All Questions
Tagged with analytic-number-theory prime-numbers
690 questions
1
vote
2
answers
182
views
Prime factors bounded by $k$
Let $S$ be the set of integers with largest prime factor bounded by a given positive integer $k$. Is there a formula for the asymptotic density of such a set $S$?
- 397
9
votes
2
answers
1k
views
On the error term of the Riemann explicit formula
Let: $\rho$ be a non-trivial zero of the Riemann zeta function, $\Lambda$ be the von-Mangoldt function and $\psi(x) =\sum_{n \leq x} \Lambda(n)$. What is the best known upper bound for
$$f(x, T) := \...
- 1,019
1
vote
0
answers
104
views
Validity of analysis of summation of function of primes using Abel–Plana summation:
Consider the analytic function $g(x)$
Define
$$f(x)=g(x)\frac{\sin^2\left(\frac{π\Gamma(x)}{2x}\right)}{\cos^2\left(\frac{π}{2x}\right)}$$
Note that:
$$f(p)=g(p) \text{ for prime } p$$
And $f(n)=0$ ...
- 790
4
votes
0
answers
262
views
Asymptotic number of "modular primes"
We can say that a number $p$ is prime modulo $N$ if for any two numbers $1<a,b<p$, $ab \not\equiv p \pmod N$. We will define $p(n)$ to be the number of primes mod $n$. I'm wondering about the ...
- 3,319
9
votes
2
answers
319
views
Representation of a residue modulo prime as a specific product
Let $p$ be a prime number. For every integer $m$, there are integers $u_1$, $u_2$, such that $\lvert u_1\rvert, \lvert u_2\rvert < \sqrt{p}$ and
$$m \equiv u_1u_2^{-1} \pmod{p}.$$
Proof of this ...
- 965
0
votes
0
answers
351
views
On a Duality between Riemann-weil explicit formula and Abel- Plana summation of trigonometric prime counting function:
Consider the analytic function $g(x)$
Now define
$f(x)=g(x)\frac{\sin^2\left(\frac{π\Gamma(x)}{2x}\right)}{\cos^2\left(\frac{π}{2x}\right)}$
Such that
$|f(x+it)|=o(e^{2πt})$
uniformly for every $x$...
- 790
2
votes
1
answer
466
views
Number of points on a surface modulo p
I am guessing that the number of solutions $(x_1,x_2,\cdots ,x_s)$ modulo $p$ of the system of polynomials
$$x_1x_2\cdots x_s=1,$$
$$(x_1-1)(x_2-1)\cdots (x_s-1)=u$$
where $u$ is non-zero modulo $p$.
...
user126352
1
vote
0
answers
148
views
Counting prime factors of polynomial functions
Let $\Omega(n)$ denote the number of prime factors (counted with multiplicity) of a non-zero integer $n$. For $f \in \mathbb Z[X]$ non-zero, let $$m(f) = \liminf_{n \to \infty} \Omega(f(n))$$
(1) Is $...
- 11.9k
-2
votes
2
answers
149
views
Calculate the great common factor between $2^{2n+1}-1$ and $2^{4m+2}+1$ [closed]
How to calculate the great common factor between $2^{2n+1}-1$ and $2^{4m+2}+1$, where $n$ and $m$ are positive numbers.
We guess that: the great common factor is $1$.
- 577
3
votes
1
answer
220
views
Bins-and-primes (prime divisors of $\prod(a_i-a_j)$, II)
This is a refined version of a question I have recently posted.
For a prime $p$, let $\varphi_p\colon\mathbb Z\to\mathbb Z/p\mathbb Z$ denote the canonical homomorphism from the integers onto the ...
- 23k
2
votes
1
answer
163
views
Prime divisors of $\prod(a_i-a_j)$
For a prime $p$, let $\varphi_p\colon\mathbb Z\to\mathbb Z/p\mathbb Z$ denote the canonical homomorphism from the integers onto the group of order $p$.
Given an integer $n\ge 3$, what is the smallest ...
- 23k
1
vote
0
answers
55
views
Convergence of Farey series integral of a "density" function as the order tends to infinity
Let $F_n$ denote the $n$-th Farey sequence, and let $q$ be a rational number such that $0 \leq q \leq 1$. I am studying the convergence of a specific integral related to Farey series, defined as ...
- 375
2
votes
1
answer
153
views
Estimating the minimum number of distinct least prime factors found in range of $c$ consecutive integers
When I look at the count of distinct least prime factors for a range of consecutive integers, I am seeing the same minimum number appear again and again. I am wondering if this number represents the ...
- 1,049
11
votes
2
answers
1k
views
Mertens-like theorem
Mertens' first theorem states that
$$
\sum_{p \leq n} \frac{\log p}{p} = \log n + O(1).
$$
I read in this paper that the following variant is "classical":
$$
\sum_{p \leq n} \frac{\log p}{p -...
