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

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0
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1answer
142 views

On odd perfect numbers $N = q{p^{2a}}{m^2}$ satisfying certain conditions

Let $N = q{p^{2a}}{m^2}$ be an odd perfect number, satisfying the conditions $$\sigma(m^2) = p^{2a}$$ $$\sigma(p^{2a}) = q$$ and $$q + 1 = 2{m^2}.$$ Note the following: $p^a m < q$ $q$ is ...
-1
votes
0answers
90 views
+50

On even almost perfect numbers other than the powers of two, as compared to odd perfect numbers given in Eulerian form

This question has been cross-posted from MSE. Antalan and Tagle (in a 2004 preprint titled Revisiting forms of almost perfect numbers) show that, if $M \neq 2^t$ is an even almost perfect number, ...
0
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0answers
112 views

Is there any significance in Heegner numbers (or class number 1) representation symmetry?

$\mathrm{A003173}(n) = 1+((1 + \sqrt{3})^{n-1} - (1 - \sqrt{3})^{n-1})/(2\sqrt{3})$ for n = 1,2,3,4. $\mathrm{A003173}(n) = 19+24((1 + \sqrt{3})^{n-6} - (1 - \sqrt{3})^{n-6)}/(2\sqrt{3})$ for n = ...
0
votes
1answer
172 views

If $N = qn^2$ is an odd perfect number, is it possible to have $q + 1 = \sigma(n)$?

The title says it all. Question If $N = qn^2$ is an odd perfect number with Euler prime $q$, is it possible to have $q + 1 = \sigma(n)$? Heuristic From the Descartes spoof, with quasi-Euler ...
32
votes
5answers
1k views

The sequence $a_{n+1}=\left\lceil \frac{-1+\sqrt{5}}{2}a_{n}-a_{n-1} \right\rceil$ is periodic

Let $(a_{n})_{n \ge 1}$ be a sequence of integers such that for all $n \ge 2$: $0\le a_{n-1}+\frac{1-\sqrt{5}}{2}a_{n}+a_{n+1} <1$. Prove that the sequence $(a_{n})$ is periodic. ...
1
vote
1answer
137 views

Weight 12 cusp forms for $\Gamma_0(p)$

Let $S_k$ be the space of weight 12 cusp forms of $\Gamma_0(p)$, ($p$ prime), then Sage tells that $\dim S_k^{\text{new}}=\dim S_k-2$. Thus the old forms spans a 2-dimensional subspace. One of the ...
-4
votes
0answers
77 views

How to prove that the only integer solutions to ${r}^{3}/\left({r-1}\right)$ are $r\in \left\{{0,2}\right\}$ [on hold]

This seams simple, how to prove that the only integer solutions to ${r}^{3}/\left({r-1}\right)$ are $r\in \left\{{0,2}\right\}$. If $r$ is odd then we have an odd/even where there is always at least ...
1
vote
0answers
41 views

Pairing for non-uniformizable Anderson T-motives

Let $M$ be an Anderson T-motive (the simplest case, i.e. abelian in the meaning of [G] Goss, Basic structures of function field arithmetic, Def. 5.4.12, over $A^1$, having $N=0$), and let $H_1(E)$ ...
5
votes
2answers
297 views

Asymptotics of product of Euler's totient function (A001088)?

Conjecture: \begin{align} \lim_{n\to \infty } \, \frac{\left(\prod _{k=1}^n \phi (k)\right){}^{1/n}}{n}\sim 0.2059\text{...} \end{align} The numerical result from 100000 terms is: My questions ...
2
votes
0answers
118 views

Computing intersection number of two arithmetic line bundles

I have some questions in Arithmetic Arakelov geometry Let $\mathcal X\to Spec(\mathcal O_K)=C$ be an arithmetric projective variety over $C$ , where $\mathcal O_K$, ring of number filed $K$ and ...
9
votes
0answers
123 views

Is the Flajolet-Martin constant irrational? Is it transcendental?

