All Questions
Tagged with analytic-number-theory prime-numbers
690 questions
3
votes
0
answers
151
views
On an inequality for the arithmetic function counting the number of primes $\lfloor n^c\rfloor$ in the spirit of Ramanujan's prime counting inequality
In page 3 of [1] (please see if you need it the book by Berndt) Axler refers an inequality that involves the prime-counting function $\pi(x)$ and that was deduced by Ramanujan. I'm curious to know if ...
2
votes
0
answers
243
views
Primes in arithmetic progression
We call a prime $p$ "good" if there is $0<k<\log p$ with $2kp+1$ prime. What is the asymptotic density of good primes?
5
votes
0
answers
349
views
Smallest prime $p$ such that $2\mid\operatorname{ord}_p(q)$, the multiplicative order of $q$ modulo $p$
$\DeclareMathOperator\ord{ord}$Let $q$ be prime. I want to upper bound the smallest odd prime $p$ such that $2\mid\ord_p(q)$ (where $\ord_p(q)$ is the multiplicative order of $q$ modulo $p$).
Using ...
- 101
13
votes
1
answer
777
views
Large sieve inequality for sparse trigonometric polynomials
Let $S(\alpha) = \sum_{n\leq N}f(n) e^{2\pi i \alpha n}$ for some arithmetic function $f$. Suppose $\alpha_1, \ldots, \alpha_R$ are real numbers that are $\delta$-spaced modulo $1$, for some $0 < \...
- 253
1
vote
0
answers
103
views
$g$-gap radius of an integer
For $n$ a large enough composite integer, define the $g$-gap radius of $n$, if it exists, for positive even $g$ as the smallest positive integer $\rho_{g}(n)$ such that both $n-\rho_{g}(n)$ and $n+\...
- 6,974
3
votes
0
answers
252
views
Counting twin primes with a sieve-like algorithm
The sequence A002822, denoted as $S$, represents all the twin primes except $\{3, 5\}$. Other than that exception, $k$ and $k+2$ are twin primes iff $(k+1)/6\in S$. Let $S(N)$ be the subset of $S$ ...
- 3,098
3
votes
0
answers
154
views
Reference request for the following results
I am looking for references on the following results. In what follows $\pi(x)$ denotes the prime counting function.
Result 1. For all real $k>1$ there exists $x^k_0 \in \mathbb{R}$ such that for ...
- 31
8
votes
1
answer
471
views
Conjecture about the density of primes
Conjecture
For any sufficiently large integer $kn$ , the sequence representing
the number of primes in each block obtained by splitting $kn$ into $k$
equal blocks, is a strictly decreasing sequence, ...
- 197
6
votes
0
answers
149
views
Dickson's conjecture for Beatty sequences
A particular case of Dickson's Conjecture states that for $a_1,q_1,a_2,q_2$ with $(a_1,q_1)=(a_2,q_2)=1$, there are infinitely many $n$ for which $q_1 n + a_1$ and $q_2 n+a_2$ are both prime, provided ...
- 1,990
2
votes
1
answer
224
views
Prime factor distribution over $\mathbb{N}$
I wonder if there is something like a general "prime component distribution pattern" of "the general natural number" $n$?
Using the following notation for the prime factorization $...
3
votes
1
answer
329
views
Fully explicit Linnik's Theorem
Linnik's Theorem states that there exist absolute constants $c$ and $L$ such that for every $m \in \mathbb{N}$ and every $a$ coprime to $m$, there is a prime $p$ with $p \equiv a \pmod{m}$ and $p < ...
- 1,663
-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 ...
- 1,776
6
votes
2
answers
723
views
Does always exist a prime number between $n(n+1)$ and $(n+1)(n+2)$?
Let $n$ is positive integer number, does always exist a prime number between $n(n+1)$ and $(n+1)(n+2)$?
- 1,776
-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 $n-r$ ...
