# Tagged Questions

**4**

votes

**0**answers

23 views

### How to compute modular forms of weight one on Shimura curves?

Classical modular forms of weight $k\geq 1$ can be computed explicitly by exhibiting their $q$-expansion. When the weight is at least two, the most standard method uses modular symbols. When the ...

**3**

votes

**0**answers

51 views

### An asymptotic formula involving the $2$-torsion subgroup of the class group of real quadratic fields

Let $R$ be an order in some number field $K$ (not necessarily maximal). Then the class number $\text{Cl}(R)$ is equal to the cardinality of the Picard group of $R$, which is the group of equivalence ...

**1**

vote

**2**answers

184 views

### About consecutive integers covered by arithmetic progressions

Help me please to solve the following problem.
There are $n$ arithmetic progressions of the form:
$$(2i+1)k + x_i,~~~~ i = 1,\ldots,n, k \geq 0$$
Initial integer terms $x_i \geq 0$ are varying.
...

**1**

vote

**2**answers

91 views

### On the automorphism group of binary quadratic forms

This question is a continuation of the following two questions:
Discriminants of indefinite integral binary quadratic forms admitting 3 or 6 torsion.
On certain solutions of a quadratic form ...

**0**

votes

**1**answer

117 views

### Clarification of the proof of the main theorem of the paper of Hulse et al

I am trying to understand some open steps in the following article The Sign of Fourier coefficients of Half-integral Weight Cusp Form by Hulse, Kiral, Kuan, and Lim, I find the following :
Let $f\...

**0**

votes

**1**answer

188 views

### Number of fixed points in Zagier's involution (Fermat's Theorem) [on hold]

Zagier's has found a famous one sentence proof for Fermat's theorem on sums of two squares. It centers on the following involution of the set $S= \lbrace (x,y,z) \in N^3: x^2+4yz=p \rbrace $ having ...

**3**

votes

**0**answers

83 views

### Elementary symmetric functions of reciprocals of monic polynomials in function fields

Let $q$ be a prime power and $\mathbb{F}_q$ the field of cardinality $q$. Let $A = \mathbb{F}_q[T]$ and let $A_+ \subset A$ be the monic polynomials. Choose any ordering $<$ of $A_+$ and let $k$ be ...

**0**

votes

**1**answer

161 views

### Do we know an upper bound for the number of possible real parts of the non trivial zeroes of $\zeta$?

Let $n_{\zeta}$ denote the number of possible real parts for the non trivial zeroes of the Riemann Zeta function. RH is equivalent to $n_{\zeta}=1$, and the symmetry arising from the functional ...

**2**

votes

**1**answer

119 views

### GCD for two Cullen numbers

The $n$'th Cullen number is $C_n = n\cdot2^n+1$.
If $m$ and $n$ are natural numbers, what can one say about $\gcd(C_n,C_m)$, where $m$ and $n$ are different positive integers?

**5**

votes

**1**answer

101 views

### On certain solutions of a quadratic form equation

This is a continuation of this question: A class of quadratic equations
Let $f(x,y) = ax^2 + bxy + cy^2$ be an irreducible and indefinite binary quadratic form. Consider the equation
$$\displaystyle ...

**-1**

votes

**0**answers

58 views

### Applications of computing the averages of arithmetical functions

I often read many papers in which the authors compute the average of certain arithmetical functions like 'On the distribution of the Euler function of shifted smooth numbers' of Shparlinski and al. ...

**21**

votes

**0**answers

637 views

### Applications of arithmetic topology to number theory

There is a well-known analogy between 3-manifolds and number fields, with knots corresponding to prime ideals. Are there any results in number theory that have been proven using topology through this ...

**0**

votes

**0**answers

99 views

### Selmer and free rank of Elliptic Curves

If I am not mistaken, the equality of the $p$-Selmer rank and the free rank of an elliptic curve are conjectured to be equal.
This is one of the many equivalent formulations of the Birch and ...

**2**

votes

**0**answers

103 views

### Are all zeros of $\dfrac{\Gamma'}{\Gamma^2}(s) \pm \dfrac{\Gamma'}{\Gamma^2}(1-s)$ either real or on the line with $\Re(s)=\frac12$?

In this post it was proven that, except for a finite few, all zeros of $\Gamma(s) \pm \Gamma(1-s)$ are either real or reside on the line $\Re(s)=\frac12$.
I have been searching for similar reflexive $...

