Note on the Prime Number Theorem

A note on Primes in Short Intervals

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].

Some Notes on Digit Strings in the Primes

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 $S$.

关于实二次域中素理想分布的一个注记

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。

A note on primes with prime indices

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…

A Note on Prime Fibonacci Sequences

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.

A note on bounded gaps between primes

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.

Notes on properties of binary strings which encode prime occurrences

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…

Note On Prime Gaps And Very Short Intervals

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.

Notes on Pair Correlation of Zeros and Prime Numbers

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.

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.

关于素数簇与素数荒漠分布的注记

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。

关于 $\mathbb{F}_p^*$ 中某子群的素数指标的注记

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 $46\\%$。

A note on the normal largest gap between prime factors

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 $n$.

A Note On Non-ordinary Primes

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…

A note on prime number races and zero free regions for $L$ functions

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 $Re(s)>1-ε$.

一则非寻常（素数）注记

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。

关于相邻素数之差的零密度方法的注记

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。

Hua's Theorem with the Primes in Piatetski-Shapiro Prime Sets

In this paper, we study the hybrid problem of Hua's theorem and the Piatetski-Shapiro prime number theorem, and obtain results in this direction of the nonhomogeneous case $k=3$, which deepen the classical result of Hua.

Note on the additive complements of primes

We extend two results of Ruzsa and Vu on the additive complements of primes

A note on the $\Sopfr(n)$ function

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.

关于黎曼R函数与殆素数的注记

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。

Mertens' theorem and prime number theorem for Selberg class

In 1874, Mertens proved the approximate formula for partial Euler product for Riemann zeta function at $s=1$, which is called Mertens' theorem. In this paper, we generalize Mertens' theorem for Selberg class and show the prime number theorem for Selberg class.

New proof of Tauberian theorem of Laplace Transform and Prime Number Theorem

We give here a new proof of a Tauberian Theorem of complex Laplace transform using the Theory of measure and theory of function with bounded variations. However we deduce the simple proof of Prime Number Theorem.

A note on primes in certain residue classes

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 $\prod_{i=1}^k \left(1-\frac{1}{a_i}\right)$.

通过单边Tauberian定理证明素数定理

The prime number theorem through one-sided Tauberian theorems

本文提供了单边Ingham-Karamata Tauberian定理的优雅证明，并以此为基础简短推导出素数定理。

Medium-sized values for the Prime Number Theorem for primes in arithmetic progression

We give two improved explicit versions of the prime number theorem for primes in arithmetic progression: the first isolating the contribution of the Siegel zero and the second completely explicit, where the improvement is for medium-sized values. This will give an improved explicit Bombieri-Vinogradov like result for non-exceptional moduli.

A prime Geodesic Theorem for SL3(Z)

We show a Prime Geodesic Theorem for the group SL3(Z), counting those geodesics whose lifts lie in the split Cartan subgroup. This is the first arithmetic Prime Geodesic Theorem of higher rank for a non-cocompact group.

A note on the distribution of normalized prime gaps

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 …

A combinatorial theorem of Schur implies infinitude of primes

Comments about the paper by Elsholz, Fermat's last theorem implies Euclid's infinitude of primes, (2021), and simplification.

An analogue of the prime number theorem for finite fields

We prove an analogue of the prime number theorem for finite fields.

Finiteness theorems for universal sums of squares of almost primes

In this paper we study quadratic forms which are universal when restricted to almost prime inputs, establishing finiteness theorems akin to the Conway--Schneeberger 15 theorem.

Fermat's Last Theorem, Schur's Theorem (in Ramsey Theory), and the Infinitude of the Primes

Alpoge and Granville (separately) gave novel proofs that the primes are infinite that use Ramsey Theory. In particular, they use Van der Waerden's Theorem and some number theory. We prove the primes are infinite using an easier theorem from Ramsey Theory, namely Schur's Theorem, and some number theory (Elsholtz independently obtained the same proof that the primes were infinite). In particular, we use the n=3 case of Fermat's last theorem. We also apply our method to show other domains have an infinite number of irreducibles.

The ternary Goldbach-Vinogradov theorem with almost equal Piatetski-Shapiro primes

In this paper, we proved a theorem that every large enough odd number can be represented as the sum of three almost equal Piatetski-Shapiro primes.

The error term in the prime number theorem

We make explicit a theorem of Pintz concerning the error term in the prime number theorem. This gives an improved version of the prime number theorem with error term roughly square-root of that which was previously known. We apply this to a long-standing problem concerning an inequality studied by Ramanujan.

Prime geodesic theorem of Gallagher type

We reduce the exponent in the error term of the prime geodesic theorem for compact Riemann surfaces from $\frac{3}{4}$ to $\frac{7}{10}$ outside a set of finite logarithmic measure.

指数和与Piatetski-Shapiro素数的Barban-Davenport-Halberstam型定理

Barban-Davenport-Halberstam type theorems for exponential sums and Piatetski-Shapiro primes

该研究在数论领域建立了三类Barban-Davenport-Halberstam型定理，分别适用于素数指数和、Piatetski-Shapiro素数及该类素数上的指数和，拓展了相关定理的应用范围。

一个涉及素数及素数乘积的不等式

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。

关于三井带Siegel零点素数定理的注记

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。

A Conditional Explicit Result for the Prime Number Theorem in Short Intervals

We furnish an explicit bound for the prime number theorem in short intervals on the assumption of the Riemann hypothesis.

