Tagged Questions

Prime numbers, Diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

learn more… | top users | synonyms (1)

5
votes
1answer
111 views

Parity of the Prime Counting Function

I am interested in the distribution of the parity of $\pi(x)$, the prime counting function, over the natural numbers. Let: $\ E_n = \{ k \in \{1,2, .., n\} : \pi(k) \equiv 0 \mod 2 \}\ $ and $\ O_n ...
6
votes
0answers
88 views

Special fiber of $X(p)$ in characteritic $p$

Let $p \geq 5$ be a prime. Let $Y(p)$ be the fine moduli space representing elliptic curves + basis of the $p$-torsion over $\mathbb{Q}_p$ and let $Y_1(p)$ be the fine moduli space representing ...
2
votes
0answers
96 views

Argument againts the $abcd$ conjecture with extra gcd conditions

Got an argument and numerical support againts the $abcd$ conjecture with extra gcd conditions (observe that this is different from the $abc$ and the $abcd$ conjectures). This thesis p. 20 defines the ...
4
votes
0answers
108 views

The sum of all the elements of every non empty subset of $A$ is not a multiple of $n$

Let $N=\{1,2,\ldots ,n\},n>1$. We wish to construct a set $A\subseteq N$ with the property: The sum of all the elements of every non empty subset of $A$ is not a multiple of $n$. ...
-5
votes
0answers
48 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
0answers
129 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
0answers
43 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
1answer
96 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
0answers
143 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, ...
6
votes
1answer
226 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
0answers
81 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 ...
13
votes
1answer
596 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
1answer
204 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
0answers
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
1answer
124 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
0answers
475 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
0answers
156 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
0answers
90 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
3answers
324 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
0answers
674 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
0answers
107 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
1answer
258 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
1answer
219 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
0answers
163 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
1answer
231 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 ...
6
votes
1answer
223 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
0answers
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
1answer
146 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
2answers
277 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
4answers
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
0answers
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
1answer
62 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
2answers
168 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
0answers
141 views

Talking about the abc-conjecture [closed]

What is the latest news about the abc-conjecture?
1
vote
1answer
123 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
0answers
53 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
1answer
144 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
0answers
99 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
0answers
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
1answer
262 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
0answers
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
1answer
121 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
1answer
256 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
0answers
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
2answers
352 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
2answers
354 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
1answer
87 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
1answer
158 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
0answers
100 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
1answer
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 ...