All Questions
Tagged with analytic-number-theory prime-numbers
690 questions
0
votes
2
answers
669
views
Does theta(n)<n for all n imply the Riemann Hypothesis and/or vice versa?
I know that better and better bounds of the Chebyshev Theta and Psi functions are implied by knowing that the first (insert large number here) zeta zeroes lie on the Critical Line. These bounds, ...
- 602
0
votes
2
answers
193
views
Space of functions f such that the number of primes in $ [x, x+f(x)] $ remains bounded
Given a positive integer $ n $ , let $ S_{b}(n) $ the set of functions $ f $ fulfilling the following conditions :
1) $ f $ is continuous, positive and increasing on $(n,+\infty) $
2) for ...
- 6,974
0
votes
1
answer
1k
views
1895 Math Trip problem on primitive roots of unity
How to prove that if $\theta _1,\theta _2,\theta _3$ be the arguments of the primitive roots of unity, $\sum \cos p\theta = 0$ when $p$ is a positive integer less than $\dfrac {n} {abc\ldots k}$, ...
- 281
0
votes
1
answer
437
views
Mertens' 3rd theorem, upper bound
Is it true that
$$\prod_{p\le x}\frac p{p-1}\le e^\gamma\ln x\left(1-\frac{0{.}011}{\ln x}+\frac{0.2}{(\ln x)^2}\right)$$
for all $x>25\,000$, where the product is over prime $p$?
user95470
0
votes
1
answer
404
views
Asymptotics of "ugly" function elucidate Goldbach's conjecture?
Question
We now define the following "ugly" function:
$$ A_c(s,r,n,m) =
\begin{cases}
1 & \text{ if only $sr+nm=2c$ } \\ 0 & \text{otherwise}
\end{cases}
$$
How does the "ugly"...
- 89
0
votes
2
answers
747
views
Has this formula about prime gaps already been conjectured and/or proven?
While playing around with prime gaps, I found out that the following formula seems to be a rather good approximation of the ratio $\dfrac{p_{b}-p_{a}}{b-a}$ where $a<b$ are positive integers:
$$H_{...
- 6,974
0
votes
2
answers
153
views
Is the abscissa of convergence of $s\mapsto\sum_{n>0}(ng_{n}/2)^{-s}$ known?
The famous Polignac conjecture posits that the $n$-th prime gap $g_{n}:=p_{n+1}-p_{n}$ attains all even positive integral values infinitely many times, which implies $\displaystyle{\zeta_{Pol}:=s\...
- 6,974
0
votes
1
answer
191
views
Are there some results which count $\sum_{p\in [x/2,x]} \log p$ or $\sum_{p\in [x,y]} \log p$ for for $x$ and $y$ positive and real?
I have seen the prime number theorem and on the of versions I know is that $\sum_{p\leq x} \log p=O(x)$ (I am counting over primes here and in the rest of the post). Are there any similar results for ...
user147650
0
votes
1
answer
256
views
Lower bound for $\prod_{p\equiv 3 \pmod 4} p^{v_p(n!)}$
What is the best lower bound known for $$\prod_{p\equiv 3 \pmod 4} p^{v_p(n!)},$$ where the product is taken over all the primes(congruent to $3$ modulo $4$) less than or equal to $n$.
0
votes
1
answer
198
views
Upper bound for $\sum r_{0}(n)$
Assuming Goldbach's conjecture, let's define for a sufficiently large integer $ n $ the quantity $ r_{0}(n) : =\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\} $.
Under GRH, what is the best upper bound ...
- 6,974
0
votes
1
answer
326
views
Prime numbers property. A Merten's third theorem like sequence
Here is a question I have asked on Math Stack Exchange https://math.stackexchange.com/questions/2290917/prime-numbers-property-mertens-theorem-related-sequence , that I would like this community to ...
- 251
0
votes
2
answers
288
views
Counting powerful integers. Lower bounds
Remark: The upper bounds are perhaps still more interesting; I may address them in another post.
PROBLEM: Find simple (numerically efficient) lower bounds for the number of powerful integers (...
- 4,786
0
votes
1
answer
148
views
On the quantity of twin prime pairs of a given form
Let $p_l$ the $l$-th prime number. I've considered the formula
$$\frac{N_{n+1}}{N_n}+\frac{N_{n+2}}{p_{n+1}N_n}\pm1$$
where $N_k=\prod_{l=1}^k p_l$ is the primorial of order $k$. Previous formula ...
0
votes
1
answer
120
views
How many integers $x$ satisfy that $x*p(x) \leq n$, where $p(x)$ means the largest prime factor of $x$?
I guess that the number of integers $x$ which satisfy the condition $x*p(x) \leq n$ is $O(n^{2/3})$ or $O(n^{3/4} / \ln n$), but I cannot prove it. I just write a program to count the number. The ...
- 601
0
votes
1
answer
184
views
Upper bound for tuple of exponents of prime factorization
Let $a(n)$ be the $k$-ary tuple of the exponents of the prime factorization of $n$. For example,
$$a(5184)=a(2^{6}⋅3^{4})=(6, 4), a(65536)=a(2^{16})=(16).$$
Formally, let $p_{1}^{a_{1}}, p_{2}^{a_{2}...
