**0**

votes

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**3**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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**

vote

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**2**answers

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

**0**answers

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**

vote

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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, ...