**-5**

votes

**0**answers

34 views

### basic : modulo and division [on hold]

how can one prove that a mod b < a/2?
I understand why is that happening:
if a mod b > a/2 that means that ...

**0**

votes

**0**answers

105 views

### Canonical identification between 3-manifolds cohomology and group cohomology [on hold]

I am trying to understand why this 3-manifold cohomology is equal to this group-cohomology.
$$ H_\ast (\mathbb{H}^3/PGL_2(\mathbb{Z})) \simeq H_\ast (PGL_2(\mathbb{Z}))$$
In both cases, use the base ...

**2**

votes

**0**answers

37 views

### Tensor product of two elements of the Selberg class

Maybe too easy a question for most members of this site, but suppose whenever $F$ and $G$ belong to the Selberg class, then so does $F\otimes G$ where the considered tensor product of $F$ and $G$ is ...

**0**

votes

**1**answer

87 views

### A conjectural convergence condition for a weakened Elliott-Halberstam conjecture

For $a$ and $q$ positive integers such that $a\lt q$ and $(a,q)=1$, let $\pi(x;q,a)$ be the number of primes $p\equiv a\pmod q$ below $x$. One can show that $\pi(x;q,a)\sim \dfrac{\pi(x)}{\varphi(q)}$ ...

**8**

votes

**0**answers

117 views

### Why $\gcd \{ {\rm ord}_p(q)\colon q\mid p-1 \}$ likes to be large?

For a prime $p$, let $F_p$ denote the greatest common divisor of the orders modulo $p$ of all prime divisors of $p-1$:
$$ F_p = \gcd \{ {\rm ord}_p(q)\colon q\mid p-1 \}; $$
thus, for instance, ...

**5**

votes

**1**answer

215 views

### Cusps forms for $\Gamma (N)$

I know how to build a basis of the vector space of cusp forms for the congruence subgroups $\Gamma_1 (N)$ and $\Gamma_0 (N)$, but I couldn't find in the literature how to build a basis for ...

**2**

votes

**0**answers

79 views

### Orders of Clifford algebra

Let $C_n$ be the Clifford algebra over $\mathbb{Q}$ associated to negative definite quadratic form $-I_n$ (i.e. $-x_1^2-\dots-x_n^2$). Let $\mathcal{O}$ be a $\mathbb{Z}$-order of $C_n$.
Q1) Is it ...

**11**

votes

**1**answer

565 views

### Proving the Irrationality of this Number

I found this problem on Math.SE:
Prove that $\log_35+\log_25$ is irrational.
http://math.stackexchange.com/q/986227/173397.
I labored on it for a few days, and couldn't find an algebraic ...

**0**

votes

**1**answer

199 views

### Is there a formula that can predict the primes in the sequence of ratios of consecutive superior highly composite numbers? : $2, 3, 2, 5, 2, 3, 7,…$

This is the sequence of prime numbers which are the elementary building blocks for the superior highly composite numbers:
$2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 2, 23, ...$
The $n^{th}$ ...

**0**

votes

**0**answers

47 views

### Odd length repetends in recurring decimals [on hold]

For any number n the reciprocal can be expressed as a decimal, which will be composed of a recurring pattern as long as n is co-prime with 2 and 5. In general terms 1/n will produce a recurring ...

**1**

vote

**1**answer

122 views

### About Section 4.2 in 'introduction to the spectral theory of automorphic forms' by Iwaniec

In Section 4.2 in 'introduction to the spectral theory of automorphic forms' by Iwaniec, the author said that the kernel of the invariant integral operator
$$
(Lf)(z)=\int_{\mathbb{H}}k(z,w)f(w)d\mu w
...

**14**

votes

**0**answers

471 views

### Positive binary quadratic form plus univariate monic cubic (giving Hilbert class field)

We have the Lucas numbers, $$ L_1 = 1, \; L_2 = 3, \; L_3 =4, \; L_4 = 7, L_5 = 11, \; L_{n+2} = L_{n+1}+ L_n \; . $$
Question: is it the case that
$$ f(x,y,z) = 4 x^2 + 3 x y + 9 y^2 + z^3 + 3 z ...

**4**

votes

**0**answers

147 views

### Visibility interpretation of Riemann zeta zeros on the critical line?

This is a long shot, but ...
The fraction of $\mathbb{Z}^2$ lattice points
visible from the origin
$1/\zeta(2)=6/\pi^2 \approx 61$%.
The fraction of $\mathbb{Z}^3$ lattice points visible
from the ...

**1**

vote

**0**answers

87 views

### Skew symmetry for the Hilbert symbol

Let $K$ be a local field containing the group $\mu_n$ of $n$th roots of 1 and the $\theta_K:K^*\to G_K^{ab}$ be the reciprocity map. The we know that the Hilbert symbol $$K^*\times K^*\to \mu_n$$ ...

**4**

votes

**3**answers

321 views

### Textbook request for class field theory [duplicate]

I am studying class field theory. I need good reference books, notes, or other materials which explain the following topics: ideles and ideals, Haar measure and integration on local fields, Fourier ...

**8**

votes

**0**answers

633 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

106 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?

**5**

votes

**1**answer

255 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 ...

**2**

votes

**1**answer

206 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 ...

**2**

votes

**0**answers

162 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

229 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

215 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

64 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

137 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 ...

**6**

votes

**2**answers

271 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 ...

**15**

votes

**4**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

61 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

**2**answers

156 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

136 views

**1**

vote

**1**answer

120 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

49 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 ...

**4**

votes

**1**answer

138 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

97 views

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

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

104 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

257 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

131 views

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

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

120 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

250 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 [closed]

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

348 views

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

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

352 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

85 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

157 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

99 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

181 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 ...

**4**

votes

**0**answers

150 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 ...

**2**

votes

**2**answers

133 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

97 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

205 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 $0<d_i \le i$. ...