Additive functions on shifted primes

平均Brun-Titchmarsh定理与具有大素因子的移位素数

An average Brun-Titchmarsh theorem and shifted primes with a large prime factor

该研究利用Maynard的三重卷积估计突破，改进了平均Brun-Titchmarsh定理，并应用于移位素数，证明无穷多个素数$p$满足$p-1$的最大素因子大于$p^{0.679}$。

带受限数位的移位素数的Sárközy定理

Sárközy's theorem for shifted primes with restricted digits

该研究针对带受限数位的移位素数，通过构造关联指数和的局部逼近函数证明其具有van der Corput性质，进而得出正上Banach密度的自然数集A中存在满足a₁+p-1=a₂的相关元素及素数p的结论。

On a conjecture on shifted primes with large prime factors

Let $\mathcal{P}$ be the set of all primes and $π(x)$ be the number of primes up to $x$. For any $n\ge 2$, let $P^+(n)$ be the largest prime factor of $n$. For $0<c<1$, let $$T_c(x)=\#\{p\le x:p\in \mathcal{P},P^+(p-1)\ge p^c\}.$$ In this note, we proved that there exists some $c<1$ such that $$\limsup_{x\rightarrow\infty}\frac{T_c(x)}{π(x)}<\frac12,$$ which disproves a conjecture of Chen and Chen.

Salié和的移位双线性和与移位素数的模平方根分布

Shifted bilinear sums of Salié sums and the distribution of modular square roots of shifted primes

研究Salié和的移位双线性和的上界，并分析移位素数的模平方根分布，揭示$b\not \equiv 0 \bmod q$时的新困难。

The shifted fourth moment of automorphic L-functions of prime power level

We prove the asymptotic formula for the fourth moment of automorphic $L$-functions of level $p^ν$, where $p$ is a fixed prime number and $ν\rightarrow \infty$. This paper is a continuation of work by Rouymi, who computed asymptotics of the first three moments at prime power level, and a generalization of results obtained for prime level by Duke, Friedlander & Iwaniec and Kowalski, Michel & Vanderkam.

On van der Corput property of shifted primes

We prove that the upper bound for the van der Corput property of the set of shifted primes is O((log n)^{-1+o(1)}), giving an answer to a problem considered by Ruzsa and Montgomery for the set of shifted primes p-1. We construct normed non-negative valued cosine polynomials with the spectrum in the set p-1, p<=n, and a small free coefficient a_0=O((log n)^{-1+o(1)}). This implies the same bound for the Poincaré property of the set p-1, and also bounds for several properties related to uniform distribution of related sets.

移位素数因数

Shifted-prime divisors

本文研究由 Prachar 提出的移位素数因数计数函数 $\omega^*(n)$ 的高阶矩及相关问题。

On shifted primes with large prime factors and their products

We estimate from below the lower density of the set of prime numbers p such that p-1 has a prime factor of size at least p^c, where c lies in between 1/4 and 1/2. We also establish upper and lower bounds on the counting function of the set of positive integers n up to x with exactly k prime factors, counted with or without multiplicity, such that the largest prime factor of gcd(p-1 : p | n) exceeds n^{1/2k}.

Divisors of shifted primes

We bound from below the number of shifted primes p+s<x that have a divisor in a given interval (y,z]. Kevin Ford has obtained upper bounds of the expected order of magnitude on this quantity as well as lower bounds in a special case of the parameters y and z. We supply here the corresponding lower bounds in a broad range of the parameters y and z. As expected, these bounds depend heavily on our knowledge about primes in arithmetic progressions. As an application of these bounds, we determine the number of shifted primes that appear in a multiplication table up to multiplicative constants.

Character sums over shifted primes

For integer $q$, let $χ$ be a primitive multiplicative character$\pmod q.$ For integer $a$ coprime to $q$, we obtain a new bound for the sums $$\sum_{n\le N}Λ(n)χ(n+a),$$ where $Λ(n)$ is the von Mangoldt function. This bound improves and extends the range of a result of Friedlander, Gong and Shparlinski

A Hardy-Ramanujan type inequality for shifted primes and sifted sets

We establish an analog of the Hardy-Ramanujan inequality for counting members of sifted sets with a given number of distinct prime factors. In particular, we establish a bound for the number of shifted primes p+a below x with k distinct prime factors, uniformly for all positive integers k.

