All Questions
Tagged with analytic-number-theory prime-numbers
690 questions
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 ...
- 50.6k
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} - (\...
- 105k
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 ...
- 50.6k
24
votes
1
answer
693
views
Gaussian primes in small boxes
The best unconditional result bounding prime gaps is due to Baker, Harman and Pintz, and states that for any sufficiently large $n$, the interval $$[n,n+Cn^{0.525}]$$ contains a prime, for some ...
- 63.9k
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 ...
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 ...
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 ...
- 146
7
votes
4
answers
1k
views
Reference for the expected number of prime factors of n larger than n^alpha is -log alpha
Let $0 < \alpha < 1$ be a constant. The expected number of prime factors of a "random" integer near $n$ which are greater than $n^\alpha$ is $-\log \alpha$.
It's my understanding that (...
- 11.4k
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
-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:
$$ \tau(n)\ :=\ \sum_{k=...
- 19.6k
-1
votes
1
answer
280
views
A question on assigning finite values to divergent sums involving expression of primes
We know the following:
$$\gamma=\lim_{n\to\infty }\left(\sum_{k=1}^n\frac{1}{k}-\ln(n)\right).$$
This could be a good candidate for renormalized sum of $\left(\sum_{k=1}^{\infty}\frac{1}{k}\right)$.
...
- 149
2
votes
1
answer
140
views
Weak Siegel–Walfisz property
Let $f:\mathbb N \to \mathbb C$ be an arithmetic function. There are various ways to define what the Siegel–Walfisz (S–W) property is for $f(n)$. One simple way is that
there exists some function $g(...
- 14.6k
2
votes
2
answers
260
views
Inequalities for two functions related to the primorial function
Added: As remarked in the answers below, my question has a negative (and well-known) answer.
We denote by $\mathcal P=\lbrace 2,3,5,7,\ldots\rbrace$ the set of prime-numbers and by
$\mathcal P^*=\...
- 18.4k
1
vote
1
answer
347
views
On equations with arithmetic functions [closed]
Is this good topic for research:
equations with arithmetic functions, for example equations like $\varphi(n)=\sigma(n)$ or $\varphi(n)+\sigma(n)=d(n)$ ?
If Anyone here have an advise please tell me ...
- 31
14
votes
0
answers
297
views
An 'onion-structure' for roots of a series associated to prime numbers?
The series $$\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
3
votes
2
answers
795
views
Estimate about primes
Can anyone give an estimate (upper bound or lower bound) for the number of divisors $d\mid P_r$ such that $\frac{\sqrt{P_r}}{2}< d < \sqrt{P_r}$, where $P_r$ is the product of the $r$ smallest ...
- 4,713
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
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 ...
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
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 ...
- 28.2k
30
votes
3
answers
4k
views
Heuristic argument for the prime number theorem?
Here is a bad heuristic argument for the prime number theorem. Let $n$ be a positive integer and assume that PNT holds up to $n$. Then $n$ itself is prime if and only if for each prime $p<n$ the ...
- 4,713
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 ...
- 1,315
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 $...
- 50.6k
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 ...
- 105k
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 $\...
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 \\
...
- 9,114
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 ...
- 12.9k
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
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 (...
- 105k
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
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 $$
where $\...
- 63.9k
6
votes
0
answers
466
views
On improvements of the GPY sieve
When $\chi_\mathbb P(n)$ denotes the characteristic function of primes and $\mathcal H=\{h_1,h_2,\dots,h_k\}$ is some admissible $k$-tuple, the GPY sieve can be formulated as follows:
$$
S(x)=\sum_{x&...
- 6,984
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
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
2
votes
1
answer
297
views
Sums over primes in arithmetic progressions
Do we know anything about sums over primes in arithmetic progressions like the following:
$$\sum_{\substack{q \equiv a (\text{mod } l) \\ q \le x}} q^{\alpha}$$
where $q$ is a prime and $\alpha > ...
- 301
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
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,
$$...
- 111
2
votes
0
answers
244
views
Lower bounding the number of Galois radii of an integer
Recall that I call $r>0$ a Galois radius of an integer $n$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ primes and positive $a$ and $b$ and a primality radius of $n$ if $a=b=1$.
Does it suffice to ...
- 6,984
1
vote
0
answers
293
views
Can a lower bound for this weakening of Goldbach's conjecture be reached?
Say a non negative integer $r$ is a primality radius of $n$ if both $n-r$ and $n+r$ are prime, and that a non negative integer $w$ is a Galois radius of $m$ if $\omega(m-w)=\omega(m+w)=1$, where $\...
- 6,984
0
votes
0
answers
89
views
A similar inequality for the Dedekind psi function, than an inequality stated by Schinzel
I would like to ask about the next question that seems to me interesting. I know an article that was written by Andrzej Schinzel in which he stated Lemma 2. In this post we denote the Dedekind psi ...
-1
votes
1
answer
250
views
Significance of $N_0(T+1)-N_0(T)\sim \frac{1}{2\pi}\log \frac{T}{2\pi}$
Let $N(T)$ be the number of zeros of Riemann zeta function upto height $T$ in the critical strip and $N_0(T)$ be the number of zeros on the critical line.
What will be the significance of proving ...
- 105k
2
votes
0
answers
263
views
Selberg's 1943 result on primes in short intervals and primality radius
This preprint: https://arxiv.org/abs/2207.05038 states in the last paragraph of the first page that a result of Selberg (1943) implies that under RH, almost all intervals of the form $(x,x+\left(\log ...
- 13k
5
votes
1
answer
611
views
Why does this convolution of the prime counting function $\pi$ look like a parabola?
In this previous question it is shown that the convolution of the prime counting function $\pi$ with itself, is related to the Goldbach conjecture:
$$\pi^*(n):=\sum_{k=0}^n \pi(k) \pi(n-k)$$
The ...
- 3,064
9
votes
1
answer
422
views
Getting prime by changing 2 digits
I just got a result, an exercise of Tenenbaum's book that one cannot get a prime from arbitrary natural number $n$ by changing only one digit of its decimal expansion. For example, you cannot get ...
- 271
2
votes
0
answers
158
views
Quadratic patterns in summands of Goldbach's conjecture
Let $n $ be even and define
$$ Q(n)=\sum_{\substack{ p,q \ \textrm{ primes} \\p+q=n }}\left(\frac{p}{q} \right),$$ where $\left(\frac{p}{q} \right)$ is the quadratic Legendre symbol.
Has this sum been ...
- 3,062
17
votes
3
answers
2k
views
About the prime divisors of values of polynomials
Let $P$ be a polynomial having integer coefficients (and degree $\geq 3$), and let $\mathscr P_P$ be the set of prime numbers dividing some value $P(n)$ with $n \in \mathbb Z$.
Is it true that $\...
- 105k
1
vote
1
answer
317
views
An explicit value for a bound proof
I saw a proof that $|p_n - li^{-1}(n)| \leq n e^{-c \sqrt{\ln(n)}} $,
without saying anything about $c$ !
My questions is, what the explicit value of $c$ ??
It just says for some number $c$ without ...
- 105k