Intereting Posts

Solve $3^y=y^3,\space y\neq1,\space y\neq3$
The Meaning of the Fundamental Theorem of Calculus
If $f: M\to M$ an isometry, is $f$ bijective?
Regular monomorphisms of commutative rings
Negative Exponents in Binomial Theorem
How to prove that $a^2b+b^2c+c^2a \leqslant 3$, where $a,b,c >0$, and $a^ab^bc^c=1$
Classification Theorem for Non-Compact 2-Manifolds? 2-Manifolds With Boundary?
Is this an outer measure, if so can someone explain the motivation
Determining axis of rotation from the rotation matrix without using eigenvalues and eigenvectors
Primes in Gaussian Integers
probability of getting 50 heads from tossing a coin 100 times
Integral involving Dirac delta: two different results?
What does P in blackboard bold type of letter stand for? ℙ?
Is there another simple way to solve this integral $I=\int\frac{\sin{x}}{\sin{x}+\cos{x}}dx$?
Curvature of planar implicit curves

I have a set S of prime numbers and I would like to find the size (in some sense, ideally some nice asymptotic expression) of the set of positive integers which are the product of with all prime divisors in S. (That is, for each prime $p$ dividing such a number, $p\in S$.)

What are good methods for going about this? The trivial cases are when $S$ is finite or cofinite (in the set $\mathcal{P}$ of primes):

- If $S$ is finite a Mertens-like product would give a relative density.
- If $\mathcal{P}\setminus S$ is finite

Some particular cases of interest:

- A conditional asymptotic for $\sum_{\text{$p,p+2$ twin primes}}p^{\alpha}$, when $\alpha>-1$
- brun's method and primitive roots
- How to derive an identity between summations of totient and Möbius functions
- Volume of first cohomology of arithmetic complex
- Bounding the integral $\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}}$
- Are these zeros equal to the imaginary parts of the Riemann zeta zeros?

- $S$ is the set of primes in a finite collection of arithmetic progressions: $S=\mathcal{P}\cap\left(\bigcup_{i=1}^k(a_i+b_in)_{n\in\mathbb{N}}\right)$.
- $S$ is not known, but $s_n$, the n-th term of $S,$ obeys $f(n)\le s_n\le g(n)$ for sufficiently nice function $f,g.$ Example: $n^3\le s_n\le n^4$ for $n>100.$

These include very nice problems like “how common are numbers which are the sum of two squares” and “how dense are abundant numbers”.

An asymptotic would be ideal, but at this point I’ll take what I can get.

- Generate a polynomial w/ integer coefficients whose roots are rational values of sine/cosine?
- Does this trigonometric pattern continue for all primes $p=6m+1$?
- Count ways to take pots
- Probability that a natural number is a sum of two squares?
- Show that $\left| \sqrt2-\frac{h}{k} \right| \geq \frac{1}{4k^2},$ for any $k \in \mathbb{N}$ and $h \in \mathbb{Z}$.
- Are primes randomly distributed?
- Largest Triangular Number less than a Given Natural Number
- If p $\equiv$ 3 (mod 4) with p prime, prove -1 is a non-quadratic residue modulo p.
- Is it possible to get arbitrarily near any acute angle with Pythagorean triangles?
- If $d>1$ is a squarefree integer, show that $x^2 - dy^2 = c$ gives some bounds in terms of a fundamental solution.

- Hensel's Lemma and Implicit Function Theorem
- $(g\circ f)^{-1}=f^{-1}\circ g^{-1}$
- What is the origin of the expression “Yoneda Lemma”?
- Number of Idempotent matrices.
- Find expected number of successful trail in $N$ times
- In $\sf{ZF}$, is every finite set also hereditarily finite?
- What's so special about sine? (Concerning $y'' = -y$)
- Combinatorial argument for $1+\sum_{r=1}^{r=n} r\cdot r! = (n+1)!$
- Are there real-life relations which are symmetric and reflexive but not transitive?
- inclusion-exclusion problem
- Why is the universal quantifier $\forall x \in A : P(x)$ defined as $\forall x (x \in A \implies P(x))$ using an implication?
- How to find a closed form solution to a recurrence of the following form?
- If $M$ and $N$ are graded modules, what is the graded structure on $\operatorname{Hom}(M,N)$?
- Proof of “the continuous image of a connected set is connected”
- Show that $\sum\limits_{n=1}^{\infty}\frac{x}{1+n^2x^2}$ is not uniformly convergent in $$