**5**

votes

**0**answers

171 views

### The zeta function and classical mechanics

In this paper, Guilherme França and André LeClair show that $$\gamma_{y}\sim 2 \pi \left(y-11/8\right)/W\left((y-11/8)e^{-1}\right)$$ where $W$ is the Lambert W function, and $\gamma_{y}$ is the ...

**2**

votes

**0**answers

56 views

### Effective version of the Bombieri-Vinogradov theorem

Is there an effective version of the Bombieri-Vinogradov Theorem, in that have bounds on the implied constant been found?

**4**

votes

**0**answers

106 views

### Is there a simple proof that Milnor $K_2$ of a number field is torsion?

This is a theorem of Garland. I had a look at the original paper which looks pretty complicated. I was wondering if the proof has been simplified over the years or if a different approach is nowadays ...

**1**

vote

**0**answers

96 views

### Can the Gaussian integers be covered by restricted recurrences?

Relaxation of the second question here.
Let $a(n)$ be recurrence of the form $a(n)=f(n,a(n-1)\ldots(a(n-k))$
with fixed initial terms.
(Observe that it might depend on $n$).
$f$ may contain ...

**-8**

votes

**0**answers

74 views

### The Riemann Zeta Function Works [on hold]

1 - Any counterexamples known for the Riemann Zeta Function?
2 - How to generalize the following?
Here we have the visualization of the Riemann Zeta Function 3D Plot and the plane. We can observe ...

**1**

vote

**0**answers

145 views

### What would be the consequences of $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log k}$?

The question is in the title: what would be the number theoretic consequences if we managed to establish the conjectured asymptotic equality $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log ...

**3**

votes

**1**answer

184 views

### Question on effective Mordell conjecture

Suppose $F(x,y,z)$ is a homogeneous polynomial over $\mathbb{Q}$, where $C:F(x,y,z)=0$ is a curve of genus $g\geq 2$.
Question: Faltings proved that $C$ has finite many rational points. Suppose that ...

**5**

votes

**1**answer

202 views

### Polynomial recurrence relation covering the integers (and then Gaussian integers)

Say that a polynomial recurrence relation (my terminology)
for $f_i$ is:
$k$ initial conditions setting $f_1,\ldots,f_k$ to integers ($\in \mathbb{Z})$.
A recurrence equation of the form $f_i =$ a ...

**3**

votes

**0**answers

58 views

### Number of orbits of $\mathrm{SL}_2(\mathcal O_A)$ on $\mathbf P^1(A)$ when $A$ is a quaternion algebra

This is a reference request.
Let $A$ be an anisotropic quaternion algebra over $\mathbf Q$. Let $\mathcal O_A$ be a maximal order in $A$. Then $\mathrm{SL}_2(A)$ acts transitively on the right on ...

**4**

votes

**1**answer

107 views

### A lower bound on the $L^2$ norm of a Dirichlet polynomial

The Question. Suppose $0 < \alpha < \beta$ are fixed, and $a_n$ is an arbitrary sequence of real numbers. Is it known how to bound from below
\begin{equation*}
\int_0^{T} \Big| \sum_{\alpha T ...

**5**

votes

**2**answers

247 views

### $j$-invariants of elliptic curves over finite fields

Let $K$ be a finite field, and $\overline{K}$ its algebraic closure. It is well known that two curves are isomorphic over $\overline{K}$ if and only if they have the same $j$-invariant. If two such ...

**12**

votes

**2**answers

1k views

### Is every number the sum of two cubes modulo p where p is a prime not equal to 7?

If p is a prime other than 7, can every integer be written as sum of two cubes modulo p?
Has Waring's problem mod p for cubes been proved simply and directly?
Thanks for your proof.
Lemi

**0**

votes

**0**answers

50 views

### 2x3 = 5+1 AND 2+3 = 5x1. How many other examples of this type? [migrated]

I noticed the following:
2x3 = 5+1.
If you switch the operators, it is still true:
2+3 = 5*1.
There is another obvious/trivial example where you can swap the operators:
2x2 = 2+2.
I think these ...

