Prime numbers, Diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
7
votes
0answers
119 views
Number of representations of an integer as an (arbitrary) sum of products
If $n$ is a positive integer, let $r(n)$ denote the number of representations of $n$ as a sum of products of pairs of positive integers. (Here, the order of the terms in the sum does not matter, but ...
0
votes
0answers
130 views
Divisibility by Euler's Totient Function
Let $n$ be an odd composite number $(>1)$ and $p$ be an odd prime such that:
$p\nmid n$,
$\phi(n)\nmid (n-1)$,
For some even part of $\phi(n)$ (say, $\phi(n_0)$), $\phi(n_0)|(n-1)$,
...
0
votes
1answer
115 views
Sharpening a bound on $\zeta'(s)$
I want to find an upper bound for $\zeta'(s)$ along a vertical line $\Re(s)=b$, where $-1<b<0$.
One way to do this is using $$\frac{\zeta'(b+iT)}{\zeta(b+iT)}=O_b(\log T)$$ and ...
4
votes
0answers
113 views
Angular equidistribution of lattice points on circles
The following question is well known:
Consider representations of a given integer as sums of two squares, i.e. solutions to $a^2 + b^2 = n$ in $a,b\in\mathbb Z$ with $n$ fixed. As $n \to \infty$, ...
3
votes
1answer
147 views
Fourier expansion of Takagi-function (everywhere non differentiable function).
Let us consider Takagi-function defined by
$T(x) \colon\!= \sum_{n=0}^{\infty}s(2^nx)/2^n$,
where $s(x) \colon\!\!= \underset{n \in {\Bbb Z}}{\mathrm{min}} \,|x-n|$.
$T(x)$ has its period $1$, so ...
0
votes
0answers
86 views
Counting function for sums of three squares [migrated]
Legendre showed that an integer is the sum of three squares if and only if it is not of the form $4^n(8m + 7)$ for some nonnegative integers $n$ and $m$. However, I have been unable to find any ...
5
votes
1answer
416 views
Main conjecture for elliptic curves
Suppose that $E$ is an elliptic curve defined over $\mathbb{Q}$, and that $p$ is a prime where $E$ has good ordinary reduction. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ ...
0
votes
0answers
75 views
upper bound for an incomplete exponential sum
Let $q$ be a large prime and $\delta\in(0,1)$. Let $k$ be an integer which is not a multiple of $q$. Define $e(x)=e^{-2\pi ix}$. Can we get any non-trivial upper bounds for ...
-3
votes
0answers
42 views
Generators of Ideals in Integers in a Number Field [migrated]
Let $R$ be the ring of integers in a number field $K$. It is known that each ideal of $R$ can be generated by two elements. In fact if $I$ is an ideal of $R$ and $a\in I$ is a nonzero element, then ...
3
votes
0answers
58 views
Congruences of ternary forms
Suppose $f(x,y,z) \in \mathbb{Z}[x,y,z]$ is an irreducible homogeneous polynomial, or a ternary form, of degree $d \geq 2$. Define
$$\displaystyle \rho_f(m) = \# \{(x,y,z) \in \{0, 1, \cdots, m-1\}^3 ...
4
votes
0answers
121 views
p-adic valuation of a sum
Let $n_1$, ..., $n_k$ denote positive integers, and let us write
$$ n_i=\prod_{j=1}^m p_j^{\alpha_{ij}}$$
for $1\le i\le k$, where the $p_j$'s are distinct prime numbers, and $\alpha_{ij}\ge 0$ for ...
0
votes
0answers
73 views
Automorphisms of $\mathbb{C}$ and L-function
I already asked this question on MSE but got no answer. Suppose $\sigma$ is an automorphism of $\mathbb{C}$ such that there is an element $F$ of the Selberg class such that $\sigma\circ ...
5
votes
2answers
206 views
Sum of digits of repeating end of reciprocal of prime over period is $\frac{9}{2}$
Take a prime other than 2,3 or 5 and look at the part of it that repeats in base 10. Is it true that the sum of the digits in the end divided by the period(number of repeated digits id always ...
-2
votes
0answers
38 views
${p^km \choose p^k} \equiv m \pmod p$ [migrated]
Let $p$ be a prime number and $m, k$ two positive integers. Then ${p^km \choose p^k} \equiv m \pmod p$.
I've been trying to demonstrate this lemme all the day. Have you got any suggestion? Thank you ...
-6
votes
1answer
97 views
Given an arbitrary composite odd integer $N$, find two integers $P$ and $Q$ such that $P-Q \neq 1$ and $N=P^2-Q^2$ [on hold]
Given an arbitrary composite odd integer $N$, find two integers $P$ and $Q$ such that:
$P-Q \neq 1$ and $N=P^2-Q^2$
I am assuming that the best known solution to this problem runs at $O(2^{|N|})$.
...
2
votes
3answers
241 views
Does this 'alternating' Euler product converge for all $\Re(s) > 0$?
