Ben Green
Let $f : \mathbf{N} \rightarrow \mathbf{C}$ be a bounded multiplicative function. Let $a$ be a fixed integer (say $a = 1$). Then $f$ is well-distributed on the progression $n \equiv a \pmod{q} \subset \{1,\dots, X\}$, for almost all primes $q \in [Q,2Q]$, for $Q$ as large as $X^{\frac{1}{2} + \frac{1}{78} - o(1)}$.
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We develop a theory of multiplicative functions (with values inside or on the unit circle) in arithmetic progressions analogous to the well-known theory of primes in arithmetic progressions.
We show that smooth-supported multiplicative functions $f$ are well-distributed in arithmetic progressions $a_1a_2^{-1} \pmod q$ on average over moduli $q\leq x^{3/5-\varepsilon}$ with $(q,a_1a_2)=1$.
Let $G$ be a multiplicative subgroup of the prime field $\mathbb F_p$ of size $|G|> p^{1-κ}$ and $r$ an arbitrarily fixed positive integer. Assuming $κ=κ(r)>0$ and $p$ large enough, it is shown that any proportional subset $A\subset G$ contains non-trivial arithmetic progressions of length $r$. The main ingredient is the Szemerédi-Green-Tao theorem.
Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.
We prove a result on the distribution of the general divisor functions in arithmetic progressions to smooth moduli which exceed the square root of the length.
We obtain the analog of the Bombieri-Vinogradov theorem for square moduli up to any power of x less than 1/2.
We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$, extending the Bombieri-Vinogradov theorem to moduli of size $x^{1/2+δ}$ which have conveniently sized divisors. The main feature of these estimates is that they are completely uniform with respect to the residue classes considered, unlike previous works on primes in arithmetic progressions to large moduli.
We prove that there exists an absolute constant $c>0$ such that if an arithmetic progression $\cP$ modulo a prime number $p$ does not contain zero and has the cardinality less than $cp$, then it can not be represented as a product of two subsets of cardinality greater than 1, unless $\cP=-\cP$ or $\cP=\{-2r,r,4r\}$ for some residue $r$ modulo $p$.
We investigate lower bounds for the variance in arithmetic progressions of certain multiplicative functions "close" to $1$. Specifically, we consider $α_N$-fold divisor functions, when $α_N$ is a sequence of positive real numbers approaching $1$ in a suitable way or $α_N=1$, and the indicator of $y$-smooth numbers, for suitably large parameters $y$. As a corollary, we will strengthen a previous author's result on the first subject and obtain matching lower bounds to some Barban-Davenport-Halberstam type theorems for $y$-smooth numbers. Incidentally, we will also find a lower bound for the vari…
We establish new Bombieri-Vinogradov type estimates for a wide class of multiplicative arithmetic functions and derive several applications, including: a new proof of a recent estimate by Drappeau and Topacogullari for arithmetical correlations; a theorem of Erd{\H o}s-Wintner type with support equal to the level set of an additive function at shifted argument; and a law of iterated logarithm for the distribution of prime factors of integers weighted by $τ(n-1)$ where $τ$ denotes the divisor function.
We establish new mean value theorems for primes of size $x$ in arithmetic progressions to moduli as large as $x^{3/5-ε}$ when summed with suitably well-factorable weights. This extends well-known work of Bombieri, Friedlander and Iwaniec, who handled moduli of size at most $x^{4/7-ε}$. This has consequences for the level of distribution for sieve weights coming from the linear sieve.
Let $μ$ be the Möbius function and let $k \geq 1$. We prove that the Gowers $U^k$-norm of $μ$ restricted to progressions $\{n \leq X: n\equiv a_q\pmod{q}\}$ is $o(1)$ on average over $q\leq X^{1/2-σ}$ for any $σ> 0$, where $a_q\pmod{q}$ is an arbitrary residue class with $(a_q,q) = 1$. This generalizes the Bombieri-Vinogradov inequality for $μ$, which corresponds to the special case $k=1$.