- 6,974
6
votes
0
answers
654
views
Generalized prime number theorem and Riemann Hypothesis for non-number math objects
My question is about some math objects (matrices, polynomials) and operators that satisfy a number of properties which can lead to a theory similar to PNT, RH, Dirichlet functions, abscissa of ...
- 3,098
5
votes
0
answers
205
views
Is there a polynomial version of Wilson's theorem which can avoid Cramer flavored conjectures?
Wilson's theorem states that a natural number $n > 1$ is a prime number if and only if the product of all the positive integers less than $n$ is one less than a multiple of $n$.
Is there a version ...
- 13.9k
2
votes
0
answers
205
views
Sum of all primes below $n$ without listing all primes below $n$
Asymptotically there is around $\frac{n}{\ln n}$ primes below a given integer $n$. Thus $\frac{n}{\ln n}$ is a lower bound for the time complexity of any algorithm that at some point finds each prime ...
- 21
3
votes
1
answer
134
views
Is it possible to find an estimate of $\sum_{k=1}^n\frac1{\varphi(k\cdot p_k)}$?
Is it possible to find an estimate of the summation
$$s(n)=\sum_{k=1}^n\frac1{\varphi(k\cdot p_k)}$$
where $\varphi(n)$ is the totient function and $p_k$ the k-th prime?
The corresponding series seems ...
- 1,008
7
votes
1
answer
635
views
Is there a Chebotarev‘s theorem for non-Galois extension over Q？
For a Galois extension $K/\mathbb{Q}$, the Chebotarev Density Theorem predicts the density of primes with a certain splitting type.
I'm wondering if there is a similar result for non-Galois extension?
...
- 547
3
votes
1
answer
247
views
Explicit bounds on number of squarefree numbers coprime to a certain number
We know that the number of squarefree integers $\le x$ that are coprime to $A$ is
$$
Q_A(x) = x \prod_{p|A} \left(1-\frac{1}{p}\right) \prod_{p \nmid A} \left(1-\frac{1}{p^2}\right) + O(\sqrt{x}).
$$
...
- 301
20
votes
1
answer
1k
views
Possible contemporary improvement to bounded gaps between primes?
In his summary of his book Bounded gaps between primes: the epic breakthroughs of the early 21st century, Kevin Broughan writes
Which brings me to my final remark: where to next in the bounded gaps ...
- 35.5k
0
votes
0
answers
50
views
k specific prime factors guess and related prime guess [duplicate]
there is no more than one group
of continuous composite sequence
of length k composed of only k different specific prime factors.
for example 2 3 5[8 9 10]just only one group. I have prove that k ...
- 29
8
votes
1
answer
937
views
On the connection between sums of prime numbers and distribution of prime numbers
As an amateur mathematician, I have always been fascinated by the magic of prime numbers, and their apparently random distribution. I was utterly amazed when I found the following connection between ...
- 189
-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 $\left(\sum_{k=1}^{\infty}\frac{1}{k}\right)$.
...
- 149
1
vote
1
answer
342
views
What is a non-trivial upper bound on the $k$th prime below a given prime $p$?
Given a prime number $p_0$, by Bertrand's postulate we know that
\begin{gather}
p_1\ge\frac{p_0}{2}\\
p_2\ge\frac{p_1}{2}\ge\frac{p_0}{2^2}\\
\vdots\\
p_k\ge\frac{p_0}{2^k}
\end{gather}
where $p_1,p_2,...
- 113
1
vote
0
answers
96
views
Smooth number pairs satisfying a congruence
Let $\mathcal P=\{p_1,\dots,p_{2t}\}$ be $2t$ primes between $2^\ell$ and $2^{\ell+1}$ and fix an exponent bound $a\in\mathbb Z_{\geq2}$.
Fix $N\in\mathbb N$ whose prime factors $p$ satisfy $p>2^{\...
- 13.9k
-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 ...
