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

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41 views

Ref request: If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal

On p. 76 of the 1996 edition of Serre's A Course in Arithmetic, one reads the following (inline) remark: One can prove that, if $A$ has natural density $k$, the analytic density of $A$ exists and ...
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18 views

If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, is $n^2$ solitary?

(Note: A similar question was asked in MSE two months ago.) Let $\sigma(x)$ be the sum of the divisors of the natural number $x$, and denote the abundancy index $\sigma(x)/x$ by $I(x)$. Here is my ...
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1answer
53 views

writing an integer as particular summation [on hold]

I think my question is an elementary question. Thanks for any help or comment. Is there any formula for the number of writting a natural number $n$ in a summation as follows, $n=a_1+\dots+a_k$, ...
0
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0answers
81 views

An elementary residue problem

Given coprime $A,B\in\Bbb N$ with $|A-B|=o(\min(A,B))$ how to find $C,D\in\Bbb N$ such that $|D-A^6|=o(A^6)$, $|D-B^6|=o(B^6)$, $A^5<C<A^6$, $B^5<C<A^6$ along with $$|(CA\bmod ...
-4
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0answers
88 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 ...
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0answers
46 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 $H_1(E)$ ...
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1answer
152 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 ...
5
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2answers
316 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 ...
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134 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 ...
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59 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 ...
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0answers
145 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
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1answer
118 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 ...
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0answers
94 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 ...
2
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0answers
122 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 ...
5
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119 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 ...
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1answer
186 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 ...
3
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0answers
49 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 $$ ...
9
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1answer
185 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 ...
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0answers
65 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 ...
12
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1answer
397 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.
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1answer
140 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 ...
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2answers
245 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 ...
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1answer
256 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 ...
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0answers
34 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
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0answers
245 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 ...
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1answer
550 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
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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 $$ \zeta(s_1, s_2)=\sum_{n>m \geq 1} \frac{1}{n^{s_1}m^{s_2}}, \qquad s_1 \geq 2, \quad s_2 \geq 1 $$ ...
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100 views

Question about the Reimann Zeta Function [closed]

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 ...
3
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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 ...
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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 ...
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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 ...
8
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1answer
258 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 ...
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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 ...
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0answers
53 views

Approximating Ehrhart Polynomial of Rational n-Tetrahedron

A set of positive integers $d_1, \dots, d_n$ describe a n-dimensional tetrahedron $T$ with the vertices $$ (0,\dots,0), (1/d_1,0,\dots,0), (0, 1/d_2,\dots,0), \dots, (0,\dots,1/d_n).$$ Let $L_T(t)$ be ...
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0answers
63 views

largest Kaprekar number [closed]

I got interested in Kaprekar number recently. I searched google scholar but there didn't seem many results and the article I wished to read was not available. Does anyone know what the largest known ...
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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. ...
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0answers
86 views

Density of ratios of an arbitrary increasing sequence of prime numbers

It is well known that the set $\left\{ \frac{p}{q} : p,q \textrm{ prime numbers }\right\}$ is dense in the positive real numbers $\mathbb{R}_{>0}$. Not having a background in number theory, I ask ...
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0answers
261 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)\ ...
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107 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, ...
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71 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, $n > 1$ ...
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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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0answers
81 views

A connection between basic hypergeometric series and number theory

I am studying functions given by the power series: $$f(z)=1+\sum_{n=1}^{\infty}\frac{z^n}{(1-q)(1-q^2)\cdots(1-q^{n})}.$$ The parameter $q$ is usually assumed to be such that $|q|<1$. Then it is ...
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0answers
117 views

Does the Tate pairing agree with the Brauer-Manin pairing

Let $X$ be a proper, smooth, geometrically integral variety over a field $k$. Let $A$ be (the identity component of) its Picard variety and let $B$ be (the identity component) of its Albanese variety. ...
12
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2answers
764 views

Proofs of Fermat-Wiles theorem for exponent 3

Apart from Wiles'proof (which I didn't read), I know 4 proofs of the Fermat-Wiles theorem for exponent 3, i. e. impossibilty of $x^{3} + y^{3} + z^{3} = 0$ with x, y and z pairwise coprime nonzero ...
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3answers
419 views

How many solutions does $\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}=1$ have?

It is well known that $\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1$ and this is the only solution to $\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}=1$ with $2\leq x_1<x_2<x_3$. My question is: Let ...
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2answers
341 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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301 views

Complete Local Ring and Fermat's Last Theorem

Let us consider the infinitely many variables formal power series ring over a finite field, viz. $R:= F_p[[S_1,...,S_∞]].$ Question: Choose an arbitrary finitely generated ideal $I$ of $R$. Is ...
11
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2answers
642 views

Upper bound on answer for Pell equation

A user on MSE, @martin , asked http://math.stackexchange.com/questions/1611411/pell-equations-upper-bound about an upper bound for $x$ in $x^2 - p y^2 = 1,$ when $p$ is prime. I checked, it appears ...
10
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1answer
416 views

Why do the Maynard-Tao weights work so well?

I am looking for an intuitive reason for why the Maynard-Tao weights work well to capture many primes of the form $n+h_1, \ldots , n+h_k$, where $(h_1, \ldots , h_k)$ is any admissible $k$-tuple. For ...
1
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1answer
265 views

Hodge-Tate representations

I know that the p-adic representaions from geometries are de Rham representations and hence they are Hodge-Tate representations. Then, are there (more than 2-dimensional) Hodge-Tate representations ...