All Questions
Tagged with analytic-number-theory prime-numbers
690 questions
6
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1
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Difficulty with "A new elementary proof of the Prime Number Theorem" by Richter
I'm studying Richter's "A new elementary proof of the Prime Number Theorem" paper, and I'm finding some problems understanding some parts of it. For example, I don't see how to get, in Lemma ...
- 2,060
-3
votes
0
answers
71
views
Is the upper bound on $H_{1}$ a decreasing function of the proportion of critical zeros of Zeta?
This question stems from https://arxiv.org/abs/2411.19762 and the numerical observation that the best unconditional upper bound for $H_{1}:=\lim\inf_{n\to\infty}p_{n+1}-p_{n}$, namely $H_{1}^{\flat}=...
- 6,984
3
votes
1
answer
596
views
Primes which are safe and Sophie Germain
If $p$ is a Sophie Germain prime then $2p+1$ is safe prime.
If $2p+1$ is safe prime then $p$ is Sophie Germain prime.
What is their conjectured distribution of primes $p$ which are both Sophie ...
- 4,985
3
votes
0
answers
192
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What smoothing to use for PNT-like results?
Consider a Dirichlet series $\sum_n a_n n^{-s}$ with desirable analytic properties (e.g., analytic extension to $\Re s>0$); one example would be $a_n=\mu(n)$. Say we want to estimate $\sum_{n\leq x}...
- 20.2k
1
vote
1
answer
172
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Are all integers up to $x$ but possibly $O_{\varepsilon}(x^{\varepsilon})$ the sum of $a$ squares and $b$ primes with $a+b\leq 3$?
This question is related to https://math.stackexchange.com/questions/3710032/conjecture-all-but-21-non-square-integers-are-the-sum-of-a-square-and-a-prime.
We know since Lagrange that every natural ...
- 1,028
11
votes
1
answer
637
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Primes such that a given number has very small order
The following came up in (a previous version of) this answer.
Question. Let $a > 1$ be a positive integer, and $f \in \mathbf Z[x]$ a polynomial with positive leading term. Does there always exist ...
- 12.9k
1
vote
1
answer
199
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Weights in the proof of Chen's theorem in Nathanson's "Additive Number Theory The Classical Bases"
I was reading Professor Nathanson's graduate texts in mathematics 164: "Additive Number Theory The Classical Bases" (http://www.alefenu.com/libri/nathansonbases.pdf), and I was wondering if ...
- 13
1
vote
1
answer
374
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$\{ x/p\} $ on average
This is a vague question: Lemma 2.2 of Friedlander and Lagarias' "On the distribution in short intervals of integers having no large prime factor" says that $$\sum _{p\leq w}\left (\{ x/p\} -...
- 14.6k
4
votes
0
answers
266
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How dense are quotients of smooth numbers?
As usual, call a positive integer $y$-smooth if it has no prime factors greater than $y$. Write $S(x,y)$ for the set of $y$-smooth integers $\leq x$. Write $R(x,y)$ for the set of quotients $\{a/b: a,...
- 20.2k
1
vote
0
answers
127
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Some property of the greatest prime factor
Let $n$ be a positive integer $\geq 2$ et denote by $ P^{+}(n)$ the greatest prime factor of $n$ my question is as follows:
If $a$ and $b$ are two numbers, is there any method to express or to bound $...
- 10.4k
9
votes
0
answers
324
views
Semi-primes represented by quadratic polynomials
According to Lemke-Oliver, irreducible quadratic polynomials $G$ with positive leading coefficient and $\rho(2)<2$, (where $\rho(m)$ denotes the number of incongruent solutions to the congruence $G(...
CommunityBot
- 1
1
vote
1
answer
729
views
Could the complex zeros of Riemann zeta function be of the form $ s=0.5+ik$ with $k$ a positive integer? [closed]
I have checked in Andrew Odlyzko, Tables of zeros of the Riemann zeta function, to know if there is an example of zeros of Riemann zeta function with integer imaginary parts, but I don't see that. I ...
- 13
3
votes
0
answers
1k
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Formula for $\pi$ involving exponents of Mersenne primes
Can someone provide a proof for the following claim?
$$\pi=\dfrac{S_0S_2}{M_3M_5} \cdot\left(\displaystyle\prod_{p \equiv 1 \pmod{4} } \frac{p}{p-1}\right) \cdot \left(\displaystyle\prod_{p \equiv 3 \...
