All Questions
Tagged with analytic-number-theory prime-numbers
690 questions
2
votes
1
answer
342
views
Can someone explain some of the steps in this paper clearly?
I'm having trouble understanding the steps this paper makes to come to the conclusion $p_{f}(d) \sim e^d\sqrt{d}$
Marek Wolf, First occurrence of a given gap between consecutive primes, preprint, ...
- 105k
0
votes
1
answer
356
views
A sufficient condition for a set of primes to be the set of reducibility of an integer polynomial
Let $P$ be the set of all positive primes. Let $S$ an arbitrary infinite subset of $P$ satisfying the following assumption: there exists a finite Galois extension $K$ of $\mathbb{Q}$ and a conjugacy ...
CommunityBot
- 1
2
votes
1
answer
551
views
Sets of primes with a given Frobenius conjugacy class
Let $a$ and $b$ be two relatively prime positive integers. Denote by $S$ the set of all positive primes of the form $a+bn$ (where $n$ is an integer, possibly zero or negative). Let $S'$ be any set of ...
- 105k
4
votes
1
answer
332
views
Estimating certain short Kloosterman sums
Recall that for the classical Kloosterman sum
$$ K(a,b,p^t):= \sum_{x \in (\mathbb{Z}/ p^t \mathbb{Z})^* } \psi \left(\frac{ax+bx^{-1}}{p^t} \right),$$
where $\psi(x)=e^{2\pi ix}$, $a,b,t$ are natural ...
- 12.3k
2
votes
1
answer
170
views
Numerical estimates for a function relating to twin primes :
Consider the following function :
$$F(s)= \sum_{\text{$p,\ p+2$ are primes}} \left({\frac{1}{p^s}}+{\frac{1}{(p+2)^s}}\right).$$
Brun's theorem tells us that $F(1)$ is finite.
We are looking for ...
- 105k
0
votes
2
answers
153
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Is the abscissa of convergence of $s\mapsto\sum_{n>0}(ng_{n}/2)^{-s}$ known?
The famous Polignac conjecture posits that the $n$-th prime gap $g_{n}:=p_{n+1}-p_{n}$ attains all even positive integral values infinitely many times, which implies $\displaystyle{\zeta_{Pol}:=s\...
- 6,476
2
votes
0
answers
156
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Questions about a certain sequence of naturals generated by primorials
I'm working on the following sequence of naturals (which is NOT listed in OEIS)
$$3,5,11,17,23,29,59,89,119,149,179,209,419,629,839,1049,1259,1469,1679,...$$
whose elements are generated this way
$$3=(...
- 63.9k
-1
votes
1
answer
246
views
A conjecture about an inequality that involve Ramanujan primes
In this post we denote for integers $n\geq 1$ the $n$-th Ramanujan prime as $R_n$ (thus the sequence A104272 from the On-Line Encyclopedia of Integer Sequences), I add a conjecture that I think can be ...
4
votes
2
answers
1k
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Calculating the infinite product from the Hardy-Littlewood Conjecture F
The Hardy-Littlewood Conjecture F [1] involves the infinite product
$$\prod\left(1-\frac{1}{\varpi-1}\left(\frac D\varpi\right)\right)$$
where $\varpi$ ranges over the odd primes and $\left(\frac D\...
CommunityBot
- 1
3
votes
0
answers
125
views
On the set $\{n>0:\ n\ \text{is a quadratic nonresidue modulo the}\ n\text{th prime}\}$
Let $S$ denote the set of positive integers $n$ with $n$ a quadratic nonresidue modulo the $n$th prime $p_n$. The first 20 elements of $S$ are
$$2,\, 3,\, 6,\, 7,\, 8,\, 10,\, 11,\, 13,\, 15,\, 18,\, ...
- 15.6k
0
votes
1
answer
137
views
A density zero set of primes dividing the values of a non-constant integer polynomial
For a given $P\in \mathbb{Z}[x]$ call a positive prime $p$ good if there exists $n\in \mathbb{Z}$ such that $p$ divides $P(n)$. Does there exist a non-constant $P$ such that the set of good primes has ...
