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Inequalities $\pi(x^a+y^b)^\alpha\leq \pi(x^c)^\beta+\pi(y^d)^\gamma$ involving the prime-counting function, where the constants are very close to $1$

Let $\pi(x)$ be the prime-counting function, I'm curious about if a suitable variant of the second Hardy–Littlewood conjecture (this corresponding Wikipedia) $$\pi(x^a+y^b)^\alpha\leq \pi(x^c)^\beta+\...
user142929's user avatar
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0 answers
133 views

Is this weaker form of Chowla's conjecture potentially interesting?

I began to read a post on Terry Tao's blog dealing with Chowla's conjecture, and doing so I came to think about a weaker form thereof. So for a given positive integer $m$, let $S_{m}(x)$ denote the ...
Sylvain JULIEN's user avatar
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0 answers
199 views

List of properties of Twin primes Dirichlet series

In a paper R. Arenstorf - There are infinitely many prime twins he stated the following Dirichlet series : $$ T(s) = \sum_{n=1}^\infty \frac{\Lambda(n)\Lambda(n+2)}{n^s} $$ Question : What are ...
TPC's user avatar
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Is it possible to get a conjecture similar to Mandl's conjecture for a different arithmetic function of number theory, mainly related to primes?

I'm curious to know if are in the literature analogous conjectures to the conjecture due to Mandl, I ask about these analogous conjectures for different sequences playing an important role in number ...
user142929's user avatar
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0 answers
128 views

Number of primes skipped by binomial coefficients?

Take $$B(l,n)=\binom{n+l}{n}$$ and $\mathcal P(t)=\{p\mbox{ prime}:p|t\}$. What is the cardinality of $\mathcal P(B(l,n))$? What is minimum cardinality of $L\subseteq\{1,\dots,n\}$ such that $$\...
Turbo's user avatar
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0 answers
114 views

The best error term for the second moment

Let $r_2(n)$ be the number of representations of a positive integer $n$ as a sum of two prime squares, i.e. $n=p^2+q^2$. Consider $S_1(x)= \sum_{n \le x} r_2(n)$ and $S_2(x) = \sum_{n \le x}r_2^2(n)$. ...
RunForrest's user avatar
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0 answers
116 views

Reference request for bounds of $n$-th composite

Motivation I will briefly elaborate here my motivations for asking the question. If you are not interested in it then please go to the questions. Recently during trying to understand and prove the ...
user avatar
0 votes
0 answers
142 views

Mobius function on values of an irreducible quadratic polynomial

Are there infinitely many integers $n$ for which $n^2 + 1$ is square-free, and has an even number of (necessarily distinct) prime factors ?
Pablo's user avatar
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0 answers
230 views

Factorial : Gamma :: Primorial :?

Is there a unique function with the following properties: f is meromorphic on the complex plane; f is log-convex for n ≥ 1 $f(n) = n\#$ for n prime and ≥ 2, where # is the primorial function, and $f(...
Zemyla's user avatar
  • 309
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0 answers
110 views

Bound for $|p_n - \operatorname{li}^{-1}(n)|$

It is well-known that $|\pi(x)-\operatorname{li}(x)| \leq \epsilon(x)$, where $\pi(x) = \sum \limits_{p \leq x} 1$ is the prime counting function, where $\operatorname{li}(x) = \int \limits_{2}^{x}\...
Ahmad's user avatar
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0 answers
164 views

Sequences that "capture their primes"

Let $g : \mathbb{N} \rightarrow \mathbb{N}$ be an increasing function, and consider the sequence $Y = \{y_n\}$ given by $y_n = g(n)$. Let $F$ be an irreducible binary form of degree $d \geq 2$ with ...
Stanley Yao Xiao's user avatar
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0 answers
112 views

Explicit formula for k-central numbers

Given a positive integer $ n $ and assuming Goldbach's conjecture, let $r_{0}(n)$ denote the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are primes. Let $k_{0}(n)$ denote 'the ...
Sylvain JULIEN's user avatar
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0 answers
349 views

Asymptotic value of sum over Möbius function

Consider the sum $$ S(x)=\sum_{\substack{{d\mid P(\sqrt{x})}\\{d\leq x}}}|\mu(d)|, $$ where $P(x)$ is the product of all primes less than or equal to $x$, and $\mu(d)$ is the Möbius function. Q: ...
user45947's user avatar
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0 answers
140 views

Bounding $\dfrac{r(x)}{\pi(x+r(x))-\pi(x-r(x))}$ with $1\ll r(x)\ll \log^{4}(x)$

I would like to know whether it is possible to obtain the bounds $\sqrt{r(x)}\ll k(x)\ll r(x)$ where $k(x)={\pi(x+r(x))-\pi(x-r(x))}$ and $1\ll r(x)\ll \log^{4}(x)$ and thus $1\ll\dfrac{r(x)}{\pi(x+r(...
Sylvain JULIEN's user avatar
-1 votes
3 answers
1k views

Which even numbers are known to be both prime gaps and the sum of 2 primes?