3
votes
0
answers
165
views
What is the density of numbers which have at least two divisors whose sum is a perfect square?
Note: This question was posted in MSE about two years ago but it not receive an answer. Hence posting in MO.
A positive integer is said to have square-sum divisors if it has at least two divisors ...
- 5,470
4
votes
1
answer
515
views
Recent works on the Hardy-Littlewood conjecture on primes represented by quadratic polynomials
I have been working on my master's thesis which is about the equivalence of the Hardy-Littlewood conjecture on primes represented by quadratic polynomials and the Lang-Trotter conjecture for CM ...
- 309
7
votes
1
answer
481
views
Some conjectures about prime gaps
I checked some relations between primes, here $1<n<10^5$ and $p_n$ is the $n$th prime.
$a) p_n^{1/3} - p_{n-1}^{1/3}<1/2$
$b) p_n^{1/n} - p_{n-1}^{1/n}<1/n $
$c) (\log p_n)^{1/2} - (\...
- 521
26
votes
0
answers
567
views
Elliptic analogue of primes of the form $x^2 + 1$
I have a project in mind for an undergraduate to investigate next quarter -- a curiosity really, but I'm surprised I can't find it in the literature. I do not want a detailed analysis here... but ...
- 13.3k
1
vote
0
answers
63
views
Set from a diophantine equation with similar statistics to primes
While doing some computational calculations with some diophantine equations, I came across with some sequences from solutions of quartic and quintic equations with slowly decreasing frequency, similar ...
3
votes
1
answer
474
views
Curious infinite product, convergence, connection to prime numbers
I have been playing with the following function:
$$
f(x)=\frac{\pi x (1-x^2)}{\sin\pi x}\prod_{k=2}^\infty \frac{\sin(\pi x/k)}{\pi x/k}
$$
It is hard to get correct numerical values. I'll start with ...
- 3,098
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
2
votes
2
answers
260
views
Inequalities for two functions related to the primorial function
Added: As remarked in the answers below, my question has a negative (and well-known) answer.
We denote by $\mathcal P=\lbrace 2,3,5,7,\ldots\rbrace$ the set of prime-numbers and by
$\mathcal P^*=\...
- 17.9k
2
votes
1
answer
140
views
Weak Siegel–Walfisz property
Let $f:\mathbb N \to \mathbb C$ be an arithmetic function. There are various ways to define what the Siegel–Walfisz (S–W) property is for $f(n)$. One simple way is that
there exists some function $g(...
- 3,062
14
votes
0
answers
297
views
An 'onion-structure' for roots of a series associated to prime numbers?
The series $$\sum_{n=1}^\infty\frac{z^{p_n-n}}{n!}$$ associated to the
sequence $p_1=2,p_2=3,p_3=5,p_4=7,p_5=11,\ldots$ of prime numbers
defines a holomorphic function in the open disc of radius $e$.
...
- 17.9k
2
votes
0
answers
70
views
Twin prime distribution centering twice a semiprime
What is the conjectured distributional behavior of semiprimes $pq$ ($p$ and $q$ are primes) having the property $2pq+1$ and $2pq-1$ are primes?
- 13.9k
1
vote
0
answers
65
views
Distribution of number of prime factors of $p^k\pm1$
What is the behavior of number of prime factors of integers of form $p^k\pm1$ where $p$ is a fixed odd prime or $2$ and $k$ varies over positive integers?
- 13.9k
2
votes
0
answers
238
views
Possible regularisation for sum of function of primes
Consider the following sum of function of primes:
$$-\sum_{p}\ln\left( 1 - \frac{1}{(ep)^{1/2}} \right){\ln(p)}$$
Here $p$ runs through all primes and $e$ is Euler's constant.
We can see that the sum ...
- 149
6
votes
0
answers
230
views
A bias for runs in Legendre symbols?
$\newcommand\Legendre[2]{\genfrac(){}{}{#1}{#2}}$An odd prime $p$ defines the sequence $\Legendre1 p,\Legendre2 p,\dotsc,\Legendre{p-1}p$
of values of the Legendre symbol describing the quadratic ...
- 17.9k
13
votes
4
answers
2k
views
Proving Mertens' theorem using the prime number theorem
Mertens' Theorem states that
$$\sum_{p \leq x}\frac{1}{p} = \log \log x + M + O(1/\log x).$$
This is weaker than the prime number theorem; in fact according to the Wikipedia page, the prime number ...
- 21.3k
6
votes
1
answer
172
views
Is the set of all solutions $x > 0$ to $ \pi(x) = \operatorname{li}(x)$ unbounded?