Facebook has a new tool to estimate the average path length between you and any other person on Facebook. An interesting aspect of their method is the use of the Flajolet-Martin algorithm. In the ...
11
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1answer
529 views

Even Galois representations “mod p”

Consider an irreducible $\mathrm{mod}$ $p$ representation: $$\rho: \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_2(\bar{\mathbb{F}}_p)$$ If $\rho$ is odd, it was conjectured by Serre in ...
-1
votes
0answers
143 views

Where I can find a Carlitz's paper? [on hold]

I am looking for a PDF of the paper: Carlitz, Lewis, Mills, Straus - Polynomials over finite fields with minimal value sets. Only I can find about it is this data: Mathematika / Volume 8 / Issue ...
-2
votes
0answers
53 views

Degree of a rational Function [on hold]

This might sound a very trivial question but I found different answers on the web. Assume on has a rational function f(x)/g(x) where f(x) and g(x) are polynomials. What is the degree of the rational ...
2
votes
1answer
114 views

Rational valued functions on the Cantor set with $\int_{C} f^{3}d\mu=1 $

Let $C$ be the Cantor set as a compact Abelian topological group, isomorphic to countable product of $\mathbb{Z}/2\mathbb{Z}$. Its normalized Haar measure is denoted by $\mu$. Is there a ...
1
vote
1answer
137 views

Methods for searching for prime generating polynomials

I am currently using a non-systematic, pseudo-random method for finding prime-generating polynomials, based on the Bateman-Horn method for finding likely candidates, and then narrowing down. I have ...
12
votes
1answer
393 views

Does the sum $\sum_{n=1}^{\infty}\frac{1}{p_n(p_{n+1}-p_n)}$ converge?

Prove, if possible in an elementary way, that $\sum_{n=1}^{\infty}\frac{1}{p_n(p_{n+1}-p_n)}$ converges/diverges, where $p_n$ denotes the $n^{\textrm{th}}$ prime.
9
votes
1answer
181 views

Elementary prime-generating sequences

A student of mine keeps coming again and again and telling "I've found a formula $n\mapsto f(n)$ giving all primes" or sometimes "infinitely many primes", where $f$ is a classical function (I mean ...
13
votes
1answer
384 views

Most discriminants are almost squarefree

Write, for $f(x) = x^d + a_2 x^{d-2} + \cdots + a_d\in \mathbb{Z}[x]$, $H(f) := \max(|a_i|^{\frac{1}{i}})$. Does anyone know of a reference that would allow me to show that the proportion of $f$ with ...
4
votes
0answers
92 views

Are prime gaps of even index essentially larger than those of odd index?

Let $g_{n}:=p_{n+1}-p_{n}$ be the $n$- th prime gap, and let's introduce the following summatory functions: $$G_{1}(x):=\sum_{1\leq n\leq x}g_{2n-1}$$ $$G_{2}(x):=\sum_{1\leq n\leq x}g_{2n}$$. Let's ...
5
votes
0answers
117 views

Comparison of sheaves of modular forms

Let $\pi:E\to X$ the universal generalized elliptic curve over the compactified modular curve, with zero section $e: X\to E$. Now consider the following two sheaves on $X$: $e^*\Omega^1_{E/X}$ and ...
3
votes
0answers
47 views

Square integral of finite Euler product

Consider the finite Euler product $$ P(t) = \prod_{r=1}^R \left(1 + p_r^{i t} \right). $$ (Here $p_1, p_2, \dots$ are of course the primes.) Question: What is a good asymptotic upper bound for $$ ...
-1
votes
0answers
64 views

value of Riemann's zeta function at even negative numbers [on hold]

The zeta function has trivial zeros at -2,-4,......But direct substitution of say -2 makes the sum diverge as the negative exponent in the denominator makes the terms 2 squared,3 squared etc. please ...
7
votes
2answers
228 views

What's in the genus of the cubic lattice?

