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

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7
votes
0answers
48 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. ...
0
votes
0answers
21 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 ...
0
votes
0answers
35 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
2answers
180 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 ...
0
votes
0answers
48 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 $d_z(n)$ ...
1
vote
1answer
102 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 ...
0
votes
2answers
564 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 ...
-2
votes
1answer
123 views

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

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 $$\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2}).$$ ...
-4
votes
0answers
45 views

Height of a tripod [on hold]

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 ...
3
votes
1answer
134 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 ...
4
votes
1answer
149 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 ...
5
votes
1answer
132 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 ...
0
votes
0answers
104 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
334 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 ...
6
votes
1answer
229 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
votes
1answer
407 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 ...
2
votes
0answers
58 views

Varieties with few monomials and the n-conjecture

The n-conjecture is a generalization of abc and basically says that the if $a_1 + \ldots + a_n=0$, no proper subsum vanishes and $a_i$ are coprime, then the radical of $a_1\cdots a_n$ can't be too ...
6
votes
1answer
190 views

p-adic Stein spaces

The higher cohomology of coherent sheaves vanish on Stein spaces (both complex and p-adic). In the case when the space ($X$) is a curve and we're working in the complex world, this shows that all ...
2
votes
1answer
82 views

Does the Lehmer quintic parameterize certain minimal polynomials of the $p$th root of unity for infinitely many $p$?

The solvable Emma Lehmer quintic is given by, $$F(y) = y^5 + n^2y^4 - (2n^3 + 6n^2 + 10n + 10)y^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)y^2 + (n^3 + 4n^2 + 10n + 10)y + 1 = 0$$ with discriminant $D = (7 + ...
5
votes
0answers
438 views
+100

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 ...
7
votes
2answers
306 views

Least supersingular prime

Given an elliptic curve over the rationals, what can one say about the size of the smallest supersingular prime?
11
votes
4answers
561 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 ...
7
votes
2answers
300 views

Numbers with all N-digit prefixes divisible by N

In base 10, the number 3816547290 contains every digit exactly once. When I take the first N digits, that substring is divisible by N. For example, 381 is divisible by 3, 38165 is divisible by 5, etc. ...
0
votes
1answer
110 views

Short arithmetic progressions in quadratic residues

Let p be a prime number of form 4k+1. I guess that there are c(d) number of 3-terms arithmetic progressions (AP) in the set of quadratic residues modulo p, where c(d) is an integer constant depending ...
-2
votes
2answers
161 views

Time estimate to determine if a number is prime [closed]

How long does it take to verify that a given number is a prime number, as a function of its number of digits, in a personal computer, say? How computationally hard is this?
11
votes
1answer
473 views

Is there a known primitive recursive upper bound on the nth “Zhang prime”

(This question is pure curiosity. Feel free to close it if you feel it is not appropriate for mathoverflow.) In 2013 Zhang showed that there are infinitely many pairs of primes which are less that ...
3
votes
1answer
213 views

“Weight-mondoromy” for open varieties

Suppose that $X/\mathbb{Q}_p$ is a smooth, projective variety, and choose a prime $\ell\neq p$. Then the weight-monodromy conjecture says that the graded pieces $\mathrm{Gr}_k^M$ of the monodromy ...
2
votes
1answer
138 views

Special values of Hecke L-function

The Dedekind zeta function for a number field $K$ is defined as $\zeta_K(s)=\sum_{I\subset O_K} (N_{K/\mathbb{Q}}(I))^{-s}$. By attaching a Hecke character $\psi$, we can define ...
3
votes
1answer
121 views

How to generate $n$ FP32 rationals s.t. no two distinct k-el. subsets have same sum?

First some Background: I have lots and lots of integer matrices, whose rows are $k$-combinations (without repetitions and sorted) of numbers from the set $S:=\{1,...,n\}$ and needed to be compared ...
3
votes
1answer
233 views

Waring's problem

What is the best known quantitative upper bound for the quantity $G(k)$? I know that it's due to Trevor Wooley and in simplest form states that $\limsup_{k \to \infty} \frac{G(k)}{k \log k} \le 1$. ...
3
votes
1answer
195 views

Representation of GL(n, F_p) over F_p, for n small

The question is related to this post Representation theory of the general linear group over a finite prime field However, I am asking for more detailed references for n small, for example, for n=2, ...
0
votes
0answers
89 views

Questions on roots of integral polynomials over $\mathbb{F}_p$

I asked earlier (A question on how polynomials split over $\mathbb{F}_p$ for large primes $p$) on when a polynomial $f(x) \in \mathbb{Z}[x]$ has a small root over $\mathbb{F}_p$, where $p$ is a large ...
11
votes
4answers
449 views

How to calculate the infinite sum of this double series?

I'm calculating this double sum: $$ \sum _{m=1}^{\infty } \sum _{k=0}^{\infty } \frac{(-1)^m}{(2 k+1)^2+m^2} $$ I know the answer is $$ \frac{ \pi \log (2)}{16}-\frac{\pi ^2}{16} $$ which can be ...
0
votes
0answers
59 views

Does this quaternary quartic form primitively represent infinitely many sufficiently large powers?