By using the $q$-analogue of van der Corput's method we study the divisor function in an arithmetic progression to modulus $q$. We show that the expected asymptotic formula holds for a larger range of $q$ than was previously known, provided that $q$ has a certain factorisation.
Primes in arithmetic progressions to smooth moduli: A minorant version
该研究提出一个函数作为素数指示函数的下界，改进了Baker--Irving和Stadlmann在算术进展中至光滑模数的素数分布结果。
Multiplicative Subgroups of $\mathbb{Z}_p^*$ that are Generalized Arithmetic Progressions
本文证明了$\mathbb{Z}_p^*$中乘法子群为广义算术进展的充要条件，并推广了Hanson和Petridis的结果。
We prove the infinitude of shifted primes $p-1$ without prime factors above $p^{0.2844}$. This refines $p^{0.2961}$ from Baker and Harman in 1998. Consequently, we obtain an improved lower bound on the the distribution of Carmichael numbers. Our main technical result is a new mean value theorem for primes in arithmetic progressions to large moduli. Namely, we estimate primes of size $x$ with quadrilinear forms of moduli up to $x^{17/32}$. This extends moduli beyond $x^{11/21}$, recently obtained by Maynard, improving $x^{29/56}$ from well-known 1986 work of Bombieri, Friedlander, and Iwaniec.
This is the second of two coupled papers estimating the mean values of multiplicative functions, of unknown support, on arithmetic progressions with large differences. Applications are made to the study of primes in arithmetic progression and to the Fourier coefficients of automorphic cusp forms.
We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$. Our main result shows that the primes are equidistributed for a fixed residue class over all moduli of size $x^{1/2+δ}$ with a 'convenient sized' factor. As a consequence, the expected asymptotic holds for all but $O(δQ)$ moduli $q\sim Q=x^{1/2+δ}$ and we get results for moduli as large as $x^{11/21}$. Our proof extends previous techniques of Bombieri, Fouvry, Friedlander and Iwaniec by incorporating new ideas inspired by amplification methods. We combine these with techniques of Zhang and P…
We obtain several asymptotic formulas for the sum of the divisor function $τ(n)$ with $n \le x$ in an arithmetic progressions $n \equiv a \pmod q$ on average over $a$ from a set of several consecutive elements from set of reduced residues modulo $q$ and on average over arbitrary sets. The main goal is to obtain nontrivial results for $q \ge x^{2/3}$ with the small amount of averaging over $a$. We recall that for individual values of $a$ the limit of our current methods is $q \le x^{2/3-\varepsilon}$ for an arbitrary fixed $\varepsilon> 0$. Our method builds on an approach due to Blomer (2008) …
Primes in arithmetic progressions to large moduli and refinements of Harman's sieve
利用多作者算术信息与Harman筛法变体，构造素数指示函数的上下界，得到模数达x^{9/17}的双线性形式和模数达x^{17/32}的三线性形式的均值定理，并给出π(x;q,a)的几乎全部模数q的新上下界。
In this paper, we give an algorithm for detecting non-trivial 3-APs in multiplicative subgroups of $\mathbb{F}_p^\times$ that is substantially more efficient than the naive approach. It follows that certain Var der Waerden-like numbers can be computed in polynomial time.
Smooth numbers in arithmetic progressions to large moduli
本文证明光滑数在大小为 $x^{66/107-o(1)}$ 的模数的算术级数中均匀分布，克服了此前 $x^{3/5-o(1)}$ 的长期障碍，核心方法基于 Drappeau 的色散法变体、Maynard 的指数和操作及优化的 Kloosterman 和估计。
Triple divisor type functions in arithmetic progressions to prime power moduli
本文研究非尖点GL_3自守形式Hecke特征值在素数幂模算术级数中的等分布，利用Kowalski-Lin-Michel和Milićević技术证明指数1/2+7/458-ε可取。
The divisor function along arithmetic progressions and binary cubic polynomials
该研究提出带两个小因子模的算术级数中除数函数的新等分布估计，给出两项应用：指数2/3的几乎所有模的算术级数上除数函数的渐近公式，以及沿非齐次二元三次多项式XY²+1的除数函数渐近公式。
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 consider some Diophantine problems of mixed modular-multiplicative type associated with the Zilber-Pink conjecture. In particular, we prove a finiteness statement for the number of multiplicative relations between singular moduli (j-invariants of elliptic curves with complex multiplication.)