**1**

vote

**1**answer

56 views

### Does restriction to an open subgroup preserve projective smooth representations?

Let $G$ be a locally profinite group and $K \le G$ an open subgroup. Does the restriction functor $\mathrm{Res}^G_K$ from the category of smooth $\mathbb{C}$-linear representations of $G$ to smooth ...

**2**

votes

**1**answer

91 views

### Compact induction as a tensor product

Let $G$ be a locally profinite (i.e., locally compact Hausdorff and totally disconnected) topological group, $H \le G$ a closed subgroup, and $(W, \sigma)$ a representation of $H$ over $\mathbb{C}$ ...

**-5**

votes

**0**answers

124 views

**1**

vote

**1**answer

115 views

### trigonometric sum and inequalities

let $x\in\mathbb{R}-\mathbb{Z}$ and $e(x)=e^{2\pi ix}$. If we have this sum $$\left|\overset{q}{\underset{h=1}{\sum}^{*}}e\left(h\, x\right)\underset{\underset{p\equiv h\,\textrm{mod}\, q}{p\leq ...

**0**

votes

**0**answers

48 views

### Studies of Specific Kinds of Beurling Primes?

I know that Beurling developed a notion of generalized primes (and integers.
However, does anyone know if Beurling, or anyone else, studied subclasses of the broader class of Beurling primes that ...

**3**

votes

**1**answer

91 views

### The sixth power integral moment of automorphic L-function attached to Maass Forms

It is known that the sixth power integral moment of automorphic L-function attached to Cusp Forms has been proved by M. Jutila, that is $\int_{0}^{T}|L(1/2+it,f)|^{6}dt \ll T^{2+\varepsilon}$.
And ...

**0**

votes

**0**answers

96 views

### A question on the Euclidean domain $\mathbb{Z}[\omega]$ [on hold]

Let $\omega=\frac{-1+i\sqrt{3}}{2}=e^{\frac{2 \pi i}{3}}$ be a complex cube root of unity, and $\mathbb{Z}[\omega]$ the Euclidean domain. In view of that $\int_0^\infty e^{ix} ...

**1**

vote

**0**answers

102 views

### Equations over $\mathbb{Z}[[T]]$ vs. equations over $\mathbb{Z}_p$

This question might be deemed totally unanswerable, unless there is an obvious counterexample. Answers to either effect would be welcome.
Question. Let $X$ be a finite-type scheme over ...

**5**

votes

**1**answer

250 views

### Parity of primes [duplicate]

While working on a completely different (combinatorial) problem, I ran a simple program to calculate the parity of the first ~50000 primes (number of 1s in their binary representation modulo 2). The ...

**2**

votes

**0**answers

130 views

### What are the minimal degrees of the real and imaginary part of an algebraic complex number? [on hold]

Let $z=a+bi\in\mathbb C$ with $b\ne0$ be an algebraic complex number of minimal degree $n$. It is obvious that $a=\dfrac {z+\bar{z}}2$ and $b=\dfrac {z-\bar{z}}{2i}$ are also algebraic. For $n=3$, it ...

**3**

votes

**1**answer

117 views

### When are all sums of the elements of a set different?

Consider a set $S = \{x_1, \dots, x_n\} \subset \mathbb{Q}\setminus\{0\}$ and assume that for any $I, J \subset [n]$ with $I \neq J$ we have that
\begin{equation}
\sum_{i \in I} x_i \neq \sum_{j \in ...

**5**

votes

**1**answer

227 views

### What does the sum of the reciprocals of all the highly composite numbers converge to?

I've calculated the sum of the reciprocals of all the $156$ first highly composite numbers up to $10^{18}$:
$\sum \dfrac{1}{HCC(n)} = \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{6} + \dfrac{1}{12} + ...

**-3**

votes

**0**answers

48 views

### Proving how many divisors of a prime factorization (including 1 and n) there are [on hold]

I'm trying to figure out this problem but I'm not sure where to start. Could anyone explain to me the question a bit more in depth or give a few hints?
The problem is,
Let n in Z+ with prime ...

