**3**

votes

**1**answer

186 views

### On partial sum of non-primitive Dirichlet characters

Consider a Dirichlet character, $\chi(n)$, and the partial sum :
$$S(\chi,x)=\bigg |\sum_{n=1}^{x} \chi(n)\bigg|$$
There are many works to bound this sum when $\chi$ is a primitive character, but ...

**9**

votes

**0**answers

272 views

### Independence between the number of prime factors of $n$ and $n+2$

I am interested in having an upper bound for the cardinality of
$\#\left\{n\leq x\,:\quad\omega(n)=k, \omega(n+2)=\ell\right\}$
for $k,\ell\geq 1$,
where $\omega(n)=\sum_{p\vert n}1$ counts the ...

**1**

vote

**0**answers

39 views

### Distribution of primes and near-primes among $\prod p_k \pm 1$

For $n\in \Bbb{Z}^+$ define the statement "$n$ is $k$-social" to mean that
$$
\prod_{i=1}^n p_i +1 \mbox{ has exactly } k \mbox{ prime factors}
$$
where $p_i$ is the $i$-th prime.
So for example $5$ ...

**0**

votes

**1**answer

102 views

### What is the relative size of the radical of an ABC-triple relative to the number of primes up to its largest element?

Write $\bf N$ for the set of natural numbers, and $P$ for the set of primes. For $x$ in $\bf N$ let $p(x)$ be the product of the primes dividing $x$ (that is, the "radical" of $x$). Also write $\#(x)$ ...

**0**

votes

**1**answer

172 views

### Is it consistent with Cramer's conjecture to conjecture that $x/\pi_{2}(x)>2B_{2}/6\times\log^{2}x$?

Brun's constant $B_{2}$ is defined as $B_{2}=1/3+1/5+1/5+1/7+1/11+1/13+...$ where the sum is taken on $p$ such that $p$ is an element of a couple of twin primes. The number of twin primes below is ...

**4**

votes

**1**answer

489 views

### Use of infinitude of primes in the Green-Tao theorem [closed]

In a video I watched last night on nuking mathematical mosquitos, Matt Parker gave the following proof of the infinitude of primes: suppose there are finitely many primes. The Green-Tao theorem says ...

**4**

votes

**1**answer

246 views

### Can we always attain another prime via inserting digits between the digits of a fixed prime?

The sequence OEIS A080437 is
For n > 10, let m = n-th prime. If m is a k-digit prime then a(n) = smallest prime obtained by inserting digits between every pair of digits of m.
I don't see why ...

**3**

votes

**2**answers

503 views

### Primary structures in $\mathbb Q$

I'll formulate a topic restricted here to the positive rational
numbers $\ \mathbb Q_{_{>0}},\ $, then will pose a question (Q2) plus some related, to which I don't know the answers nor reference. ...

**18**

votes

**1**answer

378 views

### How big are the prime factors of $2^kp - 1$?

I have already asked this question here. No answers despite the bounty (which has now ended)
Let $p$ be a prime number, $p > 3$.
Does there always exist $k \in \mathbb N_{\ge 1}$ such that the ...

**4**

votes

**0**answers

126 views

### Are prime gaps of even index essentially larger than those of odd index?

Let $g_{n}:=p_{n+1}-p_{n}$ be the $n$- th prime gap, and let's introduce the following summatory functions:
$$G_{1}(x):=\sum_{1\leq n\leq x}g_{2n-1}$$
$$G_{2}(x):=\sum_{1\leq n\leq x}g_{2n}$$.
Let's ...

**3**

votes

**0**answers

70 views

### Square integral of finite Euler product

Consider the finite Euler product
$$
P(t) = \prod_{r=1}^R \left(1 + p_r^{i t} \right).
$$
(Here $p_1, p_2, \dots$ are of course the primes.)
Question: What is a good asymptotic upper bound for
$$
\...

**12**

votes

**1**answer

239 views

### Elementary prime-generating sequences

A student of mine keeps coming again and again and telling "I've found a formula $n\mapsto f(n)$ giving all primes" or sometimes "infinitely many primes", where $f$ is a classical function (I mean ...