Landau素数理想定理的一个新的初等证明及相关结果

A New Elementary Proof of Landau's Prime Ideal Theorem, and Associated Results

本文给出Landau素数理想定理的新初等证明，推广了Richter的素数定理证明，并包含素数理想计数函数等分布的相关结果。

An alternative proof of the Dirichlet prime number theorem

Dirichlet's theorem on arithmetic progressions called as Dirichlet prime number theorem is a classical result in number theory. Atle Selberg\cite{Selberg} gave an elementary proof of this theorem. In this article we give an alternative proof of it based on a previous result of us. Also we get an estimation of the prime counting function in the special cases.

A Direct Proof of the Prime Number Theorem using Riemann's Prime-counting Function

In this paper, we develop a novel analytic method to prove the prime number theorem in de la Vallée Poussin's form: $$ π(x)=\operatorname{li}(x)+\mathcal O(xe^{-c\sqrt{\log x}}) $$ Instead of performing asymptotic expansion on Chebyshev functions as in conventional analytic methods, this new approach uses contour-integration method to analyze Riemann's prime counting function $J(x)$, which only differs from $π(x)$ by $\mathcal O(\sqrt x/\log x)$.

On Littlewood's proof of the prime number theorem

In this note we examine Littlewood's proof of the prime number theorem. We show that this can be extended to provide an equivalence between the prime number theorem and the non-vanishing of Riemann's zeta-function on the one-line. Our approach goes through the theory of almost periodic functions and is self-contained.

A prime geodesic theorem for higher rank buildings

We prove a prime geodesic theorem for compact quotients of affine buildings and apply it to get class number asymptotics for global fields of positive characteristic.

Explicit Bound for the Prime Ideal Theorem in Residue Classes

We give explicit numerical estimates for the generalized Chebyshev functions. Explicit results of this kind are useful for estimating of computational complexity of algorithms which generates special primes. Such primes are needed to construct an elliptic curve over prime field using complex multiplication method.

A Note on the Andrica Conjecture

We derive heuristically the approximate formula for the difference $\sqrt{p_{n+1}} - \sqrt{p_n}$, where $p_n$ is the n-th prime. We find perfect agreement between this formula and the available data from the list of maximal gaps between consecutive primes.

Explicit versions of the prime ideal theorem for Dedekind zeta functions under GRH, II

We have proved recently several explicit versions of the prime ideal theorem under GRH. Here we prove a version with optimal asymptotic behaviour.

The prime number theorem for primes in arithmetic progressions at large values

Assuming the Riemann hypothesis, we prove the latest explicit version of the prime number theorem for short intervals. Using this result, and assuming the generalised Riemann hypothesis for Dirichlet $L$-functions is true, we then establish explicit formulae for $ψ(x,χ)$, $θ(x,χ)$, and an explicit version of the prime number theorem for primes in arithmetic progressions that hold for general moduli $q\geq 3$. Finally, we restrict our attention to $q\leq 10\,000$ and use an exact computation to refine these results.

A prime number theorem for the majority function

In the paper, the occurrence of zeros and ones in the binary expansion of the primes is studied. In particular the statement in the title is established. The proof is unconditional.

An alternative proof of Sylvester's theorem and variations for more primes

This document presents an alternative proof of Sylvester's theorem stating that "the product of $n$ consecutive numbers strictly greater than $n$ is divisible by a prime strictly greater than $n$". In addition, the paper proposes stronger versions of Sylvester's theorem and Bertrand's postulate with more primes, as well as an approach to get more results in this direction.

A Quixotic Proof of Fermat's Two Squares Theorem for Prime Numbers

Every odd prime number p can be written in exactly (p + 1)/2 ways as a sum ab+cd of two ordered products ab and cd such that min(a, b) > max(c, d). An easy corollary is a proof of Fermat's Theorem expressing primes in 1 + 4N as sums of two squares.

Prime geodesic theorem for the modular surface

Under the generalized Lindelöf hypothesis, the exponent in the error term of the prime geodesic theorem for the modular surface is reduced to $\frac{5}{8}+\varepsilon $ outside a set of finite logarithmic measure.

Vinogradov's three primes theorem with almost twin primes

In this paper we prove two results concerning Vinogradov's three primes theorem with primes that can be called almost twin primes. First, for any $m$, every sufficiently large odd integer $N$ can be written as a sum of three primes $p_1, p_2$ and $p_3$ such that, for each $i \in \{1,2,3\}$, the interval $[p_i, p_i + H]$ contains at least $m$ primes, for some $H = H(m)$. Second, every sufficiently large integer $N \equiv 3 \pmod{6}$ can be written as a sum of three primes $p_1, p_2$ and $p_3$ such that, for each $i \in \{1,2,3\}$, $p_i + 2$ has at most two prime factors.

关于素数测地线定理

On the prime geodesic theorem

改进了模群素数测地线定理的误差项，并推广到任意不定四元数代数的同余子群。