In this paper, we consider the mean value of the product of two real valued multiplicative functions with shifted arguments. The functions $F$ and $G$ under consideration are close to two nicely behaved functions $A$ and $B$, such that the average value of $A(n-h)B(n)$ over any arithmetic progression is only dependent on the common difference of the progression. We use this method on the problem of finding mean value of $K(N)$, where $K(N)/\log N$ is the expected number of primes such that a random elliptic curve over rationals has $N$ points when reduced over those primes.
We shall give an explicit version of Bombieri-Vinogradov Theorem for moduli not divisible by an exceptional modulus.
Let $x,h$ and $Q$ be three parameters. We show that, for most moduli $q\le Q$ and for most positive real numbers $y\le x$, every reduced arithmetic progression $a\mod q$ has approximately the expected number of primes $p$ from the interval $(y,y+h]$, provided that $h>x^{1/6+ε}$ and $Q$ satisfies appropriate bounds in terms of $h$ and $x$. Moreover, we prove that, for most moduli $q\le Q$ and for most positive real numbers $y\le x$, there is at least one prime $p\in(y,y+h]$ lying in every reduced arithmetic progression $a\mod q$, provided that $1\le Q^2\le h/x^{1/15+ε}$.
We prove an asymptotic formula for squarefree in arithmetic progressions with squarefree moduli, improving previous results by Prachar. The main tool is an estimate for counting solutions of a congruence inside a box that goes beyond what can be obtained by using the Weil bound.
We consider Mertens' function M(x,q,a) in arithmetic progression, Assuming the generalized Riemann hypothesis (GRH), we show an upper bound that is uniform for all moduli which are not too large. For the proof, a former method of K. Soundararajan is extended to L-series.
Effective multiplicative independence of 3 singular moduli
本文通过有效方法证明了三个不同的非零奇异模量的乘法依赖关系仅在判别式不超过10^10时成立，推广了Riffaut对两个奇异模量的结果。
Sumsets and generalized arithmetic progressions in multiplicative subgroups
针对有限域乘法子群，研究其与广义算术级数、和集分解的关系，得出真子群为广义算术级数的充要条件，及加法不可约、和差表示数的相关结论。
Let $B$ be a set of natural numbers of size $n$. We prove that the length of the longest arithmetic progression contained in the product set $B.B = \{bb'| \, b, b' \in B\}$ cannot be greater than $O(n \log n)$ which matches the lower bound provided in an earlier paper up to a multiplicative constant. For sets of complex numbers we improve the bound to $O_ε(n^{1 + ε})$ for arbitrary $ε> 0$ assuming the GRH.
Squarefree Integers in Arithmetic Progressions to Smooth Moduli
本文证明在光滑模数下算术级数中无平方因子整数可获幂次节省误差的渐近公式，突破 $X^{3/4}$ 障碍，改进 Nunes 与 Hooley 的结果。
Multiplicative functions in short arithmetic progressions
该研究探讨有界积性函数在短算术级数中的和的方差，改进推广了相关经典定理，给出近乎最优的例外模集合，并将结果应用于三素数乘积的Linnik型问题。
We study the variance of sums of the $k$-fold divisor function $d_k(n)$ over sparse arithmetic progressions, with averaging over both residue classes and moduli. In a restricted range, we confirm an averaged version of a recent conjecture about the asymptotics of this variance. This result is closely related to moments of Dirichlet $L$-functions and our proof relies on the asymptotic large sieve.
