Ask Question

Questions on advanced topics - beyond those in typical introductory courses: higher degree algebraic number and function fields, Diophantine equations, geometry of numbers / lattices, quadratic forms, discontinuous groups and and automorphic forms, Diophantine approximation, transcendental numbers, elliptic curves and arithmetic algebraic geometry, exponential and character sums, Zeta and L-functions, multiplicative and additive number theory, etc.

38,413 questions

-1votes0answers37views

Asymptotic formula for logarithmic derivate of zeta function

My question is: find asymptotic formula for the following $$ \frac{\zeta'(s)}{\zeta(s)}$$ where $\frac{1}{2}\leq \sigma\leq 2$, $\zeta$ denotes the Riemann zeta function and $s=\sigma+it$ $$ \zeta(s)=\... user avatar user1067422 0 votes 0 answers 67 views

Can someone explain these groups of linear patterns in the dropping times of Collatz Sequences? Could this lead to a proof?

Please buckle in because this may be a long post, but I think it will be necessary to help the reader understand three things: How this data was generated. How the data is grouped into different '... user avatar dthomas

  • 123
2votes1answer42views

a digit Divisible Number

A Digit Divisible Number is a number that does not contain $0$ as a digit, and every consecutive sequence of its digits from right divides the number. For example, the number $55$ is a digit divisible ... user avatar Ahmad

  • 3,220
1vote0answers54views

Primality test for numbers of the form $\frac{a^p-1}{a-1}$?

This question is cross-linked with Mathoverflow and I didn't get an answer for the question. Here is what I observed : Inspired by Lucas-Lehmer primality test, I think I made a primality test for ... user avatar kijinSeija

  • 313
0votes1answer38views

How to find the number of options for choosing numbers from $a_1, a_2, a_3, ... a_n$ such that their sum was equal to $k$

Let our numbers $2, 5, 6, 7, 10, 15$ and $k = 15$. I need to find the number of possible options for choosing numbers that form a total of 15. It's $(5, 10), (2, 7 ,6), (15)$. So the answer is 3. user avatar Kachunskyy Igor

  • 1
0votes1answer77views

A question on the Riemann zeta function

Question: Consider a $L$-shaped path $L_\epsilon:\frac{1}{2}+\epsilon\to \frac{1}{2}+\epsilon+i\ (H+\epsilon)\to \frac{1}{2}+i\ (H+\epsilon)$ where $H>0$ is fixed and $\epsilon>0$ is arbitrarily ... user avatar user1067130 8 votes 1 answer 138 views

A new series for $\frac{1}{\pi}$

Let $C_n$ denote the $n$-th Catalan number defined by $${\displaystyle C_{n}={\frac {1}{n+1}}{2n \choose n}=\prod \limits _{k=2}^{n}{\frac {n+k}{k}}\quad \left(n\geqslant 0\right).}$$ Next, we define ... user avatar Permutator

  • 235
1vote0answers27views

Bombieri-Vinogradov Theorem and weak version of Goldbach Conjecture

I have a question related to the Bombieri-Vinogradov Theorem and the Goldbach conjecture. Bombieri-Vinogradov theorem states that if $ A > 0 $ then for all $Q$ such that $ x^{1/2} (\log x )^{-A} \... user avatar 3m0o

  • 181
4votes0answers61views

Three-term arithmetic progressions ending with a given prime

A student has asked me whether the following is true: If p is prime and bigger than 37, there exists a three-term arithmetic progression of primes ending in p. For smaller odd primes one can find 3-... user avatar Fernando

  • 89
0votes0answers90views

Perfect square equation $12\alpha^2\cdot x^3+12\alpha\cdot x^2+12\alpha\left(1-\alpha\right)\cdot x+\left(2-3\alpha\right)^2$

Well, I have the following function: $$\text{y}\left(x\right):=12\alpha^2\cdot x^3+12\alpha\cdot x^2+12\alpha\left(1-\alpha\right)\cdot x+\left(2-3\alpha\right)^2\tag1$$ Where $\alpha\in\mathbb{N}$. ... user avatar Jan Eerland

  • 27.6k
2votes1answer49views

Show that there are infinitely many positive integers $n$ for which $\phi(n)^2 + n^2$ is a perfect square.

The problem at the title is one of my elementary number theory exercises. I attempted to find a specific form of $n$ which satisfies the condition. I tried $n = p$ (prime) and $n = p^k$ case, but ... user avatar mathhello

  • 372
0votes1answer30views

Can I use a addition table with infinite length and height to define addition on the natural numbers rather than Peano's Axioms? [closed]

I'm reading David Stuart's "Foundations of Mathematics" and in chapter 8 he is building an axiomatic system for the natural numbers with addition defined using Peano's Axioms. I don't really ... user avatar drd26

  • 3
0votes0answers26views

Multiplicative function $h$ and its convergence exercise.

There is an exercise in the book on page $106$ that given a multiplicative function $h$ satisfying the conditions ($p$ denotes a prime) $$|h(p)|\le 1 \text{ if } (p|N), |h(p)|\le p^{-\delta} \text{ if ... user avatar W. Wongcharoenbhorn

  • 771
1vote1answer50views

Describing Galois groups of some local fields

We can describe the Galois group of some global fields explicitly, for example, we can describe the Galois group of splitting field of $x^n-a$ over the rationals explicitly, especially the cyclotomic ... user avatar Tireless and hardworking

  • 886
1vote1answer76views

Given $n$ integers $a_1$ to $a_n$ and an integer $K$, does there exist a solution which satisfies the following equation?

Given $n$ integers $a_1$ to $a_n$ and an integer $K$, does there exist a solution which satisfies the following equation? $$\sum_{i=1}^n a_i\cdot x_i = K $$ Note that all $x_i$ must need to be NON-... user avatar lina soares

  • 19

153050per page

1 2 3 4 5 … 2561