I'll write $\mathbf{Z}^n$ for the integral quadratic form $x_1^2 + \cdots + x_n^2$. For which values of $n$ is $\mathbf{Z}^n$ unique in its genus, i.e. isolated in Kneser's graph? In particular can ...
7
votes
1answer
867 views

Coin problem with permutations

Let $a,b,c$ be positive integers with gcd$(a,b,c)=1$, and let $\mathbb{N}$ denote the set of nonnegative integers. It is well known that $\mathbb{N} \setminus (a \mathbb{N}+b \mathbb{N} + c ...
4
votes
0answers
257 views

natural radical and an algebraic expression in $\pi$ and/or $e$

Let $\ \mathbb N:= \{1\ 2\ \ldots\}\ $ be the set of natural numbers. Let $\ \mathbb P:=\{2\,\ 3\,\ 5\,\ 7\,\ 11\,\ \ldots\}\ $ be the set of primes. Then natural radical $\ rad(n)\ $ is $$ rad(n)\ ...
4
votes
1answer
311 views

Birch's conjecture from Representation Theory

Birch has a conjecture about which automorphic forms on $PGL(2)$ are the lifts from nonsplit $O(3)$. Temporarily ignore global issues, and focus on the local nonarchimedian picture. The automorphic ...
2
votes
1answer
191 views

Generalization of a theorem of Mahler to higher dimensions

A seminal theorem of Kurt Mahler, in his papers Zur Approximation algebraischer Zahlen. I-III., is the following: Let $F(x,y) \in \mathbb{Z}[x,y]$ be a binary form of degree $d \geq 3$ and non-zero ...
1
vote
1answer
254 views

Asymptotics of “ugly” function elucidate Goldbach's conjecture?

Question We now define the following "ugly" function: $$ A_c(s,r,n,m) = \begin{cases} 1 & \text{ if only $sr+nm=2c$ } \\ 0 & \text{otherwise} \end{cases} $$ How does the ...
2
votes
1answer
251 views

Representation of rationals by quadratic form

In one paper about number theory author stated 2 lemmas Lemma 1. If $p$ is a prime $\equiv3(mod $ $4)$ then $x^2+y^2-pz^2$ represents a non-zero rational number $m$ if and only if $m$ is not of the ...
16
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0answers
901 views
+50

Is formula valid for relating $\pi$ with ALL of its OEIS A002485(n)/A002486(n) convergents?

Could anyone try to prove that the below conjectured formula is valid for relating $\pi$ with ALL of its convergents - those, which are described in OEIS via A002485(n)/A002486(n) ? $$(-1)^n\cdot(\pi ...
7
votes
2answers
908 views

Contour integration of $\zeta(s)\zeta(2s)$

I have been looking at this for days and I am going insane. I need to show that for a Dirichlet series equal to $\zeta(s)\zeta(2s)$, the sum of the coefficients less than $x$ is ...
-2
votes
0answers
33 views

equation for triangular membership funtion [on hold]

I am working on fuzzy logic. Although I know the equations for triangular membership function but I can't figure out how they are derived.Are they derived from slope of line concept or some other ...
6
votes
0answers
235 views

Interuniversal Teichmuller theory's applications

Apart from a proof of the ABC conjecture -and its accepted consequences- are there applications of Mochizuki's IUT? In particular are there already widely accepted applications? Does it shed ...
10
votes
1answer
547 views

Do we care about multiple zeta functions?

Coming from a number-theoretic background, I certainly care about $L$-functions and in particular automorphic ones. For automorphic forms on $SL_2(\mathbb{Z}) \backslash SL_2(\mathbb{R})$, ...
11
votes
1answer
1k views

What did Euler do with multiple zeta values?

