strong quantitative form of the Hardy-Littlewood prime $k$-tuple conjecture, one can assume an average form of it and still obtains the same distribution result on $ψ(x+h) - ψ(x)$ by Montgomery and Soundararajan [1].
Let $S$ be a string of $l$ decimal digits. We give an explicit upper bound on some prime $p$ whose decimal representation contains the string $S$. We also show, as a corollary of the Green-Tao theorem, that there are arbitrarily long arithmetic progressions of prime numbers all of whose decimal representations contain $S$.
A note on the distribution of prime ideals in real quadratic fields
本文总结了虚二次域素理想分布的研究，并基于相同方法建立了实二次域的类似结果。
Let $n,k\in\mathbb{N}$ and let $p_{n}$ denote the $n$th prime number. We define $p_{n}^{(k)}$ recursively as $p_{n}^{(1)}:=p_{n}$ and $p_{n}^{(k)}=p_{p_{n}^{(k-1)}}$, that is, $p_{n}^{(k)}$ is the $p_{n}^{(k-1)}$th prime. In this note we give answers to some questions and prove a conjecture posed by Miska and Tóth in their recent paper concerning subsequences of the sequence of prime numbers. In particular, we establish explicit upper and lower bounds for $p_{n}^{(k)}$. We also study the behaviour of the counting functions of the sequences $(p_{n}^{(k)})_{k=1}^{\infty}$ and $(p_{k}^{(k)})_{k=1…
In this paper, we define a variant of Fibonacci-like sequences that we call prime Fibonacci sequences, where one takes the sum of the previous two terms and returns the smallest odd prime divisor of that sum as the next term. We prove that these sequences always terminate in a power of two but can be extended infinitely to the left.
As a refinement of the celebrated recent work of Yitang Zhang we show that any admissible k-tuple of integers contains at least two primes and almost primes in each component infinitely often if k is at least 181000. This implies that there are infinitely many gaps between consecutive primes of size at most 2530338, with an improved admissible k-tuple of Andrew W. Sutherland to 2326476.
A binary string representation of prime occurrences is a sequence of bits, where $1$ entries encode positions of prime numbers. This is a convenient representation for analysis of prime distribution, since it allows for application of a broad range of existing string-analysis algorithms to problems in number theory. Binary strings of prime occurrences can be also generated with simple algorithms. Here we discuss three such algorithms and we demonstrate their applicability using the example of proving Goldbach's hypothesis for some limited sets of even numbers. This work formulates three open q…
Assuming the Riemann hypothesis, this article discusses a new elementary argument that seems to prove that the maximal prime gap of a finite sequence of primes p_1, p_2, ..., p_n <= x, satisfies max {p_(n+1) - p_n : p_n <= x} <= c1((logx)^2)/loglogx, c1 > 0 constant. Equivalently, it shows that the very short intervals (x, x + y] contain prime numbers for all y > c2((logx)^2)/loglogx, c2 > 0 constant, and sufficiently large x > 0.
These notes are based on my four lectures given at the Newton Institute in April 2004 during the Recent Perspectives in Random Matrix Theory and Number Theory Workshop. Their purpose is to introduce the reader to the analytic number theory necessary to understand Montgomery's work on the pair correlation of the zeros of the Riemann zeta-function and subsequent work on how this relates to prime numbers. A very brief introduction to Selberg's work on the moments of $S(T)$ is also given.
We continue investigations on the average number of representations of a large positive integer as a sum of given powers of prime numbers. The average is taken over a short interval, whose admissible length depends on whether or not we assume the Riemann Hypothesis.
