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

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1answer
107 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 ...
28
votes
0answers
471 views

Mod sequences that seem to become constant; and the number 316

Define a "mod sequence" of nonnegative integers based on one start parameter $s$, its first term, as follows. $A(s)=(a_1,a_2,\ldots,a_n,\ldots)$ with $a_1 = s$ and $$ a_n = \left(\sum_{k=1}^{n-1} a_k ...
0
votes
1answer
101 views

More on Vojta's exceptional set for a more general abc conjecture

Likely a mistake, but got very large exceptional set in 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 ...
2
votes
5answers
320 views

Mean of a vector

Be a set of numbers $v=(a_1, a_2, \ldots, a_n)$ I want to form the following average vector $\mu = (\frac{\sum a_i}{n}, \frac{\sum a_i}{n}, \ldots, \frac{\sum a_i}{n})$. If I do it iteratively step ...
7
votes
0answers
140 views

Is the set of numbers $\{ [n^{3/2}] \mid n\text{ an integer}\}$ a basis of order 3?

A set of integers is called a basis of order h if every nonnegative integer can be written as the sum of h not necessarily distinct elements of the set.(Nathanson, Additive Number Theory, page 7) ...
0
votes
0answers
112 views

E- and A-algorithms for finite arithmetic prime progressions and other sets

(EDIT from scratch). Let $\ \mathbf a := (a_1\ \ldots\ a_n)\ $ be an increasing non-constant arithmetic progression of odd positive numbers. The goal here is to resolve efficiently one of the two ...
0
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0answers
69 views

Abscissa of absolute convergence of the product of two Dirichlet series

I first asked the following question on Mathematics StackExchange (a few weeks ago), since the content of MathOverflow is mostly above my academic level. I didn't want to bother people on this forum ...
5
votes
1answer
459 views

Any way to prove Prime Number Theorem using Hyperbolic Geometry? [on hold]

The prime number theorem says that the density of prime numbers is inverse as the number of digits of $n$: $$\displaystyle \frac{\{1 \leq k \leq n : \text{ prime } \}}{n} \approx \frac{1}{\log n}$$ ...
19
votes
3answers
670 views

Number of elements of “$\mathrm{SL}_n(\mathbb{F}_p^\times)$” mod $p$

How many elements of $\mathrm{SL}_n(\mathbb{F}_p)$ have all nonzero entries? Just the answer mod $p$ would be fine as well. This seems like it should be easy/in the literature but I couldn't find it.
-1
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1answer
98 views

Algorithm Involving Quadratic Diophantine Equation

Let $a,b,c,d\in\Bbb Z$. $abcd\neq 0$ and $a>0$. What is the best standard low complexity procedure to find the set of solutions for $x,y,t\in\Bbb Z$ such that: $$-axy+bx+cy+d=t^2$$ where ...
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0answers
73 views

Can the divisibility-by-7 test be extended to preserve the remainder? [on hold]

The divisibility-by-7 test (in base 10) performs iterations whereby the last digit is doubled and subtracted from the number formed by all other digits. If the original number is divisible by 7, then ...
2
votes
0answers
102 views

Over which fields is a $G$-module reducible?

I have asked this question at math.stackexchange, but have not received an answer so far. Also, I'm not entirely sure that this is a suitable question for mathoverflow... See this link for the ...
21
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7answers
7k views

Collecting proofs that finite multiplicative subgroups of fields are cyclic.

I teach elementary number theory and discrete mathematics to students who come with no abstract algebra. I have found proving the key theorem that finite multiplicative subgroups of fields are cyclic ...
1
vote
1answer
137 views

Prime divisors of $p^n+1$

Let $p$ be a rational prime and $n$ be a positive integer. It can be easily deduced from Zsigmondy's theorem that $p^n+1$ has a prime divisor greater than $2n$ except when $(p,n)=(2,3)$ or ...
3
votes
1answer
228 views

Improving the bound $q < n\sqrt{3}$ for an odd perfect number $N = {q^k}{n^2}$ given in Eulerian form

(Note: This was cross-posted from MSE.) Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q, n) = 1$). Therefore, ...
4
votes
1answer
174 views

Constructing a family of convergents from continued fractions formed by a set of prime partial quotients

For a given real number $x$, the continued fraction representation $x = [a_0; a_1, a_2, \cdots]$ where $(a_n)_{n \geq 0}$is defined by setting $x = \alpha_0$, then $a_i = \lfloor \alpha_i \rfloor$, ...
0
votes
0answers
84 views

$\zeta$ function for ambiguous class group

Let $K$ be a Galoisian number field with ring of integers $O_K$. I consider $\zeta_{A,K}(s)$ the function defined by $$\zeta_{A,K}(s)=\prod_{\substack{p\in\mathbb N\\p\text{ ...
5
votes
1answer
291 views

Ruth-Aaron triples, etc

A Ruth-Aaron pair is two numbers $(n,n+1)$ such that their sum of prime factors is equal, counting repeated prime factors. (The name refers to Hank Aaron's 715 homeruns surpassing Babe Ruth's 714!) So ...
2
votes
1answer
143 views

Chebyshev polynomials factoring uniformly modulo all primes

Consider the Chebyshev polynomial of the first kind $T_n(x)$ and its factorization in $\mathbb F_p$ for a given prime $p$. Most often, this factorization is not uniform (meaning that the irreducible ...
10
votes
1answer
502 views

More elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?