关于移位半素数的最大大素因子

On the largest prime factors of shifted semiprime numbers

该研究证明了对于固定整数a和足够大的x，移位半素数乘积的最大大素因子超过x^θ，其中θ为0.5减去一个任意小的正数。

The Least Prime Primitive Root and the Shifted Sieve

We derive, for all prime moduli p except those in a very thin set, an upper bound for the least prime primitive root (mod p) of order of magnitude a constant power of log p. The improvement over previous results, where the upper bound was log p to an exponent tending to infinity with p, lies in the use of the linear sieve (a particular version called the shifted sieve) rather than Brun's sieve. The same methods allow us to rederive a conditional result of Shoup on the least prime primitive root (mod p) for all prime moduli p, assuming the generalized Riemann hypothesis. We also extend both res…

Averages of the Möbius function on shifted primes

It is a folklore conjecture that the Möbius function exhibits cancellation on shifted primes; that is, $\sum_{p\le X}μ(p+h) \ = \ o(π(X))$ as $X\to\infty$ for any fixed shift $h>0$. This appears in print at least since Hildebrand in 1989. We prove the conjecture on average for shifts $h\le H$, provided $\log H/\log\log X\to\infty$. We also obtain results for shifts of prime $k$-tuples, and for higher correlations of Möbius with von Mangoldt and divisor functions. Our argument combines sieve methods with a refinement of Matomäki, Radziwiłł, and Tao's work on an averaged form of Chowla's conject…

由移位二次型表示的素数：关于本原性与同余类

Primes represented by shifted quadratic forms: on primitivity and congruence classes

本文扩展了Iwaniec的筛法，证明了移位正定整二元二次型在附加本原性与同余类条件下表示素数的下界，并提供了经计算机验证的局部条件列表以纠正先前错误，旨在为研究Apollonian circle packings中的素数分量提供工具。

Additive decomposition of signed prime

Under the prime-tuple hypothesis, the set of signed primes is a sumset.

具有移位质数分母的单位分数

Unit fractions with shifted prime denominators

该研究证明了任何正有理数均可表示为具有特定移位质数分母的不同单位分数之和，并探讨了同余条件下的扩展情况。

Arithmetic functions at consecutive shifted primes

For each of the functions $f \in \{ϕ, σ, ω, τ\}$ and every natural number $k$, we show that there are infinitely many solutions to the inequalities $f(p_n-1) < f(p_{n+1}-1) < \dots < f(p_{n+k}-1)$, and similarly for $f(p_n-1) > f(p_{n+1}-1) > \dots > f(p_{n+k}-1)$. We also answer some questions of Sierpiński on the digit sums of consecutive primes. The arguments make essential use of Maynard and Tao's method for producing many primes in intervals of bounded length.

Multidimensional configurations in the primes with shifted prime steps

Let $\mathcal{P}$ denote the set of primes. For a fixed dimension $d$, Cook-Magyar-Titichetrakun, Tao-Ziegler and Fox-Zhao independently proved that any subset of positive relative density of $\mathcal{P}^d$ contains an arbitrary linear configuration. In this paper, we prove that there exists such configuration with the step being a shifted prime (prime minus $1$ or plus $1$).

广义移位三元除数函数的加性和

Additive Sums of Generalized Shifted Ternary Divisor Functions

本文针对特定参数范围的广义移位三元加性除数和，在假设相关函数算术级数分布水平的条件下，证明了带显式主项和幂节省误差项的渐近公式。

A new explicit formula in the additive theory of primes with applications I. The explicit formula for the Goldbach and Generalized Twin Prime Problems

A new explicit formula is proved for the contribution of the major arcs in the Goldbach and Generalized Twin Prime Problem, in which the level of the major arcs can be chosen very high. This will have many applications in the approximations of the mentioned problems.