**12**

votes

**1**answer

527 views

### Does the sum $\sum_{n=1}^{\infty}\frac{1}{p_n(p_{n+1}-p_n)}$ converge?

Prove, if possible in an elementary way, that $\sum_{n=1}^{\infty}\frac{1}{p_n(p_{n+1}-p_n)}$ converges/diverges, where $p_n$ denotes the $n^{\textrm{th}}$ prime.

**1**

vote

**1**answer

297 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"...

**3**

votes

**2**answers

244 views

### Unfamiliar prime-generating polynomials related to Heegner numbers

I just stumbled on a set of prime-generating polynomials of the form $$9 n^2-3 H n+H (H+1)/4$$ (where $H$ is a Heegner number $>11$), which generate the same number of distinct primes as their more ...

**1**

vote

**0**answers

121 views

### Primes of the form $2^{m_0}p_1^{m_1}\ldots p_r^{m_r}+1$

Is it known any example of a set of primes $\{p_1,\ldots,p_r\}$ with the following property: there are infinitely many $(m_0,\ldots,m_r)\in\mathbb N^{r+1}$ such that $2^{m_0}p_1^{m_1}\ldots p_r^{m_r}+...

**1**

vote

**0**answers

105 views

### Density of ratios of an arbitrary increasing sequence of prime numbers

It is well known that the set $\left\{ \frac{p}{q} : p,q \textrm{ prime numbers }\right\}$ is dense in the positive real numbers $\mathbb{R}_{>0}$. Not having a background in number theory, I ask ...

**5**

votes

**1**answer

372 views

### natural radical and an algebraic expression in $\pi$ and/or $e$

Let $\ \mathbb N:= \{1\ 2\ \ldots\}\ $ be the set of natural numbers. Let $\ \mathbb P:=\{2\,\ 3\,\ 5\,\ 7\,\ 11\,\ \ldots\}\ $ be the set of primes. Then natural radical $\ rad(n)\ $ is
$$ rad(n)\ :=...

**10**

votes

**1**answer

477 views

### Why do the Maynard-Tao weights work so well?

I am looking for an intuitive reason for why the Maynard-Tao weights work well to capture many primes of the form $n+h_1, \ldots , n+h_k$, where $(h_1, \ldots , h_k)$ is any admissible $k$-tuple.
For ...

**3**

votes

**0**answers

106 views

### Distribution of the inbetween prime

Let $\ \mathbb J_n\,:=\,\{1\ \ldots\ n\}\ $ be the initial interval of natural numbers, and
$$2=p_0<p_1<\ldots$$
be the increasing sequence of all primes. Let
$$ \forall_{n=1\ 2\ \ldots}\ \ d_n\...

**1**

vote

**0**answers

48 views

### On the sum of digits of primes in binary form [duplicate]

Let $s_2(m)$ be the sum of digits of $m$ in binary form.
I would like to ask the following question:
Is it true that for every $n\in \mathbb{N}$ there is at least one
prime $p$ which has $...

**0**

votes

**0**answers

71 views

### Explicit formula for $\vartheta(x)$

Is there an explicit formula for the Chebyshev Theta Function like there is for the Psi Function in terms of the zeta zeroes? I know one for theta can be derived from the one with psi using mobius ...

**2**

votes

**1**answer

157 views

### Bounds on $\pi(x)$ vs. bounds on $\vartheta(x)$

If $\pi(x) > \operatorname{Li(x)},$ is $\vartheta(x) > x$? Are the two inequalities (solutions to both of which are known to exist but not known exactly) equivalent, similar, or mostly unrelated?...

**3**

votes

**1**answer

242 views

### Integral involving the gamma function

If we define $$f(x) = 1 + \frac{\cos\big(\pi\frac{\Gamma(x) + 1}{x}\big)}{2 - \cos(2\pi{x})}$$ how would one go about evaluating
$$ \int_1^R \frac{1}{x} \log{f(xe^{i\alpha})} dx$$ for some parameter ...

**7**

votes

**2**answers

504 views

### How do i show that:$\prod\frac{p^2+1}{p^2-1}=\frac{5}{2}$ without using properties of Riemann zeta function? [duplicate]

In order to know more about product over primes ,I would like to know how do I show that :$$\prod\frac{p^2+1}{p^2-1}=\frac{5}{2}$$ without using properties of Riemann zeta function ?
Note01 : it ...