Generalized divisor functions in arithmetic progressions: I
本文研究$k$重除数函数在模数超过$\sqrt{X}$的算术级数中的分布，结合张益唐方法并在广义黎曼假设下获得与$k$无关的一致幂次节省误差项，同时避免依赖Siegel定理。
In this paper, we confirm a smoothed version of a recent conjecture on the variance of the k-fold divisor function in arithmetic progressions to individual composite moduli, in a restricted range. In contrast to a previous result of Rodgers and Soundararajan, we do not require averaging over the moduli. Our proof adapts a technique of S. Lester who treated in the same range the variance of the k-fold divisor function in the short intervals setting, and is based on a smoothed Voronoi summation formula but twisted by multiplicative characters. The use of Dirichlet characters allows us to extend …
Primes in arithmetic progressions to large moduli, and Goldbach beyond the square-root barrier
本文利用三重可良好分解权重将素数分布水平提升至0.617，并在假设Selberg猜想下达到0.625，由此为孪生素数和Goldbach问题取得新上界，实现了Goldbach问题自1966年以来的最大改进。
Multiplicative relations among differences of singular moduli
该文证明奇异模量差值的乘法关系仅存在于有限集合或有限多条代数曲线上，且曲线可显式确定。
We prove that for a large class of multiplicative functions, referred to as generalized divisor functions, it is possible to find a lower bound for the corresponding variance in arithmetic progressions. As a main corollary, we deduce such a result for any $α$-fold divisor function, for any complex number $α\not\in \{1\}\cup-\mathbb{N}$, even when considering a sequence of parameters $α$ close in a proper way to $1$. Our work builds on that of Harper and Soundararajan, who handled the particular case of $k$-fold divisor functions $d_k(n)$, with $k\in\mathbb{N}_{\geq 2}$.
Rankin--Selberg coefficients in arithmetic progressions modulo prime powers
研究模素数幂的Rankin-Selberg系数分布，基于Ramanujan-Petersson猜想，证明系数具有特定分布水平。
Part-and-parcel of the study of "multiplicative number theory" is the study of the distribution of multiplicative functions in arithmetic progressions. Although appropriate analogies to the Bombieri-Vingradov Theorem have been proved for particular examples of multiplicative functions, there has not previously been headway on a general theory; seemingly none of the different proofs of the Bombieri-Vingradov Theorem for primes adapt well to this situation. In this article we find out why such a result has been so elusive, and discover what can be proved along these lines and develop some limita…
For any $ε>0$, there exists $q_0(ε)$ such for any $q\ge q_0(ε)$ and any invertible residue class $a$ modulo $q$, there exists a natural number that is congruent to $a$ modulo $q$ and that is the product of exactly three primes, all of which are below $q^{\frac{3}{2}+ε}$. If we restrict our attention to odd moduli $q$ that do not have prime factors congruent to 1 mod 4, we can find such primes below $q^{\frac{11}{8}+ε}$. If we further restrict our set of moduli to prime $q$ that are such that $(q-1,4\cdot7\cdot11\cdot17\cdot23\cdot29)=2$, we can find such primes below $q^{\frac{6}{5}+ε}$. Final…
The study of Ramanujan-type congruences for functions specific to additive number theory has a long and rich history. Motivated by recent connections between divisor sums and overpartitions via congruences in arithmetic progressions, we investigate the existence and classification of Ramanujan-type congruences for functions in multiplicative number theory.
Under fairly general conditions, we show that families of integer-valued polynomial-like multiplicative functions are uniformly distributed in coprime residue classes mod $p$, where $p$ is a growing prime (or nearly prime) modulus. This can be seen as complementary to work of Narkiewicz, who obtained comprehensive results for fixed moduli.
We consider the multiplicative structure of sets of the form AA+1, where where A is a large, finite set of real numbers. In particular, we show that the additively shifted product set, AA+1 must have a large part outside of any generalized geometric progression of comparable length. We prove an analogous result in finite fields as well.