(Note: See also the $a^4+b^4+c^4 = 1$ version in this old MSE post.) The equation discussed in a paper by Jacobi and Madden, $$a^4+b^4+c^4+d^4 = (a+b+c+d)^4 = z^4\tag1$$ or equivalently, $$(p-2q + ...
7
votes
1answer
344 views

Tate's thesis for Artin L-functions

As far as I know, Tate's thesis has been successfully applied in two fronts: Hecke L-functions, by Tate and Iwasawa (and Teichmüller, Witt, Schmid) Automorphic L-functions, by Jacquet, Shalika, ...
2
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3answers
246 views

Growth of $r_{2}(n)$

Let $n$ be a positive integer. From Jacobi's two-square theorem we know that the number $r_{2}(n)$ of representations of $n$ as a sum of two squares is given by $$ r_{2}(n)=4(d_{1}(n)-d_{3}(n)), $$ ...
0
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0answers
78 views

Relation between cyclotomic character and fundamental character of level

The question I have is the following: is it true that the $p+1$ exponentiation of the fundamental character of level $2$ gives us the reduction (mod $p$) of the cyclotomic character? For a review of ...
6
votes
1answer
217 views

what is the intersection of all congruence subgroups of the profinite completion of SL(2,Z)?

Let $\widehat{SL(2,\mathbb{Z})}$ be the profinite completion of $SL(2,\mathbb{Z})$. Let $\Gamma(N)$ denote the typical principal congruence subgroup of $SL(2,\mathbb{Z})$ (ie, all matrices congruent ...
8
votes
1answer
546 views

What is a totient?

In addition to the Euler totient function, there are a great many generalizations and related functions which go by the "totient", usually with some name: Jordan, Lehmer*, Schemmel, Nagell, Alder, ...
1
vote
1answer
143 views

Bombieri-Vinogradov in short intervals

In 1985 Perelli, Pintz & Salerno proved a short-interval form of the Bombieri-Vinogradov theorem with $\theta \in (7/12, 1]$. Have there been any improvements on this, in particular with the ...
2
votes
3answers
729 views

Where do Set Theory and Number Theory meet together?

As all know, by absoluteness theorems in Set Theory, most of theorems in number theory are $ZFC$-provable if and only if they are consistent with $ZFC$, it's because of absoluteness of essence of ...
7
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0answers
359 views

Application of the Riemann hypothesis and the ABC conjecture to independence results

In Old Home Page of Andreas Weiermann Andreas Weiermann has stated the following: Quite recently I submitted a preprint about an application of the Riemann hypothesis and the ABC conjecture to ...
0
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1answer
702 views

A letter from J. P. Serre

Which is the letter where J. P. Serre present "Analogues Kählériens de certaines conjectures de Weil" to Weil?
10
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0answers
274 views

Applications of $p$-adic Hodge theory

I am trying to learn $p$-adic Hodge theory. I found some materials explaining main theorems (or aspects) of the theory. However, I could not find references which explaining how to use the theory. ...
2
votes
1answer
588 views

Efficient algorithms to determine the roots of: $p(x) = r^x $ in the finite field $GF(q)$, where $r$ a primitive root of the field

I need to make sure that no efficient (i.e., polynomial time) algorithm exists for the following problem: Exponentiating Polynomial Root Problem (EPRP) Let $p(x)$ be a polynomial with $\deg(p) \geq ...
3
votes
2answers
200 views

Sets of squares representing all squares up to $n^2$

Let $S_n=\{1,2,\ldots,n\}$ be natural numbers up to $n$. Say that a subset $S \subseteq S_n$ square-represents $S_n^2$ if every square $1^2,2^2,\ldots,n^2$ can be represented by adding or subtracting ...
1
vote
1answer
117 views

Cardinality of the prime divisor set of a k-power sum

Let $a_{1},\dots,a_{n}$ be positive natural numbers ($n>2$) such that $a_{i}\neq a_{j}$ if $i\neq j$. I want to prove that $$ \left\lvert \left\{ p \text{ prime} \; : \; p \mid \sum_{i=1}^n ...
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0answers
53 views