CommunityBot
- 1
3
votes
1
answer
216
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Are there any positive integers $n$, $k$ such that $n > 2, k > 6$, and all prime factors of $n^k - 1$ are less than $n$?
I noticed that the prime factorization of $68^6 - 1$ is $3^2 \cdot 7^2 \cdot 13 \cdot 19^2 \cdot 23 \cdot 31 \cdot 67$, which makes all of its prime factors less than 68. This made me wonder the ...
- 105k
5
votes
0
answers
261
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Primes generated by cyclotomic polynomials
Let $p$ be an odd prime, and let $f=\Phi_p$ be the $p$-th cyclotomic polynomial. Denote by $S_p$ the set of primes $q$ such that there exists a sequence of primes $p_1,\dots, p_g$ such that $p_1=f(1)=...
CommunityBot
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7
votes
1
answer
276
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From $\Lambda_k$ and $\Lambda$ to $\mu$ (or $\lambda$)
Let $\{a_n\}_{n=1}^\infty$, $a_n \in \mathbb{C}$, $|a_n|\leq 1$. Let $\Lambda_k = \mu \ast \log^k$; in particular, $\Lambda_1$ equals the von Mangoldt function $\Lambda$. Suppose that we have ...
- 20.2k
5
votes
1
answer
811
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A consequence of Firoozbakht's conjecture?
This is a question out of curiosity, while looking at the Firoozbakht's conjecture. It might not be research related, but as usual, I am not really sure if a question ever is research related or not, ...
- 3,064
4
votes
1
answer
403
views
Möbius square root function: existence of multiplicative and bounded function
With $\mu$ being the Möbius function, there exist infinite possibilities of square roots. For example, for each $n$ such that $\mu(n)\neq 0$ there is a choice: if $\mu(n)=-1$, we can choose to define $...
- 651
6
votes
2
answers
631
views
Rate of convergence of the prime zeta function P(2)
For an application in statistical group theory, we need explicit upper and lower bounds that an expert in number theory (I am not one) may know how to prove.
Question 1: What are "good" bounds $f_1(x)...
- 103
1
vote
0
answers
152
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On lacunary series connected with prime number theory
Consider the following lacunary sum with parameter $x$:
$$S(x)=\sum_{n=5}^{\infty}\sin^2\left(\frac{x\Gamma(n)}{n}\right).$$
As we can see for $x=\frac{\pi}{2}$
the sum becomes$$\sum_p\cos^2\left(\...
- 4,713
4
votes
0
answers
145
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Bounding an expression equivalent to Mertens function
Cross-posted from MathStackExchange, where the question is bountied but has not received any comment or answer)
Some months ago, I derived the following formula for the Merten's function $M(n)$ using ...
- 189
0
votes
0
answers
352
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On a Duality between Riemann-weil explicit formula and Abel- Plana summation of trigonometric prime counting function:
Consider the analytic function $g(x)$
Now define
$f(x)=g(x)\frac{\sin^2\left(\frac{π\Gamma(x)}{2x}\right)}{\cos^2\left(\frac{π}{2x}\right)}$
Such that
$|f(x+it)|=o(e^{2πt})$
uniformly for every $x$...
- 790
1
vote
2
answers
383
views
Is there any way to estimate this functions: $f(n)=\sum_{d|n}d\varphi(d)$ and $g(n)=\sum_{d|n}\frac{\varphi(d)}{d}$?
Let that $n$ be a natural number and $\varphi(n)$ be the Euler totient function. Is there any formula or estimation for computing functions $f,g$ such that:
$$
f(n)=\sum_{d\mid n}d\varphi(d)
$$
and
$$
...
- 1
8
votes
2
answers
393
views
Can exist a positive integer number $x$ such that $a_1=x$ and $a_n=2a_{n-1}+1$ are not prime for all $n \ge 1$?
Using my computer, I found that the most of positive integer number $x$ such that $a_1=x$ and $a_n=2a_{n-1}+1$ is prime number after a few iterations. But exist some positive integer numbers, my ...
- 39.9k
11
votes
0
answers
436
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Can we rule out the possibility that $\sqrt[3]{2}$ is small modulo every prime?