Multiplicative Functions on Shifted Primes

Let $f$ be a positive multiplicative function and let $k\geq 2$ be an integer. We prove that if the prime values $f(p)$ converge to $1$ sufficiently slowly as $p\rightarrow +\infty$, in the sense that $\sum_{p}|f(p)-1|=\infty$, there exists a real number $c>0$ such that the $k$-tuples $(f(p+1),\ldots,f(p+k))$ are dense in the hypercube $[0,c]^k$ or in $[c,+\infty)^k$. In particular, the values $f(p+1),\ldots,f(p+k)$ can be put in any increasing order infinitely often. Our work generalises previous results of De Koninck and Luca.

连续素数之间大区间中某类加性函数的行为

The behaviour of a certain additive function in large intervals between consecutive primes

本文研究连续素数之间大区间内特定整数素因子所决定的加性函数的行为，结合素数大间隙相关结果、Erdős–Kac 定理及 Kubilius 模型得到主要结论。

平移素数的最大的平方因子

Largest square divisors of shifted primes

该论文证明存在无穷多个素数 $p$，使得对任意非零整数 $a$，$p-a$ 有大于 $p^{1/2+1/700}$ 的平方因子，改进了 Merikoski 的指数，并使用了 Harman 筛法中的强力工具。

非主特征在移位素数上的取值之和

Sums of values of non-principal characters over shifted primes

本文研究非主特征在移位素数序列上的取值和，在 $x\ge D^{5/6+\varepsilon}$ 条件下证明了带指数衰减因子的非平凡上界估计。

An Additive Problem over Piatetski-Shapiro Primes and Almost-primes

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, we establish a theorem of Bombieri-Vinogradov type for the Piatetski-Shapiro primes $p=[n^{1/γ}]$ with $\frac{85}{86}<γ<1$. Moreover, we use this result to prove that, for $0.9989445<γ<1$, there exist infinitely many Piatetski-Shapiro primes such that $p+2=\mathcal{P}_3$, which improves the previous results of Lu, Wang and Cai, and Peneva.

移位素数中的和积模式

Sum-product patterns in the shifted primes

本文证明移位素数集 $\mathbb{P}-1$ 含无穷多和积模式 $\{x,x+y,xy\}$，并推广至任意长度 $k$ 的模式，涵盖 Green--Tao 定理。

On improving a Schur-type theorem in shifted primes

We show that if $N \geq \exp(\exp(\exp (k^{O(1)})))$, then any $k$-colouring of the primes that are less than $N$ contains a monochromatic solution to $p_1 - p_2 = p_3 -1$.

移位素数的乘积同时取完全幂值

Products of shifted primes simultaneously taking perfect power values

本文研究了移位素数的乘积同时取完全幂值的问题，给出了无平方因子整数n的数量下界，并探讨了特定A集合下的素因子数量下界。

On an additive prime divisor function of Alladi and Erdos

This paper discusses the additive prime divisor function $A(n) := \sum\limits_{p^α|| n} α\, p$ which was introduced by Alladi and Erd\H os in 1977. It is shown that $A(n)$ is uniformly distributed (mod $q$) for any fixed integer $q > 1.$

An invitation to additive prime number theory

The main purpose of this survey is to introduce an inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic machinery used to study them, and try to give a representative sample of the directions of current research.

On the largest square divisor of shifted primes

We show that there are infinitely many primes $p$ such that $p-1$ is divisible by a square $d^2 \geq p^θ$ for $θ=1/2+1/2000.$ This improves the work of Matomäki (2009) who obtained the result for $θ=1/2-\varepsilon$ (with the added constraint that $d$ is also a prime), which improved the result of Baier and Zhao (2006) with $θ=4/9-\varepsilon.$ Similarly as in the work of Matomäki, we apply Harman's sieve method to detect primes $p \equiv 1 \, (d^2)$. To break the $θ=1/2$ barrier we prove a new bilinear equidistribution estimate modulo smooth square moduli $d^2$ by using a similar argument as …