**1**

vote

**1**answer

189 views

### Convergence of a double sum involving prime numbers

This has been moved from math.stackexchange;
I am attempting to prove/disprove convergence of the following sum
$$ \lim_{n \to \infty} \frac{1}{n} \sum_{p \leq n} \sum_{k=0}^\infty \ln p \left\{\frac{...

**1**

vote

**0**answers

218 views

### Is this a proof of the Hardy-Littlewood inequality? [closed]

V.V. Miasoyedov posted a paper to the arXiv claiming a proof of the Hardy-Littlewood conjecture $\pi(x+y) \le \pi(x)+\pi(y)$. It seems a bit off, and not only because the conjecture is widely believed ...

**3**

votes

**0**answers

117 views

### exponential sum of primes

Fix $\alpha \notin \mathbb{Q}$. I would like to know a reference that shows $$\mathbb{E}_{n\leq N} \Lambda(n) e^{2\pi i \alpha n} \to 0,$$ as $N$ tends to infinity.
I am familiar with Vinagradov's ...

**5**

votes

**2**answers

385 views

### Does the antidiagonal in this square matrix always contain a prime?

Does the antidiagonal in the square matrix with entries $1,2,\ldots,n^2$ row by row in that order always contain a prime?
For example:
For n=2: $\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$ ...

**3**

votes

**1**answer

151 views

### Confusion regarding Riesz Function Definition

According to wikipedia:
'In mathematics, the Riesz function is an entire function defined by Marcel Riesz in connection with the Riemann hypothesis, by means of the power series
$$
{\rm {Riesz}}(x)=...

**0**

votes

**1**answer

166 views

### Pruning primitive sequences but still attaining Pillai's lower bound on sum of reciprocals

Background: The answer to a previous question I asked here specified a construction to achieve Pillai's bound on reciprocal sums of primitive sequences. A primitive sequence $1<a_1<\ldots<a_k\...

**3**

votes

**3**answers

537 views

### How do estimates on $N_\chi(\alpha,T)$ lead to the Dirichlet prime number theorem for arithmetic sequences?

Let $ N_\chi(\alpha,T)$ be the number of zeros of $L(s=\sigma+it,\chi) = \sum \frac{\chi(n)}{n^s}$ where $c > 0$ and $(\sigma,t) $ are in the rectangle $ [\alpha,1] \times [-T,T]$.
In various ...

**14**

votes

**1**answer

569 views

### Prove $4\sum_{k=1}^{p-1}\frac{(-1)^k}{k^2}\equiv 3\sum_{k=1}^{p-1}\frac{1}{k^2}\pmod{p^2}$

Wolstenholme's theorem is stated as follows:
if $p>3$ is a prime, then
\begin{align*}
\sum_{k=1}^{p-1}\frac{1}{k}\equiv 0 \pmod{p^2},\\
\sum_{k=1}^{p-1}\frac{1}{k^2} \equiv 0 \pmod{p}.
\end{align*}
...

**1**

vote

**0**answers

141 views

### The existence of solution for special equation on integer ring

I have a question which belongs to the field of number theory. Can we prove or disprove the following claim:
For all prime number $p=24t+1$ and the natural number $n=6t+1$, there is at least, one ...

**0**

votes

**2**answers

453 views

### 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 ...

**8**

votes

**0**answers

125 views

### Equation which has nontrivial solutions modulo $N$ for every $N \ge 2$ does not have any nontrivial integer solutions

Let $\alpha = \sqrt[3]{2}$ and $K = \textbf{Q}(\alpha)$. I want to show that the equation$$\text{N}_\textbf{Q}^K\left(x + 4y + z\alpha + w\alpha^2\right) - 6(x + y)\left(x^2 + xy + 7y^2\right) = 0,$$...

**5**

votes

**1**answer

462 views

### $(n+1)!_\mathbb{P}$ and the Euler-Mascheroni constant

I'm studying the following limit
$$\lim_{n\to \infty} \frac{1}{n} \ln\left( \frac{(n+1)!_\mathbb{P}}{n^n}\right) $$
where $$(n+1)!_\mathbb{P} = \prod\limits_{p \in \mathbb{P}}^{}
{p}^{\omega_p{(n)}},...