Consider a prime $p$ such that the polynomial $X^3-2$ splits into linear factors over $\mathbb{F}_p$: $X^3-2 = (X-\alpha_p)(X-\beta_p)(X-\gamma_p)$. It seems reasonable to expect that (identifying $\...
- 12.9k
2
votes
0
answers
191
views
The exponential sum over primes on average
In https://academic.oup.com/blms/article-abstract/20/2/121/266256?redirectedFrom=fulltext Vaughan shows the following bounds for the $L^1$-mean of the exponential sum over primes $$\sqrt x\ll \int _0^...
- 1,381
4
votes
1
answer
258
views
Density of numbers where a large prime factor satisfies a congruence
I am looking for an upper bound on the number of integers $n<x$ such that $n$ has a prime factor $p>\log(x)^{(1+\delta)}$ such that $p \equiv a \mod b$. Where $a,b$ are fixed and coprime and $0&...
CommunityBot
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4
votes
1
answer
266
views
Prime omega function values on a product of prime powers predecessors
Let $p_1, ... , p_n, ...$ be the prime numbers in order. Define
$$
P_n = \prod_{k=1}^n p_k^q
$$ It is known that $\omega(P_n) = n$ where $\omega(\cdot)$ is the little prime omega function. For a given,...
- 14.6k
3
votes
2
answers
465
views
Least number coprime to a given integer
For a positive integer $n$ let $$f(n):=\min\{m\in \mathbb N: m>1, \gcd(m,n)=1\} .$$
Equivalently, $f(n) $ is the smallest prime not dividing $n$.
Is there any upper bound literature for this? It is ...
- 20.8k
11
votes
2
answers
1k
views
Do consecutive integers have a big prime factor?
Let us say that three consecutive positive integers $(m-1,m,m+1)$ have a big prime factor if the largest prime factor $p$ of $N=(m-1)m(m+1)$ satisfies $e^p>N$.
I ckecked that it is true for all $m&...
CommunityBot
- 1
20
votes
2
answers
1k
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Median largest-prime-factor
Let $P(n)$ denote the largest prime factor of $n$. For any integer $x\ge2$, define the median
$$
M(x) = \text{the median of the set }\{P(2), P(3), \dots, P(x) \}.
$$
Classical results of Dickman and ...
- 1,446
13
votes
1
answer
1k
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About the number of primes which are the sum of 3 consecutive primes (OEIS A034962)
I made some numerical simulations about the number of primes which are the sum of 3 consecutive primes (OEIS A034962), that is for instance:
$$5+7+11=23$$
$$7+11+13=31$$
$$11+13+17=41$$
$$17+19+23=59$$...
- 365
4
votes
4
answers
913
views
Let $X$ be a positive integer. Then $\pi{(X+\ln^2{X})}-\pi{(X-\ln^2{X})}>\ln{X}$?
The prime-counting function is the function counting the number of prime numbers less than or equal to some real number $x$. It is denoted by $\pi{(x)}$. Using my computer I found that for any ...
- 7,182
4
votes
0
answers
335
views
The number of continuously increasing primes gaps in the interval $[2,n]$ is less than $\log n$
A prime gap is the difference between two successive prime numbers. The $n$-th prime gap, denoted $g_n$ or $g(p_n)$ is the difference between the $(n+1)$-st and the $n$-th prime numbers. Using my ...
- 12.9k
3
votes
1
answer
401
views
Probability of finding a prime number between $x-\ln(x)$ and $x+\ln(x)$
Using my computer, I found that in the interval $[1, N]$ the probability of finding a prime number between $x-\ln(x)$ and $x+\ln(x)$ is greater than constant $c$ where $N=10^2, 10^3,...,10^{9}$, $x$ ...
- 105k
8
votes
1
answer
245
views
Asymptotic density of sums of consecutive primes
Call a positive integer respectable if it is a sum of consecutive prime numbers. For example, every prime numbers is respectable. So are $3+5=8$, $2+3+5=10$, $5+7=12$, $3+5+7=15$, $2+3+5+7=17$, $7+11=...
CommunityBot
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3
votes
0
answers
145
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Roth's theorem for primes in a given arithmetic progression to a large modulus
Let $\mathbb{P}_{a, q}$ denote the set of primes congruent to $a$ modulo $q$. Are there any estimates for the number of $3$-Arithmetic Progressions in the set $\mathbb{P}_{a, q}\cap [1, X]$, where $(a,...