移位素数除数函数的矩

Moments of the shifted prime divisor function

本文证明了移位素数除数函数的矩的渐近公式，证实了 Fan 和 Pomerance 的猜想，其证明结合了最小公倍数的组合恒等式与数论解析工具。

关于长度为4的弱素数可加数

On the Weakly Prime-Additive Numbers with Length 4

本文在广义黎曼假设下证明了对任意正整数m，存在无穷多个能被m整除的长度为4的弱素数可加数，并给出了一个与长度为3情形类似的相关结果。

On Heegner Points for primes of additive reduction ramifying in the base field

Let $E$ be a rational elliptic curve, and $K$ be an imaginary quadratic field. In this article we give a method to construct Heegner points when $E$ has a prime bigger than $3$ of additive reduction ramifying in the field $K$. The ideas apply to more general contexts, like constructing Darmon points attached to real quadratic fields which is presented in the appendix.

Additive properties of product sets in fields of prime order

Let $F_p$ be the field of a prime order $p$. Then for any positive integer $n>1$, for any $ε>0$, and for any subset $A$ of $F_p$, every element of $F_p$ can be represented as a sum of $N$ elements, each of them is a product of $n$ elements from $A$, where $N$ depends on $n$ and $\espilon$.

关于具有大素因子的移位素数的一个猜想，II

On a conjecture on shifted primes with large prime factors, II

本文研究移位素数 $p-1$ 具有大素因子的密度问题，证明当 $8/9\le c<1$ 时相应比例的上极限不超过 $8(1/c-1)$，并在 $c\to1$ 时趋于零。

On a theorem of Sárközy for difference sets and shifted primes

We show that if the difference of two elements of a set $A \subseteq [N]$ is never one less than a prime number, then $|A| = O (N \exp (-c (\log N)^{1/3}))$ for some absolute constant $c>0$.

A note on an average additive problem with prime numbers

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.

关于移位素数的Sárközy定理

On Sárközy's theorem for shifted primes

该研究探讨数论中移位素数相关的Sárközy定理，给出无两元素差为素数减1的集合的大小上界。

Additive uniqueness of $\mathtt{PRIMES}-1$ for multiplicative functions

Let $\mathtt{PRIMES}$ be the set of all primes. We show that a multiplicative function which satisfies \[ f(p+q-2) = f(p) + f(q) - f(2) \text{ for }p,q \in \mathtt{PRIMES} \] is one of the following: \begin{enumerate} \item $f$ is the identity function \item $f$ is the constant function with $f(n)=1$ \item $f(n)=0$ for $n \ge2$ unless $n$ is odd and squareful. \end{enumerate} As a consequence, a multiplicative function which satisfies \[ f(a+b) = f(a) + f(b) \text{ for }a,b \in \mathtt{PRIMES}-1 \] is the identity function.

关于移位三重除数函数的 Voronoi 求和公式

A Voronoi summation formula for the shifted triple divisor function

本文证明了由加性特征扭曲的移位3重除数函数的Voronoi求和公式，主要工具是提供了移位$GL(3)$ Estermann函数的函数方程。

Additive problems with almost prime squares

We show that every sufficiently large integer is a sum of a prime and two almost prime squares, and also a sum of a smooth number and two almost prime squares. The number of such representations is of the expected order of magnitude. We likewise treat representations of shifted primes p-1 as sums of two almost prime squares. The methods involve a combination of analytic, automorphic and algebraic arguments to handle representations by restricted binary quadratic forms with a high degree of uniformity.

On additive and multiplicative decompositions of sets of integers with restricted prime factors, I. (Smooth numbers.)

In [10] the third author of this paper presented two conjectures on the additive decomposability of the sequence of ''smooth'' (or ''friable'') numbers. Elsholtz and Harper [4] proved (by using sieve methods) the second (less demanding) conjecture. The goal of this paper is to extend and sharpen their result in three directions by using a different approach (based on the theory of $S$-unit equations).

