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

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0
votes
1answer
63 views

Expression and growth bound for $r_{p^m,k}(n)$

Let's define , $$R_{p^m,k}(n)=\#\{(a_1,\dots,a_k)\in\mathbb{Z}^k:\sum_{i=1}^ka_i^2\le n \ \text{and} \ p^m|\sum_{i=1}^ka_i^2\}$$ what will be growth bound of $R_{p^m,k}(n)$? This can be thought as a ...
4
votes
1answer
75 views

Inequality due to Siegel (assumptions) and upper bounds on number field discriminants

In Siegel's 1969 paper, Abschätzung von Einheiten, on page 73, he states the inequality $$\log\sqrt d\le n-1+{n\over 2}\log\pi+r_2\log 2\qquad (*)$$ and compares with the bound due to Minkowski that ...
0
votes
0answers
32 views

Bounding the absolute value of a polynomial involving a Diophantine equation

Let $\mathbf{z}\in\mathbb{C}^n$ with entries $z_1,z_2,\ldots,z_n$. I would like to bound the following quantity \begin{equation*} ...
-5
votes
0answers
62 views

Proof that $\sqrt2$ is irrational [on hold]

Question: Using fundamental theorem of integers and the fact that every natural number that is not prime, prove that $\sqrt{2}$ is irrational unless $n=m^2$ for some $m\in\mathbb N$. Here is how I ...
0
votes
1answer
72 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 ...
-1
votes
0answers
54 views

Proof that at least one of the nontrivial zeta zeroes has an irrational height (assuming RH) [on hold]

This seems quite simple so its likely someone has done this before (a few Google searches returned empty and I would be really grateful for a relevant link), but in case it's new, I wanted to check if ...
1
vote
1answer
80 views

Strong divisibility of Lucas sequences

Let $a$ and $b$ be relatively prime integers and let $u_n$ be their associate Lucas sequence, i.e., the second order linear recurrence sequence satisfying $u_0 = 0$, $u_1 = 1$ and $u_{n+2} = au_{n+1} ...
0
votes
0answers
45 views

Bounding Random Quadratic Gauss sums

I'm interested in seeing whether the following is true. Assume $u$ is uniform on $[0,1]$ and $|\epsilon_k|=1$ for all $k=1,2,\ldots,n$. We have \begin{align*} ...
0
votes
0answers
41 views

Why the Szpiro conjecture over number fields doesn't depend on the discriminant of the number field?

Szpiro's conjecture states that the Szpiro ratio is: $$ \sigma_{E/K}=\frac{\log{|N_{K/Q}\Delta_{E/K}|}}{\log{|N_{K/Q} f_{E/K}}|}$$ Given $ \varepsilon >0$ there are only finitely many $ E/K$ with ...
3
votes
1answer
77 views

Rank one (phi,Gamma)-modules

Let $\mathbb{Q}_p$ be the field of $p$-adic numbers. Consider an unramified representation $\rho : Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p) \to \mathbb{F}_p^{\times}$ which sends the arithmetic ...
1
vote
0answers
132 views

Davenport's proof that almost all integers are the sum of 4 cubes

Where can I find a pdf that describes Davenport's proof that almost all integers are the sum of $4$ cubes?
2
votes
2answers
265 views

Is there a von Koch-type theorem for the generalized Riemann hypothesis?

Helge von Koch proved in 1901 that the Riemann hypothesis is equivalent to the error term in the prime number theorem having the bound $$ \mid\pi(x)-\textrm{li}(x)\mid=O(\sqrt{x} \log x). $$ Q1: ...
4
votes
1answer
158 views

transcendence of beta values

(1) Can anybody suggest a readable reference for Schneider's theorem that the number $$ \beta(a, b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} $$ is transcendental for $a, b \in \mathbb{Q}$ such that none ...
0
votes
2answers
254 views

What are all positive integers n for which the congruence $a^{n+1} \equiv a (mod n)$ holds? [on hold]

Fermat's little theorem says that the congruence $a^p \equiv a (mod p)$ if $p$ is a prime number. $a^{n+1} \equiv a (mod n)$ works for all integers $a$ and some positive integers $n$, how can we ...
0
votes
0answers
56 views

Are major arcs always around a fraction with small denominator? [on hold]

In the usual circle method we might have a trigonometric polynomial $F(\theta)=\sum_{n}a_n e(n\theta)$ and we need to estimate the integral $\int_0^1 F(\theta)d\theta$ by breaking the domain into ...
2
votes
0answers
57 views

p-adic height on CM abelian varieties

Let A be a CM abelian variety over a number field and p a prime of ordinary reduction. Is the p-adic height of a non-torsion point on A over a p-adic field non-zero? When A is an elliptic curve, this ...
6
votes
2answers
208 views

Holomorphic Hoffstein-Lockhart

In the article Hoffstein, Jeffrey; Lockhart, Paul "Coefficients of Maass forms and the Siegel zero." Ann. of Math. (2) 140 (1994), no. 1, 161–181, it is stablished a good bound for the Petersson norm ...
18
votes
1answer
325 views