- 129
10
votes
1
answer
694
views
Set of prime numbers $q$ such that $\sum\limits_{p\leq\sqrt{q}}p=\pi(q)$, where $p$ are prime numbers
The question is: does the set of prime numbers $q$ such that $\sum\limits_{p\leq\sqrt{q}}p=\pi(q)$, where $p$ are prime numbers, contain infinitely many elements? You can find the first elements here (...
- 189
0
votes
0
answers
118
views
Primes in many variables polynomials form
As in this question >> Primes of the form $x^2 + y^2 + 1$ There are $\asymp \dfrac{x}{\log^{3/2} x}$ primes of the form $a^2+b^2+1$ up to $x$. I read a book stated that there exist a polynomial $...
0
votes
1
answer
249
views
How differently would we model the distribution of primes if prime gap is larger?
Cramer's conjecture based on his random model provides prime gaps are bound by $O(\log^2p_n)$ where the gap is between $(n+1)$th and $n$th prime.
How differently would primes be modeled if gaps of $O(...
- 13.9k
0
votes
0
answers
99
views
On a generalised result of Mertens
Let $\varphi$ be the Euler totient function, $N_k$ denote the product of the first $k$ primes and define $$f(k, r) = \frac{(N_k)^r}{\varphi((N_k)^r) \log \log ((N_k)^r)}.$$
where $r \in \mathbb{N}$. ...
- 1,019
0
votes
1
answer
137
views
Search for gaps between primes where each composite is divisible by increasing integers (2, 3, 4, ...)
Almost every text of number theory contains in its first chapters something similar to the following:
For any integer n, the factorial n! is the product of all positive
integers up to and including n....
- 1,008
1
vote
0
answers
81
views
An upper bound for $\,m_k=\min\,\{m\in N:\,mp_k+1\;\;is\;prime\}$
For each prime $p_k$ one can define
$$m_k=\min\,\{m\in N:\,mp_k+1\;\;is\;prime\}$$
Some computations suggest that
$$m_k=O\Big(\frac{2\sqrt k}{\log k}\Big)$$
Is this estimate confirmed by analytic ...
- 1,008
2
votes
0
answers
54
views
Asymptotic growth of the collection of Miller-Rabin pseudo-primes witnessed by a set
Consider a set $S$ of positive integers[*]. Define $P(S)$ as the set of numbers $N$ for which elements of $S$ are "witnesses" for the Miller-Rabin test for primality of $N$. Explicitly $P(S)=...
- 1,566
-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 ...
- 6,018
0
votes
1
answer
191
views
Are there some results which count $\sum_{p\in [x/2,x]} \log p$ or $\sum_{p\in [x,y]} \log p$ for for $x$ and $y$ positive and real?
I have seen the prime number theorem and on the of versions I know is that $\sum_{p\leq x} \log p=O(x)$ (I am counting over primes here and in the rest of the post). Are there any similar results for ...
user147650
2
votes
0
answers
313
views
Proving that the Riemann zeta function is zero free on Re=1 using the prime number theorem
Is $\frac{-\zeta'(s)}{\zeta(s)}+\frac{-s}{s-1}$ an analytic continuation, holomorphic for $Re\ s > 0,\ s\neq 1$, of $f(s)=s\int_{1}^{\infty}\frac{\psi(x)-x}{x^{s+1}}\mathrm{d}x$?
If so: Let $s_{0}$ ...
- 129
2
votes
0
answers
94
views
How far ahead do we have to look after $p_n$ to be sure we find another prime $q$ such that $(p_n+q)/2$ is also prime?
If Goldbach's conjecture is true, given a prime $p_n$ is surely possible to find another prime $q$ such that $\frac{p_n+q}2$ is also prime. But I ask: how far ahead do we have to look after $p_n$ to ...
- 1,008
2
votes
1
answer
273
views
Primes in modular arithmetic progression
Fix a prime $p$.