- 12.8k
2
votes
1
answer
387
views
Convergence of series $\sum_{k=1}^{\infty}\frac{p_{k+1}-p_k}{(p_{k+1}+p_k)^\alpha}$
I ask if the series
$$\sum_{k=1}^{\infty}\frac{p_{k+1}-p_k}{(p_{k+1}+p_k)^\alpha}$$
where $p_k$ stands for the prime of index $k$,
has the same properties of convergence of the series $$\sum_{k=1}^{\...
- 105k
6
votes
0
answers
201
views
Smooth integers with lower bound on $\omega(n)$
Define $(b,c)$-smooth integers to be integers having all prime factors bigger than $c$ and smaller than $b$.
Probability a number is $(b,1)$-smooth is governed by the Dickman function while ...
CommunityBot
- 1
-3
votes
1
answer
178
views
Asymptotic behavior of $\sum_{k=1}^{n}\frac{p_{k+1}}{p_{k+1}-p_k}$
I refer to my previous question Asymptotic behavior of a certain sum of ratios of consecutives primes.
We can split the sum
$$\sum_{k=1}^{n}\frac{p_{k+1}+p_k}{p_{k+1}-p_k}$$
where $p_k$ stands for the ...
- 105k
10
votes
1
answer
469
views
Asymptotic behavior of a certain sum of ratios of consecutives primes
I am looking for the asymptotic growth of the following sum
$$\sum_{k=1}^{n}\frac{p_{k+1}+p_k}{p_{k+1}-p_k}$$
where $p_k$ stands for the prime of index $k$.
Manual computations show, for small values ...
- 105k
7
votes
1
answer
231
views
The asymptotic of $|\{1\leq n\leq x|\gcd(n,S(n))=1\}|$, with $S(n)$ the sum of remainders, and get idea for other miscellany problem
Let $n\geq 1$ be an integer. In this post we denote the sum of remainders function as $$S(n)=\sum_{k=1}^n n \bmod k,$$ for example $S(1)=S(2)=0+0$ and $S(5)=0+1+2+1+0=4$. In the literature there are ...
CommunityBot
- 1
4
votes
0
answers
142
views
Is it true that $|\{k^{k+1}+(k+1)^k\pmod p:\ k=0,\ldots,p-1\}|=(1-e^{-1})p+O(\sqrt{p})\ ?$
For each prime $p$, let us define
$$w_p:=|\{k^{k+1}+(k+1)^k\pmod p:\ k=0,\ldots,p-1\}|,$$
where $a\pmod p$ denotes the residue class $a+p\mathbb Z$.
Based on my computation, I conjecture that
$$w_p=...
- 15.6k
6
votes
0
answers
435
views
Average value of $\prod_{p|d}{p-1\over p-2}$ for $d=nq$, $n\in{\mathbb N}$, with $p$ odd prime
$\newcommand{\mean}{\mathop{\mathrm{mean}}}$
Define
$$
S(d) = \prod_{p|d\atop p>2}{p-1\over p-2}.
$$
Bombieri and Davenport (1966) proved that
$$
\mean\limits_{d\in{\mathbb N}} S(d) =
\mean\...
- 345
5
votes
0
answers
614
views
is there a link with the probabilistic model for prime numbers?
Let $x \in \mathbb{R}_+$ and $k \in \mathbb{N}^{*}$.
Let :
$$\mathcal{A}(x)=\#\{(a_1, a_2, \ldots, a_k) \in \mathbb{P}^k \mid (a_1, a_2, \ldots, a_k \text{ verifying some properties}) \, , a_k \...
- 521
1
vote
1
answer
867
views
$n$th prime: a better approximation
Let $p_n$ be the $n$-th prime, then from Wikipedia I got that
$p_n \approx n \left(\ln n + \ln \ln n -1 + \frac{\ln \ln n-2}{\ln n}+\frac{6\ln \ln n-( \ln \ln n)^2-11}{\ln^2 n} \right)$.
What is a ...
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28
votes
3
answers
3k
views
Expressing the Riemann Zeta function in terms of GCD and LCM
Is the following claim true: Let $\zeta(s)$ be the Riemann zeta function. I observed that as for large $n$, as $s$ increased,
$$
\frac{1}{n}\sum_{k = 1}^n\sum_{i = 1}^{k} \bigg(\frac{\gcd(k,i)}{\...
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-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 ...