Functions $f$ on $\mathbb{Z}/N\mathbb{Z}$ with $|f|$ and $|\widehat{f}|$ constant

Let $N$ be a positive integer; for simplicity I'm happy to assume it's an odd prime but I'm interested in the general case too. Let $f \colon \mathbb{Z}/N\mathbb{Z} \to \mathbb{C}$ and let ...
-1
votes
0answers
47 views

An isogeny from a split algebraic torus [migrated]

Suppose that there is an isogeny (in the category of commutative algebraic groups) from a split algebraic torus to a semi-abelian variety. Does it follows that this semi-abelian variety is also an ...
30
votes
2answers
649 views

Does iterating a certain function related to the sums of divisors eventually always result in a prime value?

Let define the following function for integers (from 2): $f(x)=\sigma(x)-1$, where $\sigma$ is the sum of the divisors of $x$. For example $f(6)=6+3+2=11$, $f(5)=5$. Note that $x$ is a fixed point for ...
0
votes
0answers
80 views

Squarefree Parts of Mersenne Numbers with prime exponent [on hold]

The $n$-th Mersenne number is $M_n=2^n−1$. Write $M_n=a_n b^2_n$ where $a_n$ is positive and squarefree. In the discussion Squarefree Parts of Mersenne Numbers , the lower bound of $a_n$ has been ...
-7
votes
0answers
149 views

Does $\pi$ encode the prime numbers? [on hold]

I have a question regarding whether or not $\pi$ encodes the sequence of primer numbers. It is common knowledge that $$ \zeta (2) = \sum_{i = 1}^{\infty} \frac{1}{n^2} = \prod_{p \in \mathbb{P}} ...
10
votes
1answer
1k views

Go I Know Not Whither and Fetch I Know Not What

Next day: apparently my original question is harder, by far, than the other bits. So: it is a finite check, I was able to confirm by computer that, if the polynomial below satisfies $$ f(a,b,c,d) ...
1
vote
0answers
44 views

square tiled surfaces: Counting Saddle Connections vs Counting Square-Tiled Surfaces

I am reading Prime arithmetic Teichmuller discs in H(2) where they discuss the Siegel-Veech constant in a stratum $\mathcal{H}(2)$ of surfaces. However I see two definitions of Siegel-Veech constant: ...
4
votes
0answers
199 views

$x^2+1$ attaining almost prime values

Iwaniec, using the linear sieve, proved that $n^2+1$ can be a product of at most two primes infinitely often and furthermore a lower bound of the correct order of magnitude for the number of such ...
5
votes
1answer
267 views

Diophantine equation

I would like to ask the broad community what is known about the solutions of diophantine equation $$\frac{u}{v} +\frac{v}{w} +\frac{w}{u} =t$$ where $t,v,u,w\in \mathbb{N}.$ I read a book of W. ...
9
votes
1answer
401 views

Is it easy to prove that $\sum_n |X(\mathbb{F}_{q^n})| t^n$ is rational?

Background: Let $X$ be an algebraic variety over a finite field $\mathbb{F}_q$. One of the successes of Etale cohomology - previously achieved by Dwork- was proving the rationality of the Zeta ...
15
votes
3answers
490 views

Representing a number close to 1 with a sum of reciprocals of natural numbers

For positive integers $n_1, \ldots, n_k$, let $H(n_1, \ldots, n_k)$ denote $1/n_1 + \ldots + 1/n_k$. Let $V(N)$ be the largest possible value of $H(n_1, \ldots, n_k)$ that is less than 1, subject to ...
2
votes
1answer
134 views

Growth of $r_k(n)$

What is the best known growth bound of $r_k(n)$, where $$r_k(n)=\#\{(a_1,\dots,a_k\in\mathbb{Z}^k:\sum_{i=1}^ka_i^2=n\}?$$ Please provide some reference if known. Thanks.
13
votes
1answer
707 views

Wrong asymptotics of OEIS A000607?

Sequence A000607 in the Online Encyclopedia of Integer Sequences is the number of partitions of $n$ into prime parts. For example, there are $5$ partitions of $10$ into prime parts: $10 = 2 + 2 + 2 + ...
2
votes
0answers
95 views

quasi-split algebraic group [migrated]

While reading papers, there usually an assumption "quasi-split" for reductive algebraic groups. To use their results I need to know which groups are quasi-split. For the case I am interested in ...
3
votes
1answer
224 views

Theorem 7b of Serre's “Propriétés galoisiennes des points d'ordre fini des courbes elliptiques”

Could someone please point me towards a proof of the statement in the second paragraph, in the proof of Theorem 7b of Serre's Propriétés galoisiennes...? The statement is as follows: Let $F$ and $F'$ ...
0
votes
1answer
109 views

sum over primes involving divisor function (variation of the Titchmarsh divisor problem)