Let $g(x,y,z,t)=(x+y+z+t)^4-a h(x,y)$ where $h(x,y) \in \{x^4,xy^3,x^2y^2\}$ and $a$ is integer. Does $g$ represent infinitely many powers $r^n$ with $n > 4$, $x+y+z+t,ah(x,y)$ take distinct ...
24
votes
3answers
727 views

Why do Pell equations appear in Ramanujan's pi formulas?

While answering this MSE question about the Pell equation $x^2-29y^2=1$, I noticed that certain fundamental solutions appeared in Ramanujan's famous pi formula. I. Given the fundamental unit, ...
7
votes
1answer
539 views

Roots of a polynomial in a finite field related to Fermat's Last Theorem

In my class, we proved the following condition: define the polynomial $P_l(x)$ as $$P_l(x) = \sum_{r=1}^{l-1}{\frac{1}{r}x^{l-1-r}}$$ Then if for all $a \in \mathbb{Z}/l\mathbb{Z}-\{0,1\},$ ...
2
votes
2answers
220 views

Large solutions to Thue equations

Suppose that $f(x,y) \in \mathbb{Z}[x,y]$ is a homogeneous polynomial, or binary form, of degree $d$. The equation $$f(x,y) = h$$ for a given integer $h$ is known as Thue's equation (so named ...
23
votes
1answer
901 views

Which natural numbers are a square minus a sum of two squares?

Question: Which natural numbers are of the form $a^2 - b^2 - c^2$ with $a>b+c$? This question came up in (Eike Hertel, Christian Richter, Tiling Convex Polygons with Congruent Equilateral ...
1
vote
1answer
222 views

A question on how polynomials split over $\mathbb{F}_p$ for large primes $p$

Suppose that $f(x) \in \mathbb{Z}[x]$ is an irreducible polynomial (over $\mathbb{Q}$). Let $p$ be a very large prime with respect to the coefficients of $f$. Then it is possible that $f(x)$ may ...
0
votes
0answers
114 views

Is the Jacobi theta function invertible?

Let $\theta$ denote the Jacobi theta function: $$\theta=\sum_{k=0}^{\infty}{(-1)^kq^{k(k+1)}sin((2k+1)\frac{2\pi}{\omega_1}Re(z))},$$ and we have a complex number $t$. Suppose that we know there ...
3
votes
0answers
89 views

Zeta functions with Brauer class

In algebraic geometry there are examples when a variety $X$ is somehow related (I call it double-mirror) to another variety $Y$, together with a 2-torsion Brauer class. By "related" I mean statements ...
6
votes
1answer
234 views

Euler's Triangular Number closure properties

Burton, in "Elementary Number Theory", states that the following problems are due to Euler 1775: If $n$ is a triangular number, then so are $9n+1$, $25n+3$ and $49n + 6$. R. F. Jordan in the J. ...
3
votes
1answer
199 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
0answers
128 views

The divisors of $p-1$ and high-degree residues modulo $p$

Here is a somewhat more explicit version of a question that I asked a while ago. Suppose that $p$ is a prime of the form $p=2n(n+1)+1$, with a positive integer $n$. Can every odd prime divisor of ...
8
votes
1answer
271 views

minimizing an integral over integer-coefficient polynomials $\displaystyle \inf_{f \in \mathbb{Z}[x]} \int_a^b f(x)^2 \, dx $ [duplicate]

Let's consider the space $L^2[a,b]$ of functions on the interval and the norm: $$ ||f(x)||^2 = \int_a^b |f(x)|^2 \, dx $$ Now what if we consider only polynomials with integer coefficients: $f(x) ...
-1
votes
0answers
69 views

How do we usually calculate the size of the 2-division field of an elliptic curve over local fields

Let $E$ be an elliptic curve over $\mathbb Q$, with rank $0$ and $E(\mathbb Q)[2]=0$. Let $p$ be a prime number. How do we usually calculate $\# E(\mathbb Q_p)[2]$ when $E$ has good, bad reduction at ...
0
votes
0answers
68 views

Lucasian Primality Criterion for Specific Class of $k \cdot 2^n-1$

Definition Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$ , where $m$ and $x$ are nonnegative integers . Conjecture Let $N=k\cdot 2^n-1$ such ...
2
votes
2answers
120 views

Number of *distinct* dot products of an integer vector by elements of a hyper-rectangle

Imagine a vector $\boldsymbol{v}$ composed of integers, and the set $S$ of all integer vectors within a hyper-rectange, with one corner at the origin and other at $\boldsymbol{m}$. In other words: $S ...
6
votes
0answers
156 views

Can these two proofs of the parametrization of pythagorean triples be unified?

I am interested in the classical parametrization of rational solutions to $x^2 + y^2 = 1$. One proof is the classical stereographic projection technique (see, e.g., here): Choose a rational point $P$ ...