**3**

votes

**2**answers

315 views

### Intuition behind $\zeta(2) = \frac{\pi^2}{6}$ [on hold]

In literature, there are many proofs of the well-known result $$\zeta(2) = \frac{\pi^2}{6}.$$
However, as far as I know, they do not offer an intuitive explanation of why this result should be true. ...

**9**

votes

**2**answers

322 views

### Splitting integers 1, 2, 3, … n to avoid least possible sum

For each positive integer n, partition the integers 1, 2, 3, … 2n into two sets of n integers each. Let g(n) be the least integer such that there is such a partition in neither of whose parts there is ...

**1**

vote

**1**answer

80 views

### On the Saito Kurokawa representation

I know Saito-Kurokawa(SK) representation is the famous non-tempered representation of $SO(5)$. But since the tempered or non-tempered terms are concerned with local phenomenon, I am wondering that ...

**3**

votes

**1**answer

153 views

### A number array related to colored necklaces and the primes

I stumbled upon entry OEIS-A208535 on the enumeration of certain kinds of colored necklaces and noticed that the integers for the odd prime rows of the table there seem to be given by the Moreau ...

**2**

votes

**0**answers

93 views

### Invariant Theory over finite adeles

Classical invariant theory, among the other things, classifies polynomial functions over a vector space $V$ endowed with a quadratic form $Q$ which are invariant under the action of $SO(V,Q)$.
I am ...

**8**

votes

**1**answer

177 views

### A curious Gauss-Sum type identity

Let $q=e^{2\pi i/m}$, $a\in\mathbb{R}$ and $1\leq j\leq m-1$. I would like to prove that:
$$(a-1)\sum_{n=0}^{m-1} q^n\frac{\prod_{k=0}^{j-2} (q^{n+k+1}-a)}{\prod_{k=0}^{j} (aq^{n+k}-1)}=0.$$
For ...

**3**

votes

**0**answers

144 views

### Monte Carlo variant of Hilbert's Tenth Problem

Let $k \in \mathbb{N}$. Given an algorithm $\mathcal{A}$ which takes
as argument a polynomial $P \in \mathbb{Z}[x_1,\dots,x_k]$ and either
returns true or false, we say that $\mathcal{A}$ works for ...

**-3**

votes

**0**answers

272 views

### College - Ecuation to solve in 3 variables (p,q,r) in [1,2] [closed]

This is the image containing the ecuation

**0**

votes

**0**answers

152 views

### Mellin transform on $\mathbb{Z}[\omega]$ [closed]

I'm eager to ensure some facts which are elementary for many experts here.
Let $\omega=\frac{-1+i\sqrt{3}}{2}$ be a complex cube root of unity. The Eisenstein integers $\mathbb{Z}[\omega]$ (a unique ...

**2**

votes

**2**answers

132 views

### Projection formula for smooth representations of locally profinite groups

Let $G$ be a locally profinite (i.e., locally compact, Hausdorff, and totally disconnected) topological group, $H \le G$ a closed subgroup, $(\pi, V)$ a smooth representation of $G$, and $(\sigma, W)$ ...

**3**

votes

**1**answer

91 views

### Lower bound of first moment of $L$-function on $\mathrm{GL}(3)$

Let $\pi$ be an automorphic form on $\mathrm{GL}(3,\mathbb{A}_{\mathbb{Q}})$.
Do we know any case that
\[\int_0^{T} \left|L(\frac{1}{2} + it, \pi)\right| dt \gg T\]
holds unconditionally?
I know the ...

**7**

votes

**0**answers

159 views

### On the ratio of Gilbreath sequences

Definitions: let $n \in \mathbb{N}_{>0} \cup \{ \infty \}$ and let $E_n$ be the set of sequences $(d_i)_{i=1}^n$ such that $d_1=1$, $d_i$ is an even integer (for $i > 1$) and $0<d_i \le i$. ...

