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6 votes
1 answer
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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 ...
rr_math's user avatar
  • 63
-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}=...
Sylvain JULIEN's user avatar
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 ...
Turbo's user avatar
  • 13.9k
3 votes
0 answers
192 views

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}...
H A Helfgott's user avatar
  • 20.2k
1 vote
1 answer
172 views

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 ...
Sylvain JULIEN's user avatar
1 vote
1 answer
200 views

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 ...
Rick's user avatar
  • 13
4 votes
0 answers
266 views

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,...
H A Helfgott's user avatar
  • 20.2k
1 vote
0 answers
127 views

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 $...
Khadija Mbarki's user avatar
1 vote
1 answer
374 views

$\{ 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\} -...
tomos's user avatar
  • 1,381
3 votes
1 answer
216 views

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 ...
William Hu's user avatar
5 votes
0 answers
261 views

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)=...
Maurizio Moreschi's user avatar
7 votes
1 answer
276 views

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 ...
H A Helfgott's user avatar
  • 20.2k
5 votes
1 answer
811 views

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, ...
mathoverflowUser's user avatar
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 $...
Virgile Dine's user avatar
1 vote
0 answers
152 views

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(\...
TPC's user avatar
  • 790
4 votes
0 answers
145 views

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 ...
Juan Moreno's user avatar
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 ...
Đào Thanh Oai's user avatar
3 votes
0 answers
1k views

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 \...
Pedja's user avatar
  • 2,661
11 votes
0 answers
436 views

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 $\...
Jakub Konieczny's user avatar
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^...
tomos's user avatar
  • 1,381
11 votes
1 answer
637 views

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 ...
R. van Dobben de Bruyn's user avatar
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,...
C Marius's user avatar
  • 251
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&...
AsksQuestionsAboutMath's user avatar
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 $N=10^2, 10^3,...,10^{9}$, $x$ ...
Đào Thanh Oai's user avatar
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 ...
Đào Thanh Oai's user avatar
6 votes
0 answers
381 views

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 ...
Mathew's user avatar
  • 81
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 ...
Charles's user avatar
  • 9,114
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 \...
Marcos Cramer's user avatar
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), ...
Felixson's user avatar
  • 232
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<...
user2284570's user avatar
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 ...
Anton Rechenauer's user avatar
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 ...
Pruthviraj's user avatar
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 $$ ...
Jamal Farokhi's user avatar
5 votes
0 answers
131 views

Taking integer values of a sequence of Beurling primes

Let $P=(p_j)_{j=1}^\infty$ be an increasing sequence of real numbers with $1<p_1$ and $\lim_{j\to\infty}p_j=\infty$. As mentioned in [1], Beurling proved that if the multiplicative group $N_P$ ...
Anon12345's user avatar
5 votes
1 answer
737 views

Smallest prime factor of numbers

The literature refers to smooth integers as \begin{equation}\Psi(x,y):=\#\{n\le x:P_1(n)\le y\},\end{equation} where $P_1(n)$ is the largest prime factor of $n$. There are lots of results studying $\...
user avatar
11 votes
1 answer
324 views

Does the mean ratio of the largest prime factor in prime gaps to the lower bound of the gap converge?

Posting in MO since this questions has been unanswered in MSE for 3 months. Let $p_n$ be the $n$-th prime and $q_n$ be largest among all the prime factors of the composite numbers between $p_n$ and $...
Nilotpal Kanti Sinha's user avatar
3 votes
1 answer
747 views

Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , where $\pi$ denotes the prime counting function and $p_n$ denotes the $n$-th prime?

Is $$1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n},$$ where $\pi$ denotes the prime counting function and $p_n$ denotes the $n$-th prime? Context: This question came out as a result in ...
mathoverflowUser's user avatar
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,\...
Sayan Dutta's user avatar
4 votes
1 answer
286 views

Density of primes $p$ where $p-1$ has a prime factor exceeding $p^{2/3}$

Fouvry proved* that primes $p$ such that the greatest prime factor, $q$, of $p-1$ is greater than $p^{2/3}$ have positive density in the primes. (The sequence is A073024 in the OEIS.) Are there any ...
Charles's user avatar
  • 9,114
2 votes
0 answers
131 views

Limit of scaled infinite sum with Dirichlet characters modulo 4: is it zero?

I am trying to get an asymptotic formula such as $$ L_4(s, n) \sim L_4(s) + \rho_n(s)\Lambda_n + \frac{\alpha(s)}{\sqrt{n}} + \frac{\beta(s)}{\sqrt{n\log n}}+\cdots$$ where $L_4(s, n)$ is the first $n$...
Vincent Granville's user avatar
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 ...
Đào Thanh Oai's user avatar
8 votes
2 answers
1k views

Natural density of the set of simple numbers

Let us call $n>1$ simple if every prime power $q$ with $q-1 \mid n-1$ is a prime number. (Please let me know if there is already an established name for these numbers.) The simple numbers $\leq 100$...
Martin Brandenburg's user avatar
2 votes
2 answers
424 views

"Squeezing" the primes?

The logical idea here is to map a curve that encodes the primes into the region $(0,1)^2$ and analyze the distribution there more easily and achieve tight bounds. To assess the distribution of primes, ...
John McManus's user avatar
4 votes
0 answers
160 views

On the asymptotic $\pi(x+h(x)) - \pi(x) \sim \frac{h(x)}{\log x} \ (x \to \infty)$

Let $h(x)$ be a function that is positive on $\mathbb{R}_{>0}$ and satisfies $h(x) = o(x)$ and $(\log x)^a = o(h(x))$ for all $a > 0$, as $x \to \infty$. Is it reasonable to expect under these ...
Jesse Elliott's user avatar
2 votes
0 answers
103 views

On equidistribution of primes in positive characteristic

In S. Lang's book "Algebraic Number Theory" (1986), page 317, Theorem 6 states essentially that given $P$ a set of primes, let $\tau:P\longrightarrow J$ be the typical idèle map taking ...
Hair80's user avatar
  • 675
2 votes
1 answer
740 views

Does the Riemann hypothesis predict a bound for this prime-counting function?

Does the Riemann hypothesis predict an upper bound for $$\left|f(x)-\left(\operatorname{li}(x)-\frac{x}{\log x} \right)\right|,\quad x\ge 2\tag{1}$$ where $$f(x)=\sum\limits_{n=2}^x \frac{\Lambda(n)}{\...
Steven Clark's user avatar
  • 1,126
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 $S$?
pallab1234's user avatar
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) := \...
Q_p's user avatar
  • 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 $f(n)=0$ ...
TPC's user avatar
  • 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 ...
Daniel Weber's user avatar
  • 3,319

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