- 25
0
votes
1
answer
413
views
Upper bound for the number of even numbers sum of a prime and a semi-prime not fulfilling Goldbach's conjecture
Chen Jing Run proved that every large enough even integer is either the sum of two primes or the sum of a prime and a semi-prime (that is, the product of two primes). Golbach's conjecture states that ...
- 6,974
0
votes
1
answer
160
views
does the ratio of the count of rational numbers on an $n\times n$ grid to $n^2$, converge as $n$ tends to infinity [closed]
Suppose we order the rational numbers using the diagonal method (used to prove they are countable) using an $n\times n$ grid. Now suppose we count the distinct rational numbers (those points on the ...
- 375
0
votes
1
answer
187
views
Expressing odd numbers as a prime minus $a^2+a$
I am looking for results about expressing odd numbers in the form
$$p-a^2-a,$$
where $p$ is a prime and $a$ is a positive integer.
Assuming Bunyakovsky conjecture this is easy as $x^2+ x+c$ are ...
0
votes
1
answer
461
views
Primes $p$ such that $432 p +1$ is prime [closed]
Is the set of prime numbers $p$ such that $432 p + 1$ is also prime infinite?
It doesn't follow from Dirichlet's theorem as far as I can tell.
- 25
0
votes
2
answers
515
views
Dirichlet's theorem on prime density [closed]
Does anyone knows where I could find the proof of Dirichlet's theorem on the analytic density of primes congruent to a certain integer $m$, which turns out to be $\frac{1}{\varphi(m)}$?
Thanks a lot.
- 1
0
votes
1
answer
204
views
On $(\prod_{\substack{1\leq s\leq X\\s\text{ semiprime}}}s)(\sum_{\substack{1\leq s\leq X\\s\text{ semiprime}}}\frac{1}{s})$ as $X\to\infty$
Few weeks ago an user from Mathematics Stack Exchange answered my question On an inequality that involves products and sums related to the sequence of semiprimes (asked May 26). It seems that for ...
0
votes
2
answers
192
views
A variant of Turán–Kubilius inequality
Let $\omega(n)$ the number of distinct prime factors of $n$ (counted without multiplicity). A famous consequence of Turán–Kubilius inequality is
$$
\sum_{n\leq x}(\omega(n)-\log\log x)^2=O(x\log \log ...
- 515
0
votes
2
answers
317
views
On a coprime generalization of Cramer's conjecture
Given a large enough integer $n\in\Bbb N$ and a real $r\in\big(0,\frac12\big]$ and $n_1\in\Bbb N_{> n}$ is the smallest integer such that $n_1=AB$ for two coprime integers $A$ bigger than but close ...
user94040
0
votes
1
answer
97
views
$l$-th power radioprimal conjecture
I would like to know if some widely believed conjecture, be it GRH, Hardy-Littlewood conjecture, or any other would imply the following statement for some $l>1$:
$l$-th power radioprimal growth ...
- 6,974
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
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 ...
user145520
0
votes
1
answer
154
views
Sergei numbers : even integers n being a prime gap at least n times
Let's introduce Sergei (for SElf-Referential Gaps Extensible to Infinity, and as a wink to a mathematician friend of mine of Russian descent whose given name is Serge and quite interested in number ...
- 6,974
0
votes
1
answer
370
views
prime counting function pi bounds [closed]
is it true that for some integer $n_0$, that all integer numbers n such that $n \geq n_0$ the following holds true for the prime counting function :
$\frac{x}{\ln x} (1+\frac{1}{\ln x}+\frac{2}{\ln^2 ...
user95470
0
votes
1
answer
150
views
Linear forms that avoid numbers with lot of factors
Is following true?
For every given $c>0$ there is an $n_c>0$ such that for every $n>n_c$ there are integers $n<a,b<2n$ such that there are two positive integers $\frac{n}{2(\log n)^c}\...
user76479
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
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 ...
user145520
0
votes
1
answer
100
views
On the integral $I_s = \int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} \mathrm{d}x$-follow up question
This is a follow up on On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$
According to the answer that i got, $I_s$ is not known to converge for any real $s<1$. But suppose $I_s$ ...
user140392
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 ...
0
votes
1
answer
284
views
Do the roots of R(x) have any significance for the prime counting function?
I'm calculating the roots of the function
\begin{equation}
R(x) = \sum_{k=1}^{\infty}\frac{\mu(k)}{k}li(x^{1/k})
\end{equation}
This function seems to have a largest and smallest positive root. Can ...
- 3,217
0
votes
0
answers
101
views
Prime races in two competing arithmetic progressions - error bound
I read an article by Andrew Granville on the subject, there's actually quite a bit of recent literature on the topic. My problem is as follows. I have two sequences of primes: $(p_{1,n})$ and $(p_{3,n}...