I want to get $k<p$ primes $p_1<\dots<p_k$ such that at every $i\in\{1,\dots,k\}$ we have
$$p_i\equiv (2i+1+c)\bmod p$$ where $c$ is fixed and $2k+1+c<p$ holds.
For a ...
- 13.9k
2
votes
0
answers
114
views
A conjectured upper bound for the mean value of prime divisors inside prime gaps
In 1969 C.A. Grimm stated this interesting conjecture: the prime gap $\,G_n=\{x\in N:p_n\lt x\lt p_{n+1}\}\,$ contains at least $\,\#G_n=(p_{n+1}-p_n)-1=g_n-1\,$ distinct prime divisors, that is if $\,...
- 1,008
4
votes
1
answer
395
views
Mertens formulas aren't enough for prime number theorem
For the primes it's true that
$$
\sum_{p \le x}\frac{1}{p} = \ln\ln x + M + O(1/\ln x)
$$
where, $M$ is suitable constant, and, moreover, the prime number theorem gives that
$$
\lim_{x\to\infty}\frac{\...
- 41
7
votes
2
answers
636
views
How to use the Prime Number Theorem in order to prove Selberg's Formula?
I`m reading Melvin B. Nathanson's "Elementary Methods in Number Theory"
and I can't think of a way of deducing Selberg's formula (9.3) from the prime number theorem.
This is one of the tasks ...
- 129
4
votes
1
answer
310
views
Question on an analytic number theory paper
My question is just a ``I don't understand what goes on in X of paper Y" so I don't know if I can post it; on the other hand it is research. I posted it in stackexchange but it received no ...
- 1,381
2
votes
0
answers
422
views
Sequences with high densities of primes: how to boost them to get even more and larger primes
I propose a methodology to help find large prime numbers with a much higher probability than picking up random numbers and testing them for primality. This would help speed up prime number generators ...
- 3,098
3
votes
1
answer
225
views
What are the best known bounds for the smallest primes larger than $n$?
Let $n>1$ be some integer. Define $p, q$ to be the smallest primes larger than $n$, where $p<q$. What are the best known effective lower and upper bounds for $p$ and $q$ ?
user160539
4
votes
1
answer
2k
views
Bounds for prime counting function
The prime counting function $\pi(x)$ is defined as
\begin{equation}
\pi(x)=\sum_{p\leq x}1
\end{equation}
where $p$ runs over primes.
I have seen many bounds for $\pi(x)$ such as
\begin{equation}
\...
- 45
3
votes
0
answers
147
views
The bias of consecutive prime numbers towards being incongruent modulo 3
Given a positive integer $n$, let $f_1(n)$ denote the number of pairs of
consecutive prime numbers $\leq n$ which are incongruent modulo 3, and let
$f_2(n)$ denote the number of pairs of consecutive ...
- 19.6k
24
votes
1
answer
2k
views
Parity of the multiplicative order of 2 modulo p
Let $\operatorname{ord}_p(2)$ be the order of 2 in the multiplicative group modulo $p$. Let $A$ be the subset of primes $p$ where $\operatorname{ord}_p(2)$ is odd, and let $B$ be the subset of primes $...
- 429
13
votes
1
answer
383
views
Numbers that don't start with (p-1) in base p for any p
Say that an integer $n$ is $p$-leading if its expansion in base $p$ starts with the digit $p-1$. My postdoc, Lifan Guan, asks: are there infinitely many positive integers $n$ that are not $p$-leading ...
- 20.2k
1
vote
1
answer
327
views
Symmetry in Hardy-Littlewood k-tuple conjecture
Assuming Hardy-Littlewood $k$-tuple conjecture, do the "dual" prime constellations $(0,h_1, h_2,\cdots, h_i,\cdots, h_{k-1}=d)$ and $(0, h_{k-1}-h_{k-2}, h_{k-1}-h_{k-3},\cdots,h'_i=h_{k-1}-...
- 6,974