When reading about multiple zeta values, I often find the claim that the case of length two $$ \zeta(s_1, s_2)=\sum_{n>m \geq 1} \frac{1}{n^{s_1}m^{s_2}}, \qquad s_1 \geq 2, \quad s_2 \geq 1 $$ ...
9
votes
1answer
352 views

Multiple zeta values at negative integers

For a tuple of integers $\underline{s}:=(s_1,\dots, s_d)$, the multiple zeta value (MZV) at $\underline{s}$ is defined as: $$ \zeta(s_1,\dots, s_d):= \sum_{n_1>\dots>n_d>0}\frac 1 ...
57
votes
0answers
4k views

Grothendieck-Teichmuller conjecture

(1) In "Esquisse d'un programme", Grothendieck conjectures Grothendieck-Teichmuller conjecture: the morphism $$ G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T}) $$ is an isomorphism. Here ...
3
votes
2answers
225 views

Unfamiliar prime-generating polynomials related to Heegner numbers

I just stumbled on a set of prime-generating polynomials of the form $$9 n^2-3 H n+H (H+1)/4$$ (where $H$ is a Heegner number $>11$), which generate the same number of distinct primes as their more ...
2
votes
1answer
163 views

Sampling from random totally unimodular matrices of a particular type?

Is there a way to parametrize totally unimodular $(3n+2)\times(2n+2)$ matrices of form $$\begin{bmatrix} \pm1 & \pm1 & 0 & 0 &\dots & 0 & 0 & 0 & 0\\ A_{2n} & ...
2
votes
2answers
339 views

Examples of p-adic representations

When reading the books or papers on p-adic Hodge theory, non trivial example of p-adic representation seems to be only the example of Tate curves. To be sure, I had read the very readable introduction ...
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votes
0answers
100 views

Question about the Reimann Zeta Function [on hold]

I recently watched a video on youtube by a group called Numberphile. Theyre were discussing the Reimann Hypothesis and I was confused when they brought up 'Analytical Continuance' and 'Holomorphic ...
7
votes
1answer
256 views

Riemann zeta function: pair correlations vs. neighbor spacings

Montgomery's pair correlation conjecture states that the distribution of the pair correlations of the zeroes of the Riemann zeta function (normalized to have average spacing 1) is given by the ...
1
vote
0answers
116 views

Primes of the form $2^{m_0}p_1^{m_1}\ldots p_r^{m_r}+1$

Is it known any example of a set of primes $\{p_1,\ldots,p_r\}$ with the following property: there are infinitely many $(m_0,\ldots,m_r)\in\mathbb N^{r+1}$ such that $2^{m_0}p_1^{m_1}\ldots ...
11
votes
2answers
2k views

Is tensor product of local algebras local?

In general, the tensor product of two local rings is not local. For example, $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C}\ $ is not a local ring. Let $\mathbb{F}_{p}$ denote the finite field with $p$ ...
33
votes
1answer
1k views

The Bourgain-Demeter-Guth breakthrough and the Riemann zeta function?

Yesterday Bourgain, Demeter and Guth released a preprint proving (up to endpoints) the so-called main conjecture of the Vinogradov's Mean Value Theorem for all degrees. This had previously been only ...
6
votes
0answers
98 views

Local character expansion for discrete series representations of $GL_n(F)$

I'm interested about what, if anything, is known about the local character expansion of discrete-series representations of $GL_n(F)$, where $F$ is a $p$-adic field. First, some notation: let $G$ be a ...
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votes
0answers
81 views

Is the Rankin-Selberg convolution of two powers of Zeta automorphic?

I've been told that, generally speaking, it is not known whether the Rankin-Selberg convolution of two automorphic L-functions is itself automorphic. I would like to know whether this property holds ...
1
vote
0answers
70 views

Simple Diophantine equations for Cartan matrices of Kac-Moody algebras

Let us consider the matrix $A$ defined as follows: $A_{i i} =2$, $A_{i j} = - \frac{2 d_i}{d - 1 - d_{j}}, \qquad i \neq j$; $i, j = 1, \dots, n$. Here $d_1,...,d_n$ are natural numbers, $n > 1$ ...
-1
votes
1answer
227 views

Consequences of Langlands functoriality conjecture

I would like to know whether Langlands' functoriality conjecture implies that the Selberg class coincides with the class of automorphic L-functions and, if so, whether this class is closed under ...