- 3,098
0
votes
0
answers
68
views
Around similar inequalities than an inequality due to Nicolas, that involve products of consecutive Ramanujan primes
This is cross-posted (and this post is a version to ask just around the veracity of Conjecture 1) as the post with identifier 3594907 and same title), that I've edited on Mathematics Stack Exchange ...
0
votes
0
answers
136
views
Bounded sums involving primes
I'm trying to generalize the Theorem 2.7.1 in [1] where they prove:
$$\sum_{p \leq x} f(p) = \int_{2}^{x} \frac{f(t)}{\log{t}} dt + \epsilon(x)f(x) - \int_{2}^{x} \epsilon(t) f^{'}(t) dt $$
where $\...
- 109
0
votes
0
answers
169
views
On $\sum_{\rho\in D} \text{dist}(\rho)=\frac{1}{2\pi i} \int_{\partial{D}}\log \zeta(s)\ ds$
Let $D$ denote a closed two dimensional figure as: $D=2+iT\to 2\to 2-\delta\to 2-\delta+i(T-\delta)\to \frac{1}{2}+\epsilon+i(T-\delta)\to\frac{1}{2}+\epsilon\to\frac{1}{2}-\epsilon\to \frac{1}{2}-\...
- 11
0
votes
0
answers
91
views
Reducing the number of terms in Waring-Goldbach problem by allowing exponents to vary
Assuming the Waring-Goldbach problem (see https://en.m.wikipedia.org/wiki/Waring%E2%80%93Goldbach_problem) has a positive solution, can we reduce the number of terms $t$ to some value $t'$ by allowing ...
- 6,974
0
votes
0
answers
91
views
How to use prime number theorem In such cases?
Let,
$$A(x)=\sum_{p\leq x}f(p)$$
Where $p$ is a prime number.
Under the Prime Number theorem we have that,
$$\pi(x)=Li(x)+O\left(\frac{x}{e^{a\sqrt{\ln(x)}}}\right) $$
as $x$ approach infinity.
Now,
$$...
- 111
0
votes
0
answers
80
views
Relevance of the deduction of similar theorems than Maier's theorem for other prime constellations
A year ago I asked this question on Mathematics Stack Exchange with identifier 4245823 and same title Relevance of the deduction of similar theorems than Maier's theorem for other constellations of ...
0
votes
0
answers
89
views
A similar inequality for the Dedekind psi function, than an inequality stated by Schinzel
I would like to ask about the next question that seems to me interesting. I know an article that was written by Andrzej Schinzel in which he stated Lemma 2. In this post we denote the Dedekind psi ...
0
votes
0
answers
177
views
Is there a link between Elliott-Halberstam and weak Hardy-Littlewood-Goldbach conjectures?
Let $\theta$ be such that $EH(\theta)$ holds, where $EH$ stands for Elliott-Halberstam. Can one get an explicit lower bound $\delta_{\theta}$ for the quantity $\delta$ appearing in the weak Hardy-...
- 6,974
0
votes
0
answers
102
views
On simple examples of unimodularity
$w=z=x+ 1 =y−1$ provides $wz−xy=w^2−(w−1)(w+ 1) = 1$. Hence if $x,y$ are odd then $w,z$ are even and all four integers are close.
Is there elementary example where only $w$ is even and all four ...
- 13.9k
0
votes
0
answers
50
views
k specific prime factors guess and related prime guess [duplicate]
there is no more than one group
of continuous composite sequence
of length k composed of only k different specific prime factors.
for example 2 3 5[8 9 10]just only one group. I have prove that k ...
- 29
0
votes
0
answers
118
views
Primes in many variables polynomials form
As in this question >> Primes of the form $x^2 + y^2 + 1$ There are $\asymp \dfrac{x}{\log^{3/2} x}$ primes of the form $a^2+b^2+1$ up to $x$. I read a book stated that there exist a polynomial $...
0
votes
0
answers
99
views
On a generalised result of Mertens
Let $\varphi$ be the Euler totient function, $N_k$ denote the product of the first $k$ primes and define $$f(k, r) = \frac{(N_k)^r}{\varphi((N_k)^r) \log \log ((N_k)^r)}.$$
where $r \in \mathbb{N}$. ...
- 1,019
0
votes
0
answers
185
views
On the asymptotics of the Chebyshev psi function
Denote by $\psi(x)$ the Chebyshev psi function over prime powers. Assuming the RH, it can be shown via the Riemann explicit formula that
$$\psi(x)-x \ll √x |\sum_{|\gamma| < T} \frac{1}{\rho} | +...
- 41
0
votes
0
answers
77
views
Construction of (general class of) function(s), which sieves out primes, w.r.t. given conditions:
Consider the function $F(x)$ defined in following manner:
$F(n)$ is finite (likely $F(x)\in[0,1]$) if $n$ is prime and zero otherwise:
It has to satisfy following conditions:
(1) $F(x)$ is ...
- 375
0
votes
0
answers
106
views
Variants of Nicholson's inequalities for prime numbers, involving the Lambert $W$ function
The purpose of this post is ask about two closely related/inspired conjectures from inequalities due to Nicholson (see [1]) and Visser [2]. If my reasonings are right should be stronger versions of ...