Goldbach's conjecture asserts that every even integer greater than $3$ is the sum of two primes, while de Polignac's one says every even positive integer is a prime gap infinitely often. My question ...
Sylvain JULIEN's user avatar
-1 votes
2 answers
371 views

Conjecture: $ x\leqslant f(x)\leqslant x+x^{\log_{113}13} \{x|1\leqslant x\leqslant+\infty,x\in \mathrm{positive~integer}\}$?

Function f(x) is the most closest prime number not less than $x$. $f(5)=5\qquad f(9)=11$ Conjecture: $ x\leqslant f(x)\leqslant x+x^{\log_{113}13} \{x|1\leqslant x\leqslant+\infty,x\in \mathrm{...
scibee's user avatar
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-1 votes
1 answer
127 views

Statement about upper density of even numbers satisfying the Goldbach condition

For $A\subseteq \mathbb{N}$, let the upper density of $A$ be defined by $$\mu^+(A) = \lim\sup_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}.$$ Let $$A = \{n\in\mathbb{N}: 2n \text{ is the sum of } 2 \...
Dominic van der Zypen's user avatar
-1 votes
1 answer
246 views

A conjecture about an inequality that involve Ramanujan primes

In this post we denote for integers $n\geq 1$ the $n$-th Ramanujan prime as $R_n$ (thus the sequence A104272 from the On-Line Encyclopedia of Integer Sequences), I add a conjecture that I think can be ...
user142929's user avatar
-1 votes
1 answer
199 views

Find the values of $n$ that satisfy this inequality involving a product over prime numbers [closed]

Inequality What values of $n$ satisfy the following inequality? $$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-2}{p_i}\right)$$ $p$ are prime numbers and the notation $p_i$ indicates the $i$th ...
Brad Graham's user avatar
-1 votes
1 answer
250 views

Significance of $N_0(T+1)-N_0(T)\sim \frac{1}{2\pi}\log \frac{T}{2\pi}$

Let $N(T)$ be the number of zeros of Riemann zeta function upto height $T$ in the critical strip and $N_0(T)$ be the number of zeros on the critical line. What will be the significance of proving ...
user avatar
-1 votes
1 answer
258 views

A number theoretical identity of exponential sum

I try to understand a number theoretical identity used by Jan-Christoph Schlage-Puchta in this answer. He defined the function $$S(\alpha)=\sum_{n\leq N}\Lambda(n) e(n\alpha)$$ where $\Lambda(n)$ is ...
user267839's user avatar
  • 6,018
-1 votes
1 answer
115 views

probability of m to be a primality radius of n

Disclaimer : this is a crosspost from MSE, as the question got one upvote but no comment or answer whatsoever. Assuming Goldbach's conjecture, let's denote by $r_{ 0}(n):=\inf\{r\geq 0,(n−r,n+r)\in\...
Sylvain JULIEN's user avatar
-1 votes
1 answer
125 views

Prove using Dyck naturals: for $n \in \mathbb{N}_{+}$ and big enough $k \in \mathbb{N}_{+}$, $p_{k-1} < \cdots < np_{k-a_{n}}$ (a is A073093)

While conducting research in connection with arXiv:2102.02777 ("Recursive Prime Factorizations: Dyck Words as Numbers"), I noticed certain interesting patterns, one of which inspired the ...
JustAsking's user avatar
-1 votes
0 answers
347 views

On a Duality between Riemann-weil explicit formula and Abel- Plana summation of trigonometric prime counting function:

Consider the analytic function $g(x)$ Now define $f(x)=g(x)\frac{\sin^2\left(\frac{π\Gamma(x)}{2x}\right)}{\cos^2\left(\frac{π}{2x}\right)}$ Such that $|f(x+it)|=o(e^{2πt})$ uniformly for every $x$...
TPC's user avatar
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-1 votes
1 answer
279 views

A question on assigning finite values to divergent sums involving expression of primes

We know the following: $$\gamma=\lim_{n\to\infty }\left(\sum_{k=1}^n\frac{1}{k}-\ln(n)\right).$$ This could be a good candidate for renormalized sum of $\left(\sum_{k=1}^{\infty}\frac{1}{k}\right)$. ...
Zaza's user avatar
  • 149
-1 votes
1 answer
243 views

Inversion shift of a Galois radius

Say a non negative $r$ is a Galois radius of $n$ of type $(a,b)$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ prime and positive $a$ and $b$. If $a\neq b$, say $r$ is "unbalanced" and say $s$ ...
Sylvain JULIEN's user avatar
-2 votes
1 answer
834 views

Is my proof a notable result? If so, where and how do I publish it? [closed]

I can prove that given $ε$ chosen arbitrarily small, if $\prod_{p \le p_k} p^{\frac{1}{p-1}} \lt \frac{1 + ε}{e} p_k$ then $∀n\ge p_k∃p∈\mathbb{P} | n \le p \lt (1 + ε)n$. Actually this result is ...
user1582006's user avatar
-2 votes
2 answers
149 views

