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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.
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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 $s=\sigma+it$ $$ \zeta(s)=\... number-theory asymptotics analytic-number-theory riemann-zetaCan someone explain these groups of linear patterns in the dropping times of Collatz Sequences? Could this lead to a proof?
Please buckle in because this may be a long post, but I think it will be necessary to help the reader understand three things: How this data was generated. How the data is grouped into different '... number-theory modular-arithmetic dynamical-systems collatz-conjecture music-theory- 123
a digit Divisible Number
A Digit Divisible Number is a number that does not contain $0$ as a digit, and every consecutive sequence of its digits from right divides the number. For example, the number $55$ is a digit divisible ... number-theory- 3,220
Primality test for numbers of the form $\frac{a^p-1}{a-1}$?
This question is cross-linked with Mathoverflow and I didn't get an answer for the question. Here is what I observed : Inspired by Lucas-Lehmer primality test, I think I made a primality test for ... number-theory prime-numbers primality-test- 313
How to find the number of options for choosing numbers from $a_1, a_2, a_3, ... a_n$ such that their sum was equal to $k$
Let our numbers $2, 5, 6, 7, 10, 15$ and $k = 15$. I need to find the number of possible options for choosing numbers that form a total of 15. It's $(5, 10), (2, 7 ,6), (15)$. So the answer is 3. sequences-and-series combinatorics probability-theory number-theory discrete-mathematics- 1
A question on the Riemann zeta function
Question: Consider a $L$-shaped path $L_\epsilon:\frac{1}{2}+\epsilon\to \frac{1}{2}+\epsilon+i\ (H+\epsilon)\to \frac{1}{2}+i\ (H+\epsilon)$ where $H>0$ is fixed and $\epsilon>0$ is arbitrarily ... number-theory prime-numbers analytic-number-theory riemann-zeta l-functionsA new series for $\frac{1}{\pi}$
Let $C_n$ denote the $n$-th Catalan number defined by $${\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).}$$ Next, we define ... sequences-and-series number-theory generating-functions pi- 235
Bombieri-Vinogradov Theorem and weak version of Goldbach Conjecture
I have a question related to the Bombieri-Vinogradov Theorem and the Goldbach conjecture. Bombieri-Vinogradov theorem states that if $ A > 0 $ then for all $Q$ such that $ x^{1/2} (\log x )^{-A} \... number-theory analytic-number-theory- 181
Three-term arithmetic progressions ending with a given prime
A student has asked me whether the following is true: If p is prime and bigger than 37, there exists a three-term arithmetic progression of primes ending in p. For smaller odd primes one can find 3-... number-theory prime-numbers arithmetic-progressions- 89
Perfect square equation $12\alpha^2\cdot x^3+12\alpha\cdot x^2+12\alpha\left(1-\alpha\right)\cdot x+\left(2-3\alpha\right)^2$
Well, I have the following function: $$\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$$ Where $\alpha\in\mathbb{N}$. ... number-theory elementary-number-theory polynomials square-numbers- 27.6k
Show that there are infinitely many positive integers $n$ for which $\phi(n)^2 + n^2$ is a perfect square.
The problem at the title is one of my elementary number theory exercises. I attempted to find a specific form of $n$ which satisfies the condition. I tried $n = p$ (prime) and $n = p^k$ case, but ... number-theory elementary-number-theory totient-function- 372
Can I use a addition table with infinite length and height to define addition on the natural numbers rather than Peano's Axioms? [closed]
I'm reading David Stuart's "Foundations of Mathematics" and in chapter 8 he is building an axiomatic system for the natural numbers with addition defined using Peano's Axioms. I don't really ... number-theory arithmetic natural-numbers peano-axioms- 3
Multiplicative function $h$ and its convergence exercise.
There is an exercise in the book on page $106$ that given a multiplicative function $h$ satisfying the conditions ($p$ denotes a prime) $$|h(p)|\le 1 \text{ if } (p|N), |h(p)|\le p^{-\delta} \text{ if ... real-analysis number-theory elementary-number-theory analytic-number-theory- 771
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 ... number-theory galois-theory algebraic-number-theory local-field- 886
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-... number-theory diophantine-equations linear-diophantine-equations- 19
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