Average estimate for additive energy in prime field

Assume that $A\subseteq \Fp, B\subseteq \Fp^{*}$, $\1/4\leqslant\frac{|B|}{|A|},$ $|A|=p^α, |B|=p^β$. We will prove that for $p\geqslant p_0(β)$ one has $$\sum_{b\in B}E_{+}(A, bA)\leqslant 15 p^{-\frac{\min\{β, 1-α\}}{308}}|A|^3|B|.$$ Here $E_{+}(A, bA)$ is an additive energy between subset $A$ and it's multiplicative shift $bA$. This improves previously known estimates of this type.

关于一个含一个素数与一个殆素数变量的分数次幂加法问题

On an additive problem involving fractional powers with one prime and an almost prime variables

本文研究分数次幂整数部分的二元加法表示，证明在指数范围内充分大整数可表为素数与素因子个数有界的殆素数之和，并改进已有结果。

给定区间内可被移位素数整除的整数

Integers divisible by a shifted prime in a given interval

本文研究了区间$(y,z]$内形如$p-1$的素数移位除数相关的整数计数函数$H^*(x,y,z)$，发现其相变规律并推广了Ford2017年的成果。

来自薄Bohr集的素数的加性问题

Additive Problems with Primes from a Thin Bohr Set

本文研究满足薄Bohr集条件的素数上的加性问题，证明当τ∈(0,1/8)时该素数集合中存在无穷多个非平凡三项算术级数，并进一步考虑了二元Goldbach型问题。

On additive and multiplicative decompositions of sets of integers with restricted prime factors, II. (Smooth numbers and generalizations.)

In part I of this paper we studied additive decomposability of the set $\F_y$ of th $y$-smooth numbers and the multiplicative decomposability of the shifted set $\g_y=\F_y+\{1\}$. In this paper, focusing on the case of 'large' functions $y$, we continue the study of these problems. Further, we also investigate a problem related to the m-decomposability of $k$-term sumsets, for arbitrary $k$.

Congruences between Ramanujan's tau function and elliptic curves, and Mazur--Tate elements at additive primes

We show that if $E/\mathbb{Q}$ is an elliptic curve with a rational $p$-torsion for $p=2$ or $3$, then there is a congruence relation between Ramanujan's tau function and $E$ modulo $p$. We make use of such congruences to compute the Iwasawa invariants of $2$-adic and $3$-adic Mazur--Tate elements attached to Ramanujan's tau function. We also investigate numerically the Iwasawa invariants of the Mazur--Tate elements attached to an elliptic curve with additive reduction at a fixed prime number.

差递增的等差数列中移位素数乘积的分布

Distribution of products of shifted primes in arithmetic progressions with increasing difference

本文推广并细化了Karatsuba公式，推导出在差递增等差数列中满足特定同余条件的移位素数乘积数量的渐近公式。

Bombieri 无对数密度估计的显式版本与平移素数的 Sárközy 定理

An explicit version of Bombieri's log-free density estimate and Sárközy's theorem for shifted primes

本文给出 Bombieri 无对数密度估计的显式版本，并结合 Green 的工作证明：若集合 A⊆{1,…,N} 中任意两元素之差不为 p−1（p 为素数），则 |A|≪N^{1−1/10^{18}}，从而加强了 Sárközy 定理。

带有任意数量素数幂与平方数的加性问题的Cesàro平均

A Cesàro average for an additive problem with an arbitrary number of prime powers and squares

该研究针对整数表示为素数幂与平方数之和的加性问题，拓展改进了带Cesàro权加权平均的现有结果，覆盖所有已研究情形并借助所选技术得到最优结论。

Anticyclotomic $p$-adic $L$-functions for elliptic curves at some additive reduction primes

Let $E$ be a rational elliptic curve and let $p$ be an odd prime of additive reduction. Let $K$ be an imaginary quadratic field and fix a positive integer $c$ prime to the conductor of $E$. The main goal of the present article is to define an anticyclotomic $p$-adic $L$-function $Ł$ attached to $E/K$ when $E/\QQ_p$ attains semistable reduction over an abelian extension. We prove that $Ł$ satisfies the expected interpolation properties; namely, we show that if $χ$ is an anticyclotomic character of conductor $cp^n$ then $χ(Ł)$ is equal (up to explicit constants) to $L(E,χ,1)$ or $L'(E,χ,1)$.