- 6,984
5
votes
2
answers
794
views
Is the result of Schmidt conditional to RH
From this page:
https://en.wikipedia.org/wiki/Chebyshev_function#Asymptotics_and_bounds
A theorem due to Erhard Schmidt states that, for some explicit positive constant $K$, there are infinitely ...
- 105k
3
votes
0
answers
323
views
If $p^2 - q^2$ is a perfect square where $p$ and $q$ are primes $> 5000$ then is one of its prime factors always greater than $17$? [closed]
Is it true that if $p^2 - q^2$ is a perfect square where $p$ and $q$ are primes $> 5000$ then it has a prime factor greater than $17$?
Note: This question was asked in MSE but did not receive an ...
- 5,470
1
vote
0
answers
133
views
On a continuous function as a substitute of the prime-counting function in the second Hardy–Littlewood conjecture satisfying certain asymptotics
It it well-known that the prime-counting function $\pi(x)$ satisfies the prime number theorem and that were in the literature two related conjectures to this arithmetic function, these are: the ...
3
votes
1
answer
336
views
A conjecture for primes $p\equiv\pm1\pmod5$
For any prime $p\equiv\pm1\pmod5$, we can write $p$ uniquely in the form $x_p^2+3x_py_p+y_p^2$ with $x_p,y_p\in\mathbb Z$ and $x_p>y_p>0$.
I have the following conjecture.
Conjecture. We have
...
- 105k
4
votes
1
answer
507
views
A weaker version of the Brocard's Conjecture
Brocard's conjecture states that: If $p_{k}$ and $p_{k+1}$ are consecutive prime numbers greater than $2$, then between $p_{k}²$ and $p_{k+1}²$ there are at least four prime numbers.
I know that is ...
- 6,969
3
votes
0
answers
194
views
Conjectures for primes $p\equiv1\pmod3$
Let $p$ be a prime with $p\equiv1\pmod3$. It is well known that we can write $p$ uniquely as $a_p^2+a_pb_p+b_p^2$ with $a_p,b_p\in\mathbb Z$ and $a_p>b_p>0$.
Note that $a_b\not \equiv b_p\pmod3$....
- 15.6k
2
votes
1
answer
146
views
On an attempt to create interesting variants of Firoozbakht's conjecture, evoking combinations of different generalized means
The idea of this post arises when I've considered simple variants of the known as Firoozbakht's conjecture (see this corresponding Wikipedia or [1]), and comparisons by trial and error with means (if ...
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56
votes
1
answer
4k
views
A mysterious connection between primes and $\pi$
The Prime Number Theorem relates primes to the important constant $e$.
Here I report my following surprising discovery which relates primes to $\pi$.
Conjecture (December 15, 2019). Let $s(n)$ be ...
- 359
2
votes
0
answers
617
views
Arithmetic progression and average of two prime numbers
Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by:
$$
\ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1.
$$
For all terms of $A$ greater than $\ \...
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-1
votes
1
answer
127
views
Statement about upper density of even numbers satisfying the Goldbach condition
For $A\subseteq \mathbb{N}$, let the upper density of $A$ be defined by $$\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 \...
- 105k
4
votes
0
answers
191
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Overview of Combinatorial Technique of "Selberg’s Symmetry Formula"
In the paper entitled "A Computational History of Prime Numbers and Riemann Zeros" (on page 4, click here) it is written about "Selberg’s symmetry formula" that-
Until 1950 it was widely believed (...
3
votes
2
answers
218
views
The graph and sign of $p_n-\operatorname{ali}(n)$, where $p_n$ is the $n$-th prime and $\operatorname{ali}(n)$ the inverse of the logarithmic integral
I'm inspired in [1] to ask the following question. My problem is that I have not an implementation of the inverse of the logarithmic integral $\operatorname{Li}(x)=\int_2^x\frac{dt}{\log t}$, that ...
- 1,126
4
votes
0
answers
884
views
Has any professional mathematician ever attempted to solve the Riemann hypothesis using only number theory? [closed]
I have often heard people saying that ''all attempts at solving the Riemann hypothesis using number theory have failed.'' But in the literature, i cannot find any failed ''purely number-theoretic'' ...
CommunityBot
- 1
0
votes
0
answers
133
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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 ...
- 6,984
0
votes
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 ...
