# Questions tagged [number-theory]

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 algebra geometry, exponential and character sums, Zeta and L-functions, multiplicative and additive number theory, etc.

35,601 questions
Filter by
Sorted by
Tagged with
646 views

### $\lim_\limits{x \to \infty} \frac1x \sum_\limits{n\leq x}\mu(n)=0 \iff$ Prime Number Theorem

I'm reading Analytic Number theory from Tom. M. Apostol's Introduction to Analytic Number Theory. In the fourth Chapter of the book he proves the equivalence of Prime number theorem with the ...
56 views

15 views

122 views

### Is the following definition of an elliptic curve correct?

Im new to algebraic geometry so I want to make sure im getting my definitions right. I know there are a few ways to state what an elliptic curve is (ex a smooth projective curve of genus one with ...
20 views

### What the probabilty of selecting different congruences of numbers?

I was reading through an article the other day regarding the probability of selecting relatively prime integers, which is known to be $\frac{6}{\pi^2}$. How would you compute the probability of ...
139 views

### Proof verification of non vanishing of $~L(1, \chi) \neq 0~$ for real valued character

I am self studying analytic number theory from Tom M Apostol introduction to analytic number theory and I am asking for solution verification for a part of Theorem 6.20 of Apostol. I am adding it's ...
102 views

### How to solve polynomial equations of more than one transcendental functions in closed form?

Let $f_1,f_2$ functions in $\mathbb{C}$ so that $f_1,f_2$ are algebraically independent, $P$ an irreducible bivariate polynomial function in $\mathbb{C}$ whose coefficients are algebraic numbers. Why ...
637 views

### How to prove $\sum_{ d \mid n} \mu(d)f(d)=\prod_{i=1}^r (1-f(p_i))$?

I have to prove for $n \in \mathbb{N}>1$ with $n=\prod \limits_{i=1}^r p_i^{e_i}$. $f$ is a multiplicative function with $f(1)=1$: $$\sum_{ d \mid n} \mu(d)f(d)=\prod_{i=1}^r (1-f(p_i))$$ How I ...
21 views

### What are some elementary bounds for Mertens' Third Theorem?

In particular, I am looking for $A$ and $B$ such that, for all $x>s$, $$\frac{e^{-\gamma}}{\log x} B< \prod_{ p \leq x} \frac{p-1}{p}<\frac{e^{-\gamma}}{\log x}A.$$ Rosser provides both ...
32 views

### Converting between different ways of representing integer partitions

I'm aware of two ways of representing integer partitions, and I'm trying to understand how to write the mapping between then in terms of mathematical notation. Firstly, we can say that an integer ...
58 views

### Is there a Prime number test that has no false positives

The primality tests that I have found, i.e. Fermat and Miller-Rabin don't return any false negatives, but do sometimes return false positives. That is, some composites are incorrectly decided as ...
39 views

18 views

### Criterion for an extension of complete fields to be unramified

Let $A$ be a complete $DVR$ with fraction field $K$. Let $L/K$ be a finite separable extension of $K$ and let $B$ be the integral closure of $A$ in $L$. Then $B$ is also a complete $DVR$ and let us ...
50 views

show that $2b^2+ 3$ cannot be a divisor of $a^2-2$ I have tried working with residue of $2$ modulo $p$ where $p$ is a divisor of $2b^2+3$ but couldn't stablish the result.
32 views

### Determine the remainder of p1 + p2 + p3 + … + p2021 when divided by 16 [closed]

For any natural number n, it is defined as the smallest prime factor of n^8 + 1. Determine the remainder of p1 + p2 + p3 + ... + p2021 when divided by 16. What should i do? I need help to work this ...
41 views

### Prime zeta function and squarefree divisors

Let $x_n$ and $y_n$ be two random numbers, drawn uniformly and independently from $\{1,\ldots,n\}$. Write $d_{\text{sf}}(x_n,y_n)$ for the number of squarefree divisors of $\gcd(x_n,y_n)$. We are ...
48 views

### Importance of Riemanns paper [closed]

Riemann found a connection between complex analysis and number theory and found an exact formula for $\pi(x)$. As we can see in this question we have a way easier formula for the prime counting ...
### Using Quadratic Residue to show that $2^m-1$ doesn't divide $3^n-1$
Show that if $m$ is congruent to $n$ modulo $2$ and $m>1$ then $2^m-1$ cannot be a divisor of $3^n-1$ What I have tried: I showed that if $p$ is a divisor of $2^m-1$ then if $m$ and $n$ are odd ...
How many boxes are crossed by a diagonal in a rectangular table formed by $199 * 991$ small squares ? Well, the answer is $a+b-\gcd(a, b)$. but, How can I prove that? I tried drawing different ...