A note on the distribution of clusters and deserts of prime numbers
本文利用三角多项式值分布公式，研究加权 von Mangoldt 函数值之和的分布，并给出其服从类正态分布的证据。
A note on prime index of a certain subgroup in $\mathbb{F}_p^*$
在广义黎曼假设下，研究 $\mathbb{F}_p^*$ 中由 $-1$ 和 $2$ 生成的子群具有奇素数指标的素数比例，证明其与指标大于 $2$ 的素数之比趋近 $46\\%$。
Let $\{p_j(n)\}_{j=1}^{ω(n)}$ denote the increasing sequence of distinct prime factors of an integer $n$. We provide details for the proof of a statement of Erdős implying that, for any function $ξ(n)$ tending to infinity with $n$, we have $$f(n):=\max_{1\leqslant j<ω(n)}\log \Big({\log p_{j+1}(n)\over \log p_j(n)}\Big)=\log_3n+O(ξ(n))$$ for almost all integers $n$.
Suppose that $O_L$ is the ring of integers of a number field $L$, and suppose that $f(z)=\sum_{n=1}^\infty a_f(n)q^n\in S_k\cap O_L[[q]]$ (note: $q := e^{2πiz}$) is a normalized Hecke eigenform for $\mathrm{SL}_2(\mathbb{Z})$. We say that $f$ is non-ordinary at a prime $p$ if there is a prime ideal $\mathfrak{p}\subset O_L$ above $p$ for which $a_f(p)\equiv 0 \ (mod\ {\mathfrak{p}})$. For any finite set of primes $S$, we prove that there are normalized Hecke eigenforms which are non-ordinary for each $p\in S$. The proof is elementary and follows from a generalization of work of Choie, Kohnen a…
Chebyshev's bias for Fermat curves of prime degree
本文证明素数次 Fermat 曲线及其部分商的素数竞赛渐近公式与 DRH 相关部分等价，并据此在 DRH 下计算其二阶矩 L 函数在 s=1 处的零点阶数。
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 $…
A non-ordinary (prime) note
本文通过若干权为4的新形式例子，说明非寻常素数条件p整除b(p)可蕴含更强的整除性，并指出这种性质可能较为罕见。
A note on zero-density approaches for the difference between consecutive primes
本研究推广了短区间素数相关的Ingham定理与Heath-Brown和Iwaniec的结果，明晰了无零点区域、零密度估计和短区间素数的关联，还在密度假设下改进了素数定理成立的短区间范围。
The $\Sopfr(n)$ function is defined as the sum of prime factors of $n$ each of which is taken with its multiplicity. This function is studied numerically. The analogy between $\Sopfr(n)$ and the primes distribution function is drawn and some conjectures for prime numbers formulated in terms of the $\Sopfr(n)$ function are suggested.
A note on Riemann's $R$-function and almost primes
本文通过对素zeta函数及殆素数生成函数应用相同操作，构造了黎曼R函数的光滑族R_k，证明其展开为Selberg-Delange展开，并利用有限Perron分解确定了精确计数所需的额外围道项。
Given positive integers $a_1,\ldots,a_k$, we prove that the set of primes $p$ such that $p \not\equiv 1 \bmod{a_i}$ for $i=1,\ldots,k$ admits asymptotic density relative to the set of all primes which is at least $\prod_{i=1}^k \left(1-\frac{1}{φ(a_i)}\right)$, where $φ$ is the Euler's totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer $n$ such that $n \not\equiv 0 \bmod a_i$ for $i=1,\ldots,k$ admits asymptotic density which is at least $\prod_{i=1}^k \left(1-\frac{1}{a_i}\right)$.
Let us denote the nth difference between consecutive primes by d_n. The Prime Number Theorem clearly implies that d_n is logn on average. Paul Erdős conjectured about 60 years ago that the sequence d_n/logn is everywhere dense on the nonnegative part of the real line. He and independently G. Ricci proved in 1954-55 that the set J of limit points of the sequence {d_n/logn} has positive Lebesgue measure. The first and until now only concrete known element of J was proved to be the number zero in the work of Goldston, Yildirim and the present author. The author of the present note showed in 2013 …
An inequality involving primes and the product of primes
本文证明了一个涉及素数及连续素数乘积的不等式。
Notes on Mitsui's Prime Number Theorem with Siegel zeros
本文通过纳入Hecke L-函数的Siegel零点，改进了三井1957年的素数定理，使模的范数能以伪多项式速率增长，为后续研究素元素线性模式提供关键支持。
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