Calculate the great common factor between $2^{2n+1}-1$ and $2^{4m+2}+1$ [closed]

How to calculate the great common factor between $2^{2n+1}-1$ and $2^{4m+2}+1$, where $n$ and $m$ are positive numbers. We guess that: the great common factor is $1$.
C. Simon's user avatar
  • 577
-2 votes
1 answer
181 views

Polynomials of minimum degree that interpolate primes in intervals

Given an interval $[a,b]$ what is the minimum degree of univariate polynomials in $\mathbb Q[x]$ that passes through all primes between $a$ and $b$ (denoted by $\mathbb P[a,b]$ with total number of ...
VS.'s user avatar
  • 1,826
-3 votes
1 answer
271 views

Is this Goldbach conjecture related quantity equal to the number of Goldbach decompositions up to a bounded quantity?

This question is a follow-up to About Goldbach's conjecture and as such deals with the notion of primality radius of a composite integer $n$, that is, a positive integer $r$ such that both $n-r$ ...
Sylvain JULIEN's user avatar
-3 votes
1 answer
178 views

Asymptotic behavior of $\sum_{k=1}^{n}\frac{p_{k+1}}{p_{k+1}-p_k}$

I refer to my previous question Asymptotic behavior of a certain sum of ratios of consecutives primes. We can split the sum $$\sum_{k=1}^{n}\frac{p_{k+1}+p_k}{p_{k+1}-p_k}$$ where $p_k$ stands for the ...
Augusto Santi's user avatar
-3 votes
1 answer
245 views

Can this weakening of Polignac's conjecture be proven?

Let $ A $ be a set of odd primes such that between any two consecutive elements thereof there is at least one prime gap that occurs infinitely often, i.e. an even integer $ g $ such that the ...
Sylvain JULIEN's user avatar
-3 votes
1 answer
250 views

Twin prime based Dirichlet series

Assuming there are infinitely many twin primes, one can consider a Dirichlet series $ \sum_{n>0}a_{n}{n^{-s}} $ and replace the sequence of positive integers with the sequence of twin primes. That ...
Sylvain JULIEN's user avatar
-3 votes
1 answer
237 views

L. Gegenbauer's proof of Infinitude of Primes [closed]

I was going through the paper 'Euclid’S theorem on the infinitude of primes: A historical survey of its proofs' by Romeo Mestrovic where he mentioned that L. Gegenbauer proved Infinitude of Primes by ...
math is fun's user avatar
-3 votes
1 answer
269 views

Negative Dirichlet Pigeonhole Principle [closed]

From Dirichlet Pigeonhole Principle if $p$ is a prime and if $a,b\in\mathbb Z$ are in $(0,p/2)$ then there is a $t\in(0,p)\cap\mathbb Z$ such that $\|(x,y)\|_\infty<\lceil\sqrt p\rceil$ holds where ...
Turbo's user avatar
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-3 votes
0 answers
70 views

Is the upper bound on $H_{1}$ a decreasing function of the proportion of critical zeros of Zeta?

This question stems from https://arxiv.org/abs/2411.19762 and the numerical observation that the best unconditional upper bound for $H_{1}:=\lim\inf_{n\to\infty}p_{n+1}-p_{n}$, namely $H_{1}^{\flat}=...
Sylvain JULIEN's user avatar
-4 votes
3 answers
670 views

Remarkable articles about the distribution of prime numbers that were written by contemporary physicists [closed]

I would like to ask about if you know papers containing remarkable achievements that were written by contemporary physicists concerning the distribution of prime numbers (or closely related, maybe the ...
-4 votes
1 answer
465 views

Asymptotic formula for $\prod_{p\leq x} (1-p^{-1})$ [closed]

Does there exists a good asymptotic formula for $$A(x) := \prod_{p\leq x}(1-\frac 1p).$$ By using a heuristic argument one can guess: $$A(x) \sim \frac{1}{2\,\mathrm{ln}(x)}.$$ Here is the argument:...
Mostafa - Free Palestine's user avatar
-4 votes
1 answer
229 views

A generalization Bertrand's postulate [closed]

Let $n, k$ are integers number such that $1<n \le k$, does always exist a prime number between $kn$ and $k(n+1)$? When $n=1, k>1$ always exist a prime number between $k$ and $2k$ the question ...
Đào Thanh Oai's user avatar
-10 votes
1 answer
407 views

Summatory functions for fractional parts

Notation: $$ \{x\}\ :=\ x-\lfloor x\rfloor $$ APF-functions $\ \tau(n)\ $ for $\ 2<n\in\mathbb N,\ $ and $\ \xi(n)\ $ for $\ 3<n\in\mathbb N,\ $ are defined as follows: $$ \tau(n)\ :=\ \sum_{k=...
Wlod AA's user avatar
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