I'm studying theorem III.7.5 from Hartshorne's book. There are a few things I don't understand in this proof. Theorem. Let $X$ be a projective scheme over a field $k$. Then $X$ has a dualizing sheaf $\omega_X^\circ$. Proof. Let $\dim(X) = n$. Embed $X$ as a closed subscheme of $P:=\mathbb{P}^N$...

Let $p$ be an odd prime and let $n \ge 3$. I have been considering the following family of polynomials. Set $$ r_n:=\frac{1}{2\sqrt{n-1}}, $$ and define $$ B(p,n) := r_n+\frac{p+1}{r_n}+\frac{2p}{r_n^{n-1}}. $$ Let $$ A(p,n) := 2+2p\left( \left\lfloor \frac{B(p,n)-2}{2p} \right\rfloor+1 \right)....

Could someone please provide information about the best possible known bounds of the sum $$A(x)=\sum_{n\leq x}\frac{\mu(n)}{n}?$$ Unconditionally, $A(x)=O(e^{-c\sqrt{\log x}})$ is known to me whose reference I need. Does there exist any better bound unconditionally? Any help will be highly...

We consider the odd-step block transitions of the Collatz mapping by collapsing intermediate even divisions: $$3n_i + 1 = 2^{k_i} n_{i+1}, \quad \text{where } k_i = v_2(3n_i + 1)$$ We model potential upward expansion chains ($1^r$, where $k_i = 1$ for $r$ consecutive steps) followed by...

Does the proposition "if f(n)=an+b has two distinct primes f(c),f(d), with a>0 then it has infinitely many primes" imply Dirichlet's Theorem?

Let $n$ be a positive integer and let $p>3$ be a prime number such that $p\mid n^2+n+1$. Prove that $(n+1)^p\equiv n^p+1\pmod{p^3}$. Set $P(x)=\dfrac{(x+1)^p-x^p-1}{p}$. First use $p\mid n^2+n+1$ to show that $n^3\equiv1\pmod p$ and hence $p\equiv1\pmod3$. Now let $\omega$ be a primitive third...

I know of Conway's chained arrow notation. I have read the Wikipedia article on it, but it still didn't give me a rigorous proof that it exists and is unique. So, to make my question precise, suppose we are given a finite nonempty sequence $S$ of positive integers. How does one rigorously define...

I have been doing a computational/visual exploration of the Riemann zeta function and its completed xi function. I am a software developer and mathematics enthusiast rather than a professional mathematician, and my main aim here is to identify the established theory behind the following...

If $$S=\sum_{b\geq 2, \ \mu^2(b)=1} \frac{\mu(b)}{b^2}\sum_{c|b, c<\sqrt{b}}\frac{1}{\left(\frac{1}{c^2}+\frac{c^2}{b^2}\right)^{3/2}}$$ Prove that $S$ is conditionally convergent. Since $\mu(b)\neq0$ only for squarefree $b$, write $b=ck$, $c<k$, $(c,k)=1$. Then $ \mu(b)=\mu(ck)=\mu(c)\mu(k), $...

I came up with the following Diophantine equation: $$ \frac{z}{x+y}+\frac{y}{x+z} +\frac{z}{x-y}+\frac{y}{x-z}=9, $$ where $x,y,z\in\mathbb Z,$ $xyz\ne0,$ and $x+y\ne0,$ $x-y\ne0,$ $x+z\ne0,$ $x-z\ne0.$ Does this equation have a nonzero integer solution? If a solution exists, I would be...

Let $V$ be a finite dimensional complex vector space and $G$ be a finite subgroup of $G<\operatorname{GL}(V)$. My question is, essentially, when do the orbit space $V/G$ have finitely many symplectic leaves? My interest lies, specially, in the case when $V=h \oplus h^*$, and $G$ is given by a...

I developed this theory. Is it correct? What do you think? Formal definition of the space $L$ as a local ring: $$\mathcal{L} \cong \mathbb{C}[\varepsilon]/(\varepsilon^2)$$ Fundamental axioms of the basis units $\{1, c, b\}$: $$c^2 = -1, \quad b^2 = 0, \quad cb = 0$$ General representation of an...

I am investigating a novel arithmetic function $f: \mathbb{Z}^+ \to \{0, 1, \dots, 9\}$ that maps an integer to a terminal state on a $\mathbb{Z}/10\mathbb{Z}$ cycle based on its distinct prime signature. Definition Let $n \in \mathbb{Z}^+$ have the unique prime factorization $n = p_1^{a_1}...

I have constructed an arithmetic function that exhibits a chaotic behavior reminiscent of Collatz-like dynamical systems, but with a feedback loop driven by the partitions of parity-shifted divisor geometries. The system appears to inherently embed the Halting Problem within standard...

Definition 1: An element of $\operatorname{PSL}_2(q)$ is semisimple if it is diagonalisable in $\operatorname{PSL}_2(\overline{\Bbb F_q})$, where $\overline{\Bbb F_q}$ is the algebraic closure of $\Bbb F_q$. Definition 2: Let $q$ be a power of a prime. Then we say $n\in \Bbb N$ is $q$-good if:...

So I am studying the arithmetic derivative and I stumbled upon following approach: We are considering here a $\Bbb{Q}$-sub-vector-space of $\Bbb{R}$. Define the set $$\log(\Bbb{P}):=\{\log(p) : p\in\Bbb{P}\}$$ and the $\Bbb{Q}$-vector-space generated by $\log(\Bbb{P})$...

Let \chi be a Dirichlet character of prime order r, and define [ A_\chi(n)=\sum_{d\mid n}\chi(d). ] For a prime p, one trivially has [ A_\chi(p)=1+\chi(p). ] I have been looking at composites n that satisfy the same identity [ A_\chi(n)=1+\chi(n). \tag{1} ] I would like to know whether...

Let $E \subset \mathbb{R}^n$ be a Besicovitch (Kakeya) set, i.e., a compact set containing a unit line segment in every direction $e \in \mathbb{S}^{n-1}$. The Kakeya conjecture asserts that $\dim_{\text{H}}(E) = \dim_{\text{M}}(E) = n$ for all $n \ge 4$. Given the geometric obstructions in $n...

Let $f$ be a polynomial of degree $n$ over the finite field $\mathbb{F}_p$. We aim to find its Galois group. If $f$ is irreducible and separable, then $\mathbb{F}_{p^n}$ is its splitting field, and its Galois group is $Z/nZ$, which is generated by $\sigma:x\mapsto x^p$. Now, suppose we do not...

I want to find the Galois group $G$ of $f=x^5+2$ over $\mathbb{Q}$. I will write my approach, and I would like to know if there is a faster approach or a more standard one that does not require creativity. Let $\alpha$ be such that $\alpha^5=-2$ and $\zeta$ be the fifth root of unity. Then, the...