- 31
6
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0
answers
381
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A possible variant of Zagier's one-sentence proof for Fermat's sum of two squares theorem?
Is it possible to modify Zagier's one-sentence proof of Fermat's sum of two squares theorem (see here) to prove certain non-trivial cases of Jacobi's four-square theorem (see here)?
Let $p$ be a prime ...
1
vote
0
answers
128
views
Effective Erdős–Kac theorem
I have some number $N$ and some integer $k>0$. I want to know what fraction of numbers up to $N$ have more than $k$ prime factors. (In my application, with repetition, but the $\omega$ version is ...
- 9,114
-1
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3
answers
1k
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Which even numbers are known to be both prime gaps and the sum of 2 primes?
Goldbach's conjecture asserts that every even integer greater than $3$ is the sum of two primes, while de Polignac's one says every even positive integer is a prime gap infinitely often. My question ...
- 4,713
10
votes
2
answers
3k
views
Can every integer be written as a sum of squares of primes?
This question is mainly inspired from a different problem I was working on.
Is there a value of $k$ such that, for each $n\in \mathbb N$, the equation
$$\sum_{i=1}^{k}x_i^2=n$$
is solvable in $x_1,\...
3
votes
1
answer
329
views
Fully explicit Linnik's Theorem
Linnik's Theorem states that there exist absolute constants $c$ and $L$ such that for every $m \in \mathbb{N}$ and every $a$ coprime to $m$, there is a prime $p$ with $p \equiv a \pmod{m}$ and $p < ...
- 105k
5
votes
1
answer
750
views
Geometric mean of prime factors of all numbers up to n
Through numerical calculations I have discovered that for any natural number $n \geq 2$, the geometric mean of the prime factors of all natural numbers $\leq n$ can be approximated well by $1.6653 \...
- 105k
12
votes
2
answers
1k
views
Prime differences and zero multiplicity
Concerning gaps between consecutive primes, Paul Erdős conjectured that:
$$\sum_{p_n < x} (p_n -p_{n-1})^2 = O(x \log x)$$
Let's call this hypothesis EH. Assuming the Riemann hypothesis (RH), ...
- 105k
1
vote
0
answers
71
views
Tuples of natural numbers with no mutual divisibility and large reciprocal sums
Standard apology in case this is something simple, as I'm not a number theorist.
Let $E_1, \dots, E_n$ be disjoint finite sets of natural numbers, such that for any $a_1 \in E_1, \dots, a_n \in E_n$, ...
- 695
2
votes
0
answers
66
views
How to check that a number probably/likely has a divisor having a specific bit length/in range?
Given a randomly generated $\alpha\in\Bbb N$ where $\alpha$ is large thus hard to factor (no small prime composites). How to check that a divisor $F\in\Bbb N$ with a specific bitlengh $n\in\Bbb N∧n<...
- 87
4
votes
0
answers
200
views
Effective bound for odd numbers expressed as sums of three primes
I am interested in the representation of odd numbers greater than five as sums of three primes, inspired by Harald Helfgott's seminal proof of the ternary Goldbach conjecture and the nuanced findings ...
CommunityBot
- 1
2
votes
0
answers
155
views
Electrostatic potential energy of point-charges at primes up to $x$
Given a positive real (or integral) number $x$ we consider the
electrostatic potential energy of equal point charges at all primes up to $x$
given by
$$E(x)=\sum_{p_1<p_2\leq x}\frac{1}{p_2-p_1}$$
...
- 6,367
4
votes
0
answers
446
views
There are infinitely many prime which have arbitrary large gap in their digits in particular base expansion
Consider $m$ and $r$ is any fixed positive integer and $t$ is a variable $(t=0,1,2,3,...)$. Below, $[a]$ denotes the greatest integer function of $a$ (or floor function).
Claim 1 : There exists a ...
CommunityBot
- 1
8
votes
1
answer
863
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On the least prime in arithmetic progressions
My question concerns the least prime (denoted $p(a, q)$) in the arithmetic progression $a \pmod q$ where $a$ and $q$ are coprime. Quite a time ago Linnik demonstrated that
$$p(a, q) \ll q^L$$
for some ...
- 1,446