This question was also asked on MSE. Does there exist an asymptotic estimate for the following sum over primes $$ \sum_{p\leq x} \frac{\tau(p-1)}{p}\;, $$ where $\tau(n)=\sum_{d|n}1$ is the divisor ...
2
votes
0answers
214 views

Periodicity with irrational numbers [migrated]

Recently, I invented the following theorem and found a proof, it seems strange since it is very counter-intuitive to me. The proof is long and non-conceptual. Is there a place or a branch of math ...
8
votes
2answers
455 views

Prove that the Dirichlet eta function is monotonic

Let us consider $\eta(p):= \sum\limits_{n=1}^\infty \frac{(-1)^{n+1}}{n^p}$ for $p>0$. Has anyone come along with an elementary proof that $\eta(x)$ is monotonically increasing on this set? By ...
2
votes
0answers
103 views

State-of-the-art for the descent principle in relation to surfaces over a number field

I'll start with some motivating remarks (edit: as pointed out in the comments, these motivational remarks do not hold for surfaces: there is an example of a conic bundle surface over a real quadratic ...
6
votes
1answer
308 views

Is Gauss sum a p-adic measure?

Let $\Gamma$ be Galois group of cyclotomic $\mathbb{Z}_p$ extension over $\mathbb{Q}$. Consider the function $G$ which sends each finite order character $\chi$ of $\Gamma$ to the Gauss sum $G(\chi)$, ...
3
votes
0answers
121 views

Subgroup cliques in the Paley graph

It is a famous open problem to estimate non-trivially, for a prime $p\equiv 1\pmod 4$, the largest size of a subset $A\subset{\mathbb F}_p$ such that the difference of any two elements of $A$ is a ...
-3
votes
0answers
82 views

Tripet prime reciprocals series [closed]

Does any body know if the series of reciprocals of triplet primes of form p,p+2,p+6 or p, p+4,p+6 converges or diverges. Could this be used as a proof of infinity of twin primes
1
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0answers
33 views

All all hypo-multiplicative functions linear combinations of quasi-multiplicative functions?

A function is called quasi-multiplicative by many authors if $f(m)f(n)=f(1)f(mn)$, a slight generalization of multiplicativity. (Basically a multiplicative function times a constant is a ...
4
votes
0answers
161 views

What is the density of the reciprocal of the set of cubes?

In his MathOverflow question "How thick is the reciprocal of the squares?" Kevin O'Bryant asks if a certain set, the reciprocal of the set of squares (identifying sets with power series in ...
0
votes
1answer
274 views

Covering a finite subset of $\mathbb{N}$ with prime arithmetic progressions

Because of a problem I ran into I am trying to get a quick start in covering with arithmetic progressions. First I want to say I am aware of this previously asked question: Covering $\mathbb{N}$ with ...
-1
votes
1answer
164 views

Squarefree numbers with all digits equal 1 [closed]

Is this true that numbers of the form $$\frac{10^k -1}{9}$$ are squarefree for all values of $k\in\mathbb{N}$?
6
votes
2answers
262 views

Field of definition of Galois representations of weight 1 modular forms

Let $f$ be a weight 1 modular form (let's say cuspidal, new, normalized, and a Hecke eigenform). Then there's an associated Artin representation $\rho_f: \operatorname{Gal}(\overline{\mathbf{Q}} / ...
1
vote
2answers
158 views

Looking for a reference for a paper by Mordell

On page 384 of the book "Number Theory:Volume 1:tools and Diophantine Equations" by Henri Cohen there is reference to the fact that: "It has been proved by Schinzel, Mordell nd successors that such an ...
3
votes
0answers
103 views

multiplicity of automorphic representation of unitary similitude group

Let $G$ be a unitary similitude group over $\mathbb{Q}$ (as in the book of Harris-Taylor), $\pi$ an irreducible automorphic representation of $G(\mathbb{A})$. I'm looking for some results on its ...
1
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0answers
147 views

Is it trivial to get $200$ algebraic abc triples of equal quality $1.6978…$ over isomorphic number fields?

Got $200$ algebraic abc triples over distinct though isomorphic number fields of equal quality $1.6978...$ Strongly suspect I can get as many as I like (assuming the computations are correct). Is ...
0
votes
1answer
129 views

Hasse principle and twists of $\mathbb{P}^n$ [closed]

Let $X$ be a twist of the $n$-th projective space, seen as a $K$-variety for some number field $K$. For $n = 1$, the Hasse principle holds for $X$. My question is: for which $n >1$ does the ...
4
votes
0answers
94 views

The number of representations of the positive integer $n$ as $a^{2}+b^{2}+p^{2}c^{2}$

Let $n$ be a positive integer and $p$ a prime number. I know that there are formulas by which one can compute the number of representations of $n$ as the sum of two or three squares etc. I would to ...
0
votes
0answers
64 views

Metric defined over Galois extensions of the rationals [duplicate]

I don't know if this of interest, but I'd be curious to know if the following idea has been pursued. In this question (Metric on the set of subsets of the rational primes) I proposed a metric, d, ...