Primes in Arithmetic Progressions to Large Moduli and Siegel Zeroes
本文在例外特征假设下，将素数在算术级数中的渐近均匀分布推广到模数 $q$ 可达 $x^{2/3-\varepsilon}$，并证明对几乎所有 $a$ 或 $q$ 成立，优于此前 $\theta=30/59$ 或 $16/31$ 的结果。
A pretentious proof of Linnik's estimate for primes in arithmetic progressions
本文采用pretentious方法，基于pretentious大筛法思想与积性函数处理思路，证明了算术级数上von Mangoldt函数之和的强一致估计，该估计与Linnik的相关经典估计类似。
Primes in arithmetic progressions and short intervals without $L$-functions
本研究提出一种可检测乘法结构集合中素数的筛法，给出了Linnik最小素数问题和短区间素数存在性的无$L$-函数新证明，为相关数论问题提供了新的研究工具。
Halász's Theorem gives an upper bound for the mean value of a multiplicative function $f$. The bound is sharp for general such $f$, and, in particular, it implies that a multiplicative function with $|f(n)|\le 1$ has either mean value $0$, or is "close to" $n^{it}$ for some fixed $t$. The proofs in the current literature have certain features that are difficult to motivate and which are not particularly flexible. In this article we supply a different, more flexible, proof, which indicates how one might obtain asymptotics, and can be modified to short intervals and to arithmetic progressions. W…
We describe a straightforward method to generate a random prime q such that the multiplicative group GF(q)* also has a random large prime-order subgroup. The described algorithm also yields this order p as well as a p'th primitive root of unity. The methods here are efficient asymptotically, but due to large constants may not be very useful in practical settings.
On the exponent of distribution for convolutions of $\mathrm{GL(2)}$ coefficients to smooth moduli
研究GL(2)系数与光滑模数卷积在算术级数中的分布指数，证明当模数q为平方自由时，该指数可达到1/2+1/46。
Let $f_1,\ldots,f_k : \mathbb{N} \rightarrow \mathbb{C}$ be multiplicative functions taking values in the closed unit disc. Using an analytic approach in the spirit of Halász' mean value theorem, we compute multidimensional averages of the shape $$x^{-l} \sum_{\mathbf{n} \in [x]^l} \prod_{1 \leq j \leq k} f_j(L_j(\mathbf{n}))$$ as $x \rightarrow \infty$, where $[x] := [1,x]$ and $L_1,\ldots, L_k$ are affine linear forms that satisfy some natural conditions. Our approach gives a new proof of a result of Frantzikinakis and Host that is distinct from theirs, with \emph{explicit} main and error te…
Uniform distribution of polynomially-defined additive function to varying moduli
本文研究一类模可变的多项式定义加性函数的一致分布，借鉴Pollack和Singha Roy的方法，结合Cochrane和Loh的指数和界技术进行证明。
Arithmetic progressions in dense subsets of primitive elements of finite fields
本文证明有限域本原元稠密子集必含非平凡k项算术级数，并构造无此类级数的大子集，结合特征和估计与相对Szemerédi定理。
Mean values of multiplicative functions and applications to residue-class distribution
本文给出了乘性函数部分和的一致界，并据此对 Alladi-Erdős 函数和 Euler totient 函数在不同模数下的剩余类分布给出了近乎最优的估计与均匀性量化分析。
On the Mean Value of $D_k(n)$ in Arithmetic Progressions
本文研究了在算术进展中 $D_k(n)$ 的均值，通过建立渐近公式推广了之前的研究成果。
Let $τ(n)$ be Ramanujan's tau function, defined by the discriminant modular form \[ Δ(z) = q\prod_{j=1}^{\infty}(1-q^{j})^{24}\ =\ \sum_{n=1}^{\infty}τ(n) q^n \,,q=e^{2πi z} \] (this is the unique holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer's conjecture asserts that $τ(n)\neq 0$ for all $n\geq 1$; since $τ(n)$ is multiplicative, it suffices to study primes $p$ for which $τ(p)$ might possibly be zero. Assuming standard conjectures for the twisted symmetric power $L$-functions associated to $τ$ (including GRH), we prove that if $x\geq 10^{50}$, then \[ \#\{x < p\leq…
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