- 790
2
votes
1
answer
134
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Results relating prime numbers with extremely abundant numbers
A positive integer $n$ is extremely abundant if either $n=10080$, or $n>10080$ and
$$σ(n)/(n×log(log (n)))>σ(m)/(m×log(log (m)))$$
for all $10080≤m<n$. Here $σ(n)$ is the sum-of-divisors ...
- 188k
0
votes
1
answer
148
views
On the quantity of twin prime pairs of a given form
Let $p_l$ the $l$-th prime number. I've considered the formula
$$\frac{N_{n+1}}{N_n}+\frac{N_{n+2}}{p_{n+1}N_n}\pm1$$
where $N_k=\prod_{l=1}^k p_l$ is the primorial of order $k$. Previous formula ...
- 12.8k
4
votes
2
answers
472
views
Sharp estimates for Meissel-Mertens constant
I wondered if it is possible to get a similar inequality like $(1.1)$ of Michael D. Hirschhorn, Approximating Euler's Constant, The Fibonacci Quarterly, Volume 49, Number 3 (August 2011) for the ...
CommunityBot
- 1
2
votes
2
answers
489
views
On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$
Define $\pi(x)$ to be the prime counting function and Li(x) the logarithmic integral. Let $I_s$ be defined as above.
Is $I_s$ known to be convergent for any real number $s<1$ ?
- 221
2
votes
1
answer
339
views
Is there a constant $\alpha$ such that: $P_{n+1} < P_n.\left(\frac{n+1}{n}\right)^\alpha$?
Is there a constant $\alpha$ such that:
$$P_{n+1} < P_n \cdot \left(\frac{n+1}{n}\right)^\alpha$$
Or
$$\lim_{n\to\infty}\frac{\ln\frac{P_{n+1}}{P_n}}{\ln\frac{n+1}{n}} < +\infty$$
Where $...
6
votes
0
answers
211
views
some problems on sum of two squares
During my experiments with "Mathematica" I arrived to the following observations. My question is that are they interesting, known, solved or not. If they are known could you please give me a reference....
- 841
2
votes
0
answers
195
views
Error bounds for $\pi(x)-Li(x)$ and convergence of the associated Dirichlet integral
Following On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$, define
$$I_{s}=\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} \mathrm{d}x,$$ where $\pi$ denotes the prime counting function ...
- 1,019
0
votes
1
answer
474
views
An upper bound for $\sqrt{p_{n+1}}$
Let $C$ be a positive constant. Is it true that for all sufficiently large integers $n$ the inequality $$\prod_{i=1}^n (1+\frac{1}{\sqrt{p_i}})>C\sqrt{p_{n+1}}$$ holds? (Here with $p_k$ is denoted ...
CommunityBot
- 1
5
votes
1
answer
208
views
applications of finding least quadratic nonresidue mod $p$?
I saw some papers from famous mathematicians (assuming GRH or without it) which are devoted to finding bound for least quadratic nonresidues modulo prime number $p$.
My question is that why it is so ...
- 66.3k
3
votes
1
answer
231
views
Races that involve odd semiprimes: a first statement or conjecture
While I am studying the famous article [1], in English this is Andrew Granville and Greg Martin, Prime Number Races, The American Mathematical Monthly, vol. 113, (2006), I wondered what about a race ...
- 12.8k
1
vote
0
answers
111
views
Upper bound for $\alpha_{n}$ from Mertens' third theorem
This question is a follow-up to About Goldbach's conjecture.
I would like to know if an unconditional upper bound for $\alpha_{n}$, defined as $n(N_{2}(n)-\dfrac{nN_{1}(n)}{P(n)})$ (where $N_{2}(...
- 6,984
2
votes
0
answers
99
views
A problem in modular roots
We have three mutually coprime integers $r,t,M$ where $M\asymp K^{\frac12-2\epsilon}$ and $r,t\asymp K^{\frac14+\epsilon}$ holds with some fixed $\epsilon>0$ and $K>0$ is a large parameter. ...
- 13.9k
2
votes
1
answer
280
views
On a problem that equates $\frac{\text{prime}-1}{\operatorname{rad}(\text{prime}-1)}$ with the sequence of primorials
We denote for integers $m>1$ the product of the distinct prime numbers dividing $m$ as $$\operatorname{rad}(m)=\prod_{\substack{p\mid m\\p\text{ prime}}}p,$$
with the definition $\operatorname{rad}(...
- 105k