**8**

votes

**0**answers

210 views

### Generating prime numbers

By a theorem of Mills, 1947, there is a real number $c$ such that for every $n$, $[c^{3^n}]$ is a prime number.
Is there a real number $d$ such that $[d^n]$ is prime, for every $n$ ?

**2**

votes

**0**answers

112 views

### Is there an odd number which has no prime to prime matchings when compared with its reverse order? [closed]

For example look at the number 9. It has prime-prime matching at 3,5, and 7.
For example the sequence of 13 has matchings at 1,3,7,11,13.
For example 15 has the matchings(crossings) at 3,5,11,13.
...

**1**

vote

**1**answer

182 views

### Least simultaneous quadratic non-residue

Suppose $p,q$ are distinct primes with least quadratic non-residues $n_p$ and $n_q$ respectively. Can one bound the least $n$ for which $\left(\frac{n}{p}\right)=\left(\frac{n}{q}\right)=-1$ in terms ...

**6**

votes

**2**answers

270 views

### Does the limit of this product over primes converge for all $\Re(s) > \frac12$?

Numerical evidence suggests that:
$$\displaystyle F(s):= \lim_{N \to \infty}\, \ln^s\left(p_N\right)\, \prod_{n=1}^N \left(\dfrac{\left(p_n-1\right)^s}{p_n^s-1} -\frac{1}{p_n^s}\right)$$
with $p_n$ ...

**0**

votes

**0**answers

81 views

### Probability distribution associated with total divisors of an integer

Is there a generalization to https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theorem which gives distribution function for $$\omega(n)=\big|\{d\in\mathsf{prime}:d|n\}\big|$$ where $\mathsf{prime}$...

**0**

votes

**1**answer

180 views

### When is $a^{2^n}+1$ prime finitely often unconditionally?

Define generalized Fermat numbers following OEIS and mathworld.
For natural $a,n$ and $a$ even, the generalized Fermat number (GFN) is
$F_n(a)=a^{2^n}+1$.
Very large GFN primes are known (in the ...

**0**

votes

**1**answer

129 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}\...

**4**

votes

**1**answer

205 views

### Goldbach for certain classes of $n$

Asked on MSE without response here.
$\#$ of ways even $n$ can be represented by prime additions is heareafter denoted $G(n)$.
The Wiki article on the Goldbach conjecture states that
In 1975, ...

**2**

votes

**1**answer

133 views

### Gradual monotonic morphing between two natural numbers

Let $a < b$ be two natural numbers. I will use these as an example:
\begin{align*}
a & = 2^5 \cdot 3^2 \cdot 5^2 = 7200\\\
b & = 2^3 \cdot 3^5 \cdot 7^1 = 13608
\end{align*}
I seek to "...

**0**

votes

**1**answer

49 views

### Function for number of integers having only prime factors >= x? [closed]

Is there a function for the number of integers below n having only prime factors greater than or equal to p? For example, how do i determine the number of integers below 1000 only having prime ...

**3**

votes

**2**answers

284 views

### Primes $P_{2n-1}$ that are $2$ mod $3$

Are infinitely many primes $P_{2n-1}$ expressible as $3k-1$?
The primes $P_{2n-1}$ are every other prime beginning with $2$: $2,5,11,17,23,31,\cdots$. The first few are of the form $3k-1$, but $31$ ...

**0**

votes

**0**answers

47 views

### Estimates related to sum over a primes from a fixed, possibly sparse set

Let $E$ be a fixed infinite sequence of primes such that $\sum_{p \in E} \frac{1}{p} = \infty$. Assume that $\sigma > 1$ depends on some parameter $x \rightarrow \infty$ in such a way that $\sigma ...

**1**

vote

**1**answer

196 views

### Radical of the sum $=$ radical of the product

My question is:
Has it been proved/disproved or studied the following?
For every $k\geq 4$ there are $k$ pairwise relatively prime numbers $a_1,a_2,...,a_k$ all greater than $1$ such that $$\...