The hypertriangular function of $n$

I'm looking for papers or recent results on the hypertriangular function of $n$: $$H_t(n)= \displaystyle\sum\limits_{k=1}^{n} k^k$$ This is A001923 in the OEIS. I don't have much experience with ...
3
votes
1answer
136 views

Pathological behavior of Lie algebra under a map of abelian schemes

I am trying to understand Example 7.5/9 from the book "Neron models". There one has a discrete valuation ring $R'$ that is the localization of $\mathbb{Z}[\zeta_p]$ at $p$, so that the absolute ...
0
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0answers
66 views

Arithmetic functions associated with Hurwitz Zeta function raised to arbitrary complex powers, $\zeta(s,q)^z$ for $q \in \mathbb{N}$?

If $\zeta(s)$ is the Riemann Zeta function, then $\zeta(n)^z$, with $z \in \mathbb{C}$, $\Re(s)>1$, can be represented as $$\zeta(s)^z=\sum_{n=1}^\infty \frac{d_z(n)}{n^{-s}}$$ where $d_z(n)$ ...
-2
votes
1answer
138 views

Prove that $\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2})$ [closed]

Let $a,b\in\mathbb{Z}$, and $f\in C^2([a,b])$ such that $|f''(t)|\asymp \lambda$ for $a\le t\le b$. Prove that $$\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2}).$$ ...
6
votes
1answer
250 views

abelian $\ell$-adic representations in $\widehat{SL(2,Z)}$

In Grothendieck's Esquisse he claims that the action of $$\text{Gal}(\mathbb Q)\to\text{Out}(\pi_1(M_{1,1})=\text{Out}(\widehat{SL(2,Z)})$$ obtained from the homotopy exact sequence of the étale ...
-4
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0answers
45 views

Height of a tripod [closed]

I am trying to determine the height of a tripod when the length on the tripod's legs (81") and the distance between the ends that touch the ground are 57" apart. My thought process so far: When the ...
1
vote
1answer
232 views

Number of solutions in a sum of squares Diophantine equation

Let $n$ be an integer. I'm interested in upper bounding the number of integer solutions of \begin{equation*} x_1^2+x_2^2+\ldots+x_p^2=y_1^2+y_2^2+\ldots+y_p^2, \end{equation*} where for all ...
5
votes
1answer
139 views

When is $\vartheta(x)>x$? [Skewes number analog]

Let $\vartheta(x)=\sum_{p\le x}\log p$. What is known about the first time $\vartheta(x)>x?$ Bays & Hudson give good upper bounds (slightly improved by Chao & Plymen) on the first crossing ...
5
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0answers
510 views

Zeta function double product

Is it possible to write the following double product in terms of the zeta function? \begin{align} &\prod_{i=1}^{\infty}\prod_{j=1}^{\infty} \frac{1}{1-(p_i\ p_j)^{-s}} \end{align} Extending the ...
5
votes
1answer
161 views

Which criteria for “uniformly splitting” polynomials?

Let $P(x)$ be an irreducible monic polynomial of degree $\ge4$ with integer coefficients. We all know that over a finite field $\mathbb F_p$, $P$ will often split, and I am interested in polynomials ...
4
votes
1answer
705 views

Possibly an easy application of Tate-Nakayama duality and local CFT

Let $F$ be a non-archimedian local field and $F^{ur}$ the maximal unramified extension. Let $T$ be a torus defined over $F$ such that $T$ splits over a tamely ramified extension and is anisotropic ...
4
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1answer
426 views

Modular forms and “too many symmetries”

How do we interpret Barry Mazur's quote of Modular forms are functions on the complex plane that are inordinately symmetric. They satisfy so many internal symmetries that their mere existence ...
0
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0answers
110 views

Shimura reciprocity law

I have spent quite some time understanding class fields generated by Kummer extensions and class fields generated by modular forms. Now, I am turning the notch of sophistication a bit to study class ...
1
vote
1answer
351 views

Numerical evidence and argument against Littlewood conjecture

This is joint work with someone. We got numerical evidence and argument against Littlewood conjecture, though mistakes are certainly possible. Littlewood conjecture states that for any two real ...
11
votes
4answers
579 views

Zeros of the derivative of Riemann's $\xi$-function

The Riemann xi function $\xi(s)$ is defined as $$ \xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s). $$ It is an entire function whose zeros are precisely those of $\zeta(s)$. Since $\xi$ is real ...
17
votes
10answers
3k views

New proofs to major theorems leading to new insights and results?

I am wondering, historically, when has a new proof of an old theorem been particularly fruitful. A few examples I have in mind (all number theoretic) are: First example is classical... which is ...