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**6**

votes

**1**answer

458 views

### Are there effective small intervals in which primes are dense?

As mentioned in Terry Tao's comment to this question, it is constructively known
that there are primes between sufficiently large cubes. $\:$ According to wikipedia,
"there exists a constant $\: ...

**2**

votes

**1**answer

207 views

### Explicit bound on $\sum_{N\mathfrak p \leq x}\chi(\mathfrak p)\ln(N\mathfrak p)$

I'm looking for an explicit bound for $f(x) = \sum_{N\mathfrak p \leq x}\chi(\mathfrak p)\ln(N\mathfrak p)$, where $\chi$ is a Hecke character for a number field $K$ of degree $n$, on the ideals ...

**2**

votes

**2**answers

435 views

### explicit large gap for consecutive zeros of the Riemann zeta function

In Theorem 9.12, Titchmarsh (The Theory of the Riemann Zeta Function) proved that
For every large positive T, $\zeta(s)$ has a zero $\beta+i\gamma$ satisfying
$$
|\gamma-T|<\frac{A}{\log\log\log ...

**4**

votes

**1**answer

772 views

### effective/constructive/algorithmic probability theory

What sort of "alternative" probability theories are out there in which the methods of proof are inherently constructive?
I know of a number of theorems that say that if you take an infinite sequence ...

**9**

votes

**0**answers

368 views

### Effective lower bound for class numbers of cyclotomic fields

Let $K=\mathbb{Q}(\mu_p)$ with class number $h=h^+h^-$, where as usual $h^+$ is the class number of the maximal real subfield of $K$. My question is whether there is an effective lower bound for $h$ ...

**5**

votes

**2**answers

562 views

### Upper bounds on the difference of consecutive zeta zeros

There are many results on the spacing of the gaps between nontrivial zeros of the $\zeta$ function, from trivial (average value is $\frac{2\pi}{\log\gamma_n}$) to difficult (bounds on max and min ...

**5**

votes

**1**answer

373 views

### Determining the exceptional set in the theorem of Ax & Kochen

Ax & Kochen [1] proved that for every $d\in\mathbb{N}$ there exists a finite set $A(d)$ such that for every prime $p\not\in A(d),$ every homogeneous polynomial of degree $d$ over $\mathbb{Q}_p$ in ...

**10**

votes

**3**answers

781 views

### Density Ramsey theorems with explicit asymptotics

I wonder what interesting and non-trivial examples of density Ramsey theorems with explicit asymptotics are there?
I'm aware of two examples: Szemerédi's theorem and density Hales-Jewett theorem.
...

**11**

votes

**3**answers

834 views

### Bounds on squarefree numbers

Let $q_1,q_2,\ldots$ denote the squarefree integers 1, 2, 3, 5, .... What effective bounds are known for $q_n$? Clearly
$$q_n\sim\zeta(2)n$$
but I need hard inequalities. Of course from the above ...

**5**

votes

**1**answer

452 views

### Effective bounds on Euler's totient

Quick question: It's known that
$$\limsup\frac{n}{\varphi(n)\log\log n}=e^\gamma$$
but are there known C and N such that
$$\varphi(n)>\frac{Cn}{e^\gamma\log\log n}$$
for all $n>N$?
Failing ...

**7**

votes

**2**answers

674 views

### An effective way to tell if the saturation of a homogeneous ideal is the irrelevant ideal

Let $\Bbbk$ be an algebraically closed field, let $R$ denote the graded ring $\Bbbk[x_0, \dotsc, x_N]$, and let $f_1, \dotsc, f_n \in R_m$ be nonconstant homogeneous polynomials. Then the common ...

**12**

votes

**5**answers

1k views

### Advances and difficulties in effective version of Thue-Roth-Siegel Theorem

A fundamental result in Diophantine approximation, which was largely responsible for Klaus Roth being awarded the Fields Medal in 1958, is the following simple-to-state result:
If $\alpha$ is a real ...

**6**

votes

**2**answers

690 views

### Question related to Diophantine approximations and Roth's theorem

The following question came up in my arithmetic geometry course yesterday. Suppose $\alpha$ is an irrational real algebraic integer, and suppose $\epsilon >0$ is given. Then by Roth's theorem there ...

**13**

votes

**2**answers

2k views

### Effective Chebotarev Density

Let $K$ be a number field, and $p$ be a rational prime. Then the Chebotarev Density Theorem implies we can find primes $v$ and $w$ of $K$ of degree 1 which are split and nonsplit respectively in ...

**2**

votes

**0**answers

777 views

### Effective upper bound on large prime gaps; or, what is the first prime after a googolplex?

Question
What is the best known effective upper bound on the prime gap following x?
Motivation
Suppose you needed to show a good bound for the gap between a fixed large constant, say ...

**11**

votes

**0**answers

883 views

### Effective proofs of Siegel's theorem using arithmetic geometry

This is a speculation and perhaps naive. The theorem of Siegel that
There exist only finitely many integral points on a curve of genus $\geq 1$ over a number ring $\mathcal O_{K, S}$ where $S$ is ...

**10**

votes

**1**answer

944 views

### (Good) effective version of Kronecker's theorem?

Thm (Kronecker).- If all conjugates of an algebraic integer lie on the unit circle, then the integer is a root of unity.
Question: Can one provide a good effective version of this? That is: given ...