Does the following 'alternating' Euler product, with $p_n$ the $n$-th prime number, converge for $\Re(s)>0$ ?
$$\displaystyle \prod_{n=1}^\infty \left( \dfrac{1}{1-\frac{1}{p_{n}^{s}}} ...
0
votes
3answers
489 views
Definition of Prime Numbers [duplicate]
The first time I heard of prime numbers, they were defined as natural numbers $n$ that can only be divided by 1 and themselves without remainder; later, when prime factorization was introduced, I ...
0
votes
0answers
86 views
Consistency of the u-invariant under field extension
A algebraic field extension L/k induces of homomorphism between the Wittrings. We get
$\phi: W(k) -> W(L)$. If every anisotropic isometry class of $W(k)$ stays anisotropic, the kernel of $\phi$ ...
3
votes
1answer
136 views
operations on ideals in a subring of number field
For three ideals $I, J$ and $K$ of a subring $R$ in a number field $L$,
does this equality hold in general?
$(I+J) \cap K = (I \cap K) + (J \cap K)$
I have no counterexample yet but I couldn't prove ...
1
vote
0answers
49 views
Irreducibility of reflexive compositions of polynomials (II)
Given an unknown polynomial $p \in \mathbb{Z}[x]$, write on the blackboard a finite expression that defines a polynomial $q \in \mathbb{Z}[x]$ by using only constants $1, x, p$ of the ring ...
10
votes
1answer
249 views
Is $\lfloor \log(n!)\rfloor \alpha$ equidistributed on the unit circle?
In this question $\lfloor a\rfloor$ means the greatest integer not exceeding $a$.
Using van der Corput's inequalities one is able to show that $\log(n!)\alpha$ is equidistributed on the unit circle ...
0
votes
1answer
94 views
Arithmetic progression and most significant digits in different bases
Given a number $x \geq 3$, let $b(x) \in \{0,1\}$ be the second most significant digit (bit) of its binary representation, and $t(x)\in \{1,2\}$ the most significant digit of its ternary ...
-4
votes
0answers
109 views
Exact Prime Counting Function [closed]
There have been dozens of prime counting functions $\pi(n)$ that have been created over the years and yet the most common ones I see are either estimates (prime number theorem) or they require the ...
2
votes
0answers
139 views
square-free parts of values of polynomials
Given a polynomial $f(x) \in \mathbb{Z}[x]$ of degree $d$, consider the following three sets:
$$N_1(x) = \#\{k \leq x: f(k) \text{ is square-free}\}$$
$$N_2(x) = \#\{n \leq x: n = f(k) \text{ is ...
8
votes
0answers
158 views
Connections between Riemann hypothesis for curves over finite fields and Ramanujan property for graphs
this question relates to the beautiful construction of expander graphs using Cayley graphs of $PGL_2(\mathbb{F}_q)$, as exposited by Davidoff-Sarnak-Valette in their book, Elementary Number Theory, ...
0
votes
0answers
77 views
Linear system with many solutions from a finite set
Basically I am looking for a linear system with
many solutions from a finite set.
Choose a finite set of rationals $S$ and fix
positive integer $k$.
Let $A$ be a linear system with $n$ variables ...
7
votes
1answer
343 views
Hamming weight of Fibonacci numbers
The Hamming weight $w(n)$ is the number of 1s in $n$ when written in binary. Is there some effective bound on Fibonacci numbers $F_n$ with $w(F_n)\le x$ for a given $x$?
Clearly only $F_0=0$ has ...
4
votes
1answer
165 views
Asymptotic behavior of $\{\text{#}(a,x,b,y)\in \mathbb{N^4}| \text{ }ax+by=n\}$ for large $n$
For a fixed natural number $n$ is it possible to obtain an asymptotic for the number of solutions $(a,x,b,y)\in\mathbb{N^4}$ to $ax+by=n$ or equivalently an asymptotic for ...
27
votes
0answers
571 views
“Gross-Zagier” formulae outside of number theory
The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer conjecture. It relates the value at $1$ of the ...
19
votes
1answer
697 views
Is the Modularity Theorem (currently) effective?
The Modularity Theorem says every elliptic curve over $\mathbb{Q}$ can be gotten from the classic modular curve $X_0(N)$ by a rational map. Here $N$ is the conductor, easily calculable from a ...
7
votes
2answers
301 views
Forms of algebraic varieties
Let $X$ be an algebraic variety (say, projective, irreducible and smooth), defined over a field $K$, and let $L$ be a Galois extension. I am interested in algebraic varieties $Y$, defined over $K$, ...
13
votes
2answers
548 views
Groups which are only defined up to conjugation
I'm trying to understand what the right way is to think about "groups which are only well-defined up to conjugation." Since this is somewhat vague let me clarify it by pointing out the main examples ...
5
votes
0answers
132 views
Irrationality of the sum of the reciprocal of perfect powers
A couple of days ago I was trying to remember a classical exercise (which I now find out goes by the name of Goldbach-Euler theorem). Eventually I figured out that it asked to prove that ...
9
votes
3answers
1k views
An algebraic number is not a root of unity?