Is the set of all solutions $x > 0$ to the equation $\pi(x) = \operatorname{li}(x)$ unbounded? Is $\liminf_{x \to \infty} |\pi(x)-\operatorname{li}(x)|$ equal to $0$?
Here, $\pi(x)$ denotes the ...
- 4,985
6
votes
2
answers
575
views
Average value of the prime omega function $\Omega$ on predecessors of prime powers
For a positive integer $n$, the prime omega function value $\Omega(n):=\sum_{p\mid n}{\nu_p(n)}$ counts the number of prime divisors of $n$ with multiplicities. A result of Hardy and Wright, [1, ...
- 492
1
vote
1
answer
286
views
GRH and the Euler product
Let $L(\chi, s)$ be the Dirichlet L-Function of a primitive character $\chi$. I believe, if I’m not mistaken, the convergence of the Euler product of $L(\chi, s)$ in the critical strip is known to be ...
- 565
2
votes
1
answer
283
views
Explicit bounds on number of primes of given size
How many prime numbers of $b$ bits are there?
Beyond the prime number theorem, one can give explicit bounds on the number of primes below some integer $n$, or in a given interval. For instance, Rosser ...
- 456
2
votes
0
answers
352
views
An approximation for the prime counting function
NOTE: I've edited the question one last time, to be much simpler, in the hopes of getting more responses.
SETUP: Let $p_n$ denote the $n$th prime, let $p_x = p_{\lceil x \rceil}$ for all $x > 0$, ...
- 4,985
10
votes
0
answers
416
views
Are prime numbers among sums of prime numbers distributed as $\frac n{2\ln(n)}$?
Let $(s_n)_{n\in\mathbb N}$ be defined as follows:
For $n\in\mathbb N$, $s_n:=2+3+5+\cdots+p_n$ is the sum of the first $n$ prime numbers (e.g.: $s_1=2$, $s_2=5$, $s_3=10$, $s_4=17$, $\ldots$).
Let $\...
15
votes
0
answers
365
views
Do primes of the form $4k+1$ ever lead the greatest prime factor race?
Analogous to Chebyshev's race between primes, I examined the race between primes in the greatest prime factors, GPF, of natural numbers. Similar to the regular prime race, in the GPF race, the ...
- 5,470
1
vote
0
answers
155
views
Function involving argument of the Riemann zeta function
When $t$ is an ordinate of a zero of Riemann zeta function, we define \begin{equation}
f(t):=\frac{t}{2\pi}\log\left(\frac{t}{2\pi e}\right)+S(t)-\frac{1}{8}+\frac{1}{48 \pi t}+\frac{7}{5760 t^3}+...
- 19
1
vote
1
answer
131
views
Consecutive non-powerful integers
Pair of sequences $\ v_n\ $ and $\ U_n\ $ of integers start as in the following table:
[\begin{array}{rrrrrrrrrr}
n= & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & \ldots \\
...
- 4,786
3
votes
1
answer
293
views
Best available bounds for $\pi(Y)-\pi(Y-X)$?
I don't know much (anything) about sieves, but as I read the section on the Selberg upper bound sieve from Greaves's Sieves in Number Theory, there is a theorem 4 which says that
If $Y\ge X \ge 2$, ...
- 107
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
7
votes
1
answer
596
views
What consequences would follow from the density hypothesis?
Let $N(\sigma,T)$ denote the number of zeros $\rho=\beta+\gamma i$ of the Riemann zeta function satisfying $\beta\ge \sigma$ and $0<\gamma\le T$, counted with multiplicity. Then the "Density ...
- 435
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 ...
1
vote
1
answer
347
views
On equations with arithmetic functions [closed]
Is this good topic for research:
equations with arithmetic functions, for example equations like $\varphi(n)=\sigma(n)$ or $\varphi(n)+\sigma(n)=d(n)$ ?
If Anyone here have an advise please tell me ...
- 31
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 ...
- 6,984
-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 $s$ ...
- 6,984
6
votes
0
answers
465
views
On improvements of the GPY sieve
When $\chi_\mathbb P(n)$ denotes the characteristic function of primes and $\mathcal H=\{h_1,h_2,\dots,h_k\}$ is some admissible $k$-tuple, the GPY sieve can be formulated as follows:
$$
S(x)=\sum_{x&...
- 1,315
-10
votes
1
answer
407
views
Summatory functions for fractional parts
Notation:
$$ \{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=...
- 4,786
4
votes
0
answers
135
views
Average of $\lambda(n+1)$ for $n$ smooth, or smooth-and-rough? What follows?
Let $\lambda$ be the Liouville function, i.e., $\lambda(p_1\dotsb p_k)=(-1)^k$ for $p_1,\dotsc,p_k$ not necessarily distinct.
There is a conjecture (due to whom?) that there are infinitely many primes ...
- 20.2k