According to the book Number Theory II: Iwasawa Theory and Automorphic Forms by Nobushige Kurokawa, Masato Kurihara and Takeshi Saito, one way to prove the functional equation of the Dedekind eta function $\eta(-\frac{1}{z})=\sqrt{-iz}\eta(z)$ is as follows: Consider this :...

Following is an attempt at a proof (as an exercise) that the average of primes on an interval $[1,n]$ is asymptotically equal to the midpoint of the interval. While I think the idea is correct (see data below), there may be a better way, and notational improvements welcome. To show:...

Let $a, b$ be positive integers. In the Cartesian coordinate plane, consider the rectangular region $S$ (including the boundary) enclosed by the points $(1,1)$, $(1,b)$, $(a,1)$, and $(a,b)$. The line $$ l: (2a+1)y - (2b+1)x = 0 $$ divides $S$. Let $f(a,b)$ be the absolute value of the...

Let $\lambda(n) = (-1)^{\Omega(n)}$ be the Liouville function and $$ C(x) = \sum_{n \le x} \frac{\lambda(n)}{n}. $$ By Turán's classical approach, $C(x)$ is controlled by $$ \sum_{n=1}^\infty \frac{\lambda(n)}{n^{s+1}} = \frac{\zeta(2s+2)}{\zeta(s+1)}, $$ and each nontrivial zero $\rho = \beta +...

This question was asked in my assignment and I am stuck on it. Question:If $A$ is infinite integral domain , $F,G\in A[X_i|i\in I]$, $F\neq 0$ and if $A^I \setminus V_A(F)\subseteq V_A(G)$,then show that $G=0$. Attempt: Let $D(F)= A^I \setminus V_A(F)=${$a\in A^I : F(a)\neq0$} . Given...

I realize this may be an unusual request, but I am trying to find out whether this style of studying mathematics already exists, or whether there are resources that come close to it. I am not looking for a standard textbook on Galois theory, nor for a historical overview followed by the modern...

My name is Kadin Shah, and I am a rising senior at Arizona State University studying Applied Mathematics with an emphasis in mechanical engineering. I'm interested in using creative problem-solving solutions in a collaborative setting with engineers at a large company such as a National...

I am doing my MSc project on "p-adic numbers". In my first chapter of the project, I want to include the construction of $\mathbb{Z}_p$ with both algebraic and analytic approach. But I'm not able to find any detailed explanation of analytic approach i.e.,...

I extend $\mathbb Q$ to first adjoin the roots of $x^2-1=9$ and then the roots of $(x^2-1)^2-1=9$, i.e. $x^4-2x^2-9=0$. In the second extension, I have to determine whether the prime ideal $(3)$ ramifies. I use the Newton Polygon to only deduce that $1$ determines the ramification index, which...

I had currently passed 10th grade. I want to go in pure mathematics and research and publish paper in that field (especially Number Theory) please tell me roadmap and the books to follow one after one. Please Guide me.

For even number $n$, if $n$ is not a totient (i.e. not in the range of Euler totient function), then $n$ is not in OEIS sequence A252502, but if $n$ is a totient, must be $n$ in A252502? For odd number $n$, if $n-1$ is not a totient (i.e. not in the range of Euler totient function), then $n$ is...

Erdős Problem 252:(https://www.erdosproblems.com/252) Let $k\geqslant 1$ and $\sigma_{k}(n):= \sum_{d|n} d^{k}$. Is $\displaystyle \sum_{n=1}^{\infty} \frac{\sigma_{k}(n)}{n!}$ irrational? Currently this problem is open for $k \geqslant 5$. I am considering a different version of this problem...

I know that Schur and Murty proved that an Euclidean proof (hence a "non-analytic" proof) for the existence of infinite primes $p \equiv \ell \mod q$ with $q$ and $\ell$ coprime can be given if and only if $\ell^2 \equiv 1 \mod q$. Are there any cases where $\ell^2 \not\equiv 1 \mod q$, but we...

In this related question, @Gerry Myerson asked if the conjecture fails for composites. Upon examining for composites, I found a pattern for module $n^2$ instead of $n^3$ in the linked original question. I have verified these conjectures for $n \le 5 \times 10^6$. Can they be proved or disproved....

Let $F_1,F_2$ be degree $1$ and $F_3$ be degree $2$ in $\mathbb Z[x_1,\dots,x_4]$. Let there be an unique common integer to $F_i$. Let them be algebraically independent of a fourth polynomial $G$ which also has the same common integer root. Is it possible for the system to not form a complete...

I had posted this question in MO that has remained unanswered in MO for more than two years now. While working on it, I accidently found an unexpected result. Let $0<x\leq y\leq z$ be the ordered side lengths of the triangle determined by three independent uniformly distributed points on a...

It is a classical result that the roots of a general cubic polynomial $x^3 + ax^2 + bx + c = 0$ can be expressed via Cardano’s formula using radicals of the form: $$x=\sqrt[3]u+\sqrt[3]v+k$$ where $u$ and $v$ depend on the coefficients and the discriminant $\Delta$. ​I am curious about the...

My experimental data suggests that the following binomial sum congruence involving primes and Fibonacci numbers hold. I have experimentally verified them for all primes $\le 1.5 \times 10^6$. Can they be proved or disproved? Conjecture 1. Let $p$ be a prime, $ p\equiv 1 \text{ or } 19 \pmod{30}....

I. Question Consider the following independent and symmetric mixed equations, $\quad x_1^8+x_2^8+x_3^4 = y_1^8+y_2^8+y_3^4$ $\quad x_1^8+x_2^8+x_3^4+x_4^4 = y_1^8+y_2^8+y_3^4+y_4^4$ $\quad x_1^8+x_2^8+x_3^8+x_4^4 = y_1^8+y_2^8+y_3^8+y_4^4$ $\quad x_1^8+x_2^8+x_3^8+x_4^8+x_5^4 =...