This problem is related to my study of the Burau representation of the braid group $B_3$: I was trying to show that certain "congruence subgroups" are of infinite index.
There is an approach that ...
3
votes
1answer
150 views
Higher dimensional analogue of Thue's equation
The classical Thue equation is
$$\displaystyle F(x,y) = h,$$
for a binary form $F(x,y) \in \mathbb{Z}[x,y]$. Recall that a binary form is a polynomial in two variables which is homogeneous, and $h$ ...
1
vote
0answers
61 views
Question about the zeros of the sum/difference of two finite Euler products
The conjecture Are all zeros of $\zeta(0+s) \pm \zeta(0-s)$ except a finite few on the line $\Re(s)=0$? was shown to be unconditionally true.
The proof can even be extended towards the domain ...
1
vote
0answers
34 views
Would countability conjecture and degree conjecture imply unique factorization for the Selberg class?
I already asked this question on MSE but didn't get any comment or answer, so I ask it here.
Assuming countability conjecture as stated in ...
2
votes
1answer
88 views
Solutions of $rad(\sigma(m))=2rad(m)$
For $m$ a positive integer greater than $1$, let $rad(m)$ be the product of all distinct primes dividing $m$. If $n$ is an odd perfect number (conjectured not to exist), one would have $\sigma(n)=2n$, ...
5
votes
1answer
218 views
Open subgroups of the etale fundamental group of $P^1_\mathbb Q\setminus\{0,\infty\}$
Let $G$ be the etale fundamental group of $P^1_\mathbb Q\setminus\{0,\infty\}$. Then $G$ is isomorphic to a semidirect product of $\widehat {\mathbb Z}(1)$ by $ Gal_\mathbb Q$.
Is it true that ...
2
votes
1answer
198 views
on the prime divisors of $(p^2+1)/2 $
The following question is equivalent to a problem in group theory.
Let $ p > 13$ be a prime number distinct from 239. Let $ a=(p^2+1)/2 $. Is there any prime divisor $r$ of $a$ such that $r\mid ...
2
votes
2answers
193 views
What is the minimal range $[f(n),g(n)]$ that contains a prime number for every integer $n>0$?
I know the following:
Proven: There is a prime number between $n$ and $2n$ for every integer $n>0$.
Conjectured: There is a prime number between $n^2$ and $(n+1)^2$ for every integer $n>0$.
...
2
votes
2answers
134 views
injective implies completion injective?
Background and definitions.
Let $k$ denote a field complete with respect to a non-trivial non-archimedean norm. Let $R$ be the integers in $k$, and say $\pi\in R$ with $0<|\pi|<1$ ($\pi$ ...
33
votes
2answers
669 views
A curious identity related to finite fields
To three elements $a_1$, $a_2$, $a_3$ in the finite field $\mathbb F_q$
of $q$ elements we associate the number $N(a_1,a_2,a_3)$
of elements $a_0\in \mathbb F_q$ such that the polynomial
...
10
votes
1answer
252 views
Degeneration of the Hodge spectral sequence for scheme over truncated Witt ring?
Let $f:X\to S$ be a smooth proper morphism of schemes. If $S$ is of characteristic zero (i.e., $S$ is a $\mathbb{Q}$-scheme), then Deligne has shown:
The Hodge-De Rham spectral sequence ...
10
votes
1answer
388 views
Examples of elliptic curves over $\mathbb{Q}$
I need examples of two non-isogenous elliptic curves $E_{1}, E_{2}$ over $\mathbb{Q}$ having the following 2 properties -
1) $E_{1}, E_{2}$ have no rational torsion points.
2) $E_1[9] \cong E_2[9]$ ...
5
votes
1answer
266 views
Integer numbers of the form $m = x^n + y^n$
First of all, I am no number theorist, so this question may be a little dummy.
The two squares theorem imply that $m = x^2 + y^2$ for some (possible zero) integer numbers $x,y$ iff $m$ factors as $m ...
1
vote
1answer
100 views
A special curve with points of order 3(or 6)
Could you please tell me what is the points of order 3 (or 6) Hon the elliptic curve $y^2=x^3+sx^2-x$ where
$$s = -\frac{1}{432}\frac{(81k^8-2592k^4-6912)}{k^6}$$ and $k$ is rational?
20
votes
3answers
601 views
Consecutive square values of cubic polynomials
Let $P(x)$ be a cubic polynomial with integer coefficients. Does there exist a constant $c$ such that at least one of the following values $P(0),P(1),...,P(c)$ is not a square?
It is known that the ...
6
votes
0answers
78 views
Joint representation of the semi-direct product of the metaplectic group and Heisenberg group
Given a symplectic space $W$ over a local field $F$ and a additive character $\psi$ of $F$, we can construct the Weil representation $\omega_\psi$, which can be viewed as a representation of the ...
10
votes
0answers
251 views
Unramified Galois cohomology of number fields
I'm trying to understand unramified Galois cohomology of number fields a bit better.
Set-up: Let $k$ be a number field and $S$ a finite set of places of $k$ which contains all the archimedean ...