**1**

vote

**0**answers

118 views

### is there an analogy between fractals and automorphic forms? [closed]

Disclaimer: this question is rather vague and thus might not be suitable for this site. Still, as I already asked a similar question on MSE and got no feedback, I finally decided to take the plunge ...

**-4**

votes

**0**answers

84 views

### an question about number theory [closed]

Let $s_i=\frac{(q^n-1)...(q^n-q^{i-1})}{(q^{i-1})...(q^i-q^{i-1})}$, where $q$ is prime and $n$ is a positive integer.
Now can anyone tell me this, $\lim_{n\mapsto \infty}\frac{\sum_{1\leq i\leq ...

**1**

vote

**0**answers

87 views

### Clarifications on twisted forms

Suppose $F = F(\bar{k})$ is a finite algebraic group over a number field $k$. The absolute Galois group $\Gamma_k$ of $k$ acts on $F$ by group automorphisms via a homomorphism $\rho: \Gamma_k \to ...

**7**

votes

**0**answers

374 views

### Number Theory over $\mathbb{F}_q [t]$, why is it important/interesting?

This is something I was curious about, but not really sure why. I was hoping maybe someone could give me a good explanation. I understand that there are
similarities between $\mathbb{Z}$ and ...

**3**

votes

**2**answers

187 views

### Inequality for an integral involving $ \exp $, $ \sin $ and $ \cos $

Let $ t > 0 $ and $ k \in \{ 0,1,2,\ldots \} $. Does the following inequality hold?
$$
\int_{k + 1/2}^{k + 3/2}
\frac{x \sin(2 \pi x)}{1 + 2 e^{2 \pi t} \cos(2 \pi x) + e^{4 \pi t}}
...

**3**

votes

**1**answer

291 views

### A good book on adeles and ideles

Many results in number theory are stated either in a classical language or in an adelic one. I am often impressed of the efficiency and the satisfactory computational properties of the adelic setting, ...

**0**

votes

**2**answers

155 views

### Equidistribution of rational points on an algebraic variety

Suppose that we have a variety $X \subset \mathbb{P}^{n}$ defined over $\mathbb{Q}$. Suppose we have $S$ many rational points on $X$ inside the box defined by $|x_i| \leq B_i$ for $i = 0, \cdots, n$, ...

**2**

votes

**0**answers

131 views

### Tate's conjecture and symmetry of Hodge-Tate weights

I'm reading Bellaiche's notes on the Block-Kato conjecture (Hawaii summer school). Here is the link http://people.brandeis.edu/~jbellaic/BKHawaii5.pdf
At page 10 he claims that an indirect ...

**5**

votes

**0**answers

80 views

### The topology on the Robba ring

I've been reading Kedlaya's paper http://arxiv.org/abs/math/0208027 on finiteness of rigid cohomology and there's something I can't quite resolve in my understanding of the topology on the Robba ring.
...

**1**

vote

**1**answer

180 views

### Is the Cassels-Tate pairing defined for elliptic curves over function fields?

The Cassels-Tate pairing is typically defined for elliptic curves (or abelian varieties) over number fields, but it seems like it should be defined for elliptic curves over function fields as well. ...

**7**

votes

**1**answer

357 views

### What is the decomposition group at $p$ in the Galois group unramified outside $\ell?$

Let $p\ne\ell$ be two prime numbers, and let $K_{\ell}$ be the maximal extension of $\mathbb Q$ ramified only at $\ell$ and $\infty$ (i.e. it is the fixed subfield of $\overline{\mathbb Q}$ by all the ...

**0**

votes

**0**answers

59 views

### What is the proper Zariski-closed subset in these examples for Vojta's more general abc conjecture?

In A more general abc conjecture, p. 7 Paul Vojta conjectures:
If $x_0,\ldots x_{n-1}$ are nonzero coprime integers satisfing $x_0 + \cdots x_{n-1}=0$
$$ \max\{|x_0|,\ldots |x_{n-1}|\} \le C ...