**6**

votes

**2**answers

300 views

### Simple proof for $\sum_{i=1}^{n} a^{gcd(i,n)} $ is divisible by n

Burnside's Lemma Deduce That:
$$\sum_{i=1}^{n} a^{gcd(i,n)} $$
is divisible by n
it's a beautiful result. but i want to prove it without any abstract algebraic tools such as Burnside's Lemma...
is ...

**5**

votes

**1**answer

392 views

### Uniqueness of sums of roots of unity

Let $\zeta:=e^{\frac{2\pi i}{n}}$, with $n\geq4$, and let $2\leq k\leq n-2$.
Let us suppose that the prime factorization of $n$ is $n=p_1^{\alpha_1}\cdot\dots\cdot p_s^{\alpha_s}$, with $\alpha_i>...

**4**

votes

**1**answer

263 views

+50

### Intuition behind the definition of the Siegel-Eichler transformation

Recently I am reading Wall's paper "On the Orthogonal Groups of Unimodular Quadratic Forms II". In this paper, I encountered with the map $E^1_\omega$, which now I am interested in.
Let $X$ be an ...

**16**

votes

**2**answers

340 views

### Can something finite over $\mathbb{C}(q)$ be a modular form?

If $f\in\mathbf{C}[[q]]$ is non-constant, and algebraic over $\mathbf{C}[q]$ (in the sense that it is a root of a polynomial with coefficients in in $\mathbf{C}[q]$) then can $f$ be the $q$-expansion ...

**3**

votes

**2**answers

183 views

### Examples of Sets with Positive Upper Density

While reading the statement of Roth's theorem I started asking myself what are examples of sets of positive upper density? It's not hard to come up with a few:
Flip a coin with probability $\mathbb{...

**32**

votes

**2**answers

2k views

### What did Yu Jianchun discover about Carmichael numbers?

There's a news story going around (see for example [1]; other accounts are even more breathless) about an amateur mathematician, Yu Jianchun, finding an "alternative method to verify Carmichael ...

**4**

votes

**1**answer

145 views

### Asymptotics on the number of ways to pair off $\{1, 2, \dots, 2n\}$ into primes

Given $S = \{1, 2, \dots, 2n\}$, we can always pair off elements into $n$ pairs such that each sum to a prime. The proof of this fact is easy and follows from Bertrand's postulate.
Now, let $\gamma(n)...

**3**

votes

**0**answers

100 views

### A question about smooth convex lattice polygons

Let $P$ be a smooth convex lattice polygon in $\mathbb{R}^2$ (the lattice being $\mathbb{Z}^2$). Here smooth means that at any vertex of $P$, the two primitive integer vectors (i.e. vectors whose ...

**3**

votes

**0**answers

174 views

### Artin conjecture on L-functions

Artin conjecture on Artin $L$-functions asserts that the Artin $L$-function $L(\rho,s)$ of a non-trivial irreducible representation $\rho$ of the Galois group $\Gamma$ of a number field admits ...

**0**

votes

**0**answers

80 views

### What is the sharpest bound of this sum?

Fix $y\geq 1$ and let $\delta$ be a small enough positive real number. Put
$$\mathcal{D}^{+}=\left\{d=p_{1}...p_{l}: p_{l}<...<p_{1},\ p_{m} \leq y_{m} \ \textrm{for all odd} \ m \right\},$$
...

**1**

vote

**0**answers

66 views

### Eigenvalues of the Laplacian modulo primes

Let $d\geq 1$ be an integer and consider $M = \mathbb{R}^d / \mathbb{Z}^d$. Let $\Delta$ be the Laplace operator on $M$. Then the eigenvalues are $$\frac{1}{4 \pi^2}\lambda_{\vec{k}} = |\vec{k}|^2.$$
...

**6**

votes

**3**answers

368 views

### linear independence of $\sin(k \pi / m)$

I have tried searching the literature for a result like the following, but have not found anything.
For a positive integer $m$, is it known that
$$\{ \sin (k \pi / m): 1 \leq k \leq m/2, (k,m)=1 \}$$
...

**0**

votes

**1**answer

154 views

### Characters and Galois stability

Let $G$ be a finite abelian group and $\widehat{G}$ the character group.
Let $S \subset \widehat{G}$ be a Galois-stable subset i.e. if $\chi \in S$, then the Galois conjugates $\chi^{\sigma} \in S$ ...

**29**

votes

**2**answers

672 views

### $x_1 = 2$, $x_{n + 1} = {{x_n(x_n + 1)}\over2}$, what can we say about $x_n \text{ mod }2$?

This question was asked on MathStackexchange here, but there was no answer, so I am asking it here.
Let$$x_1 = 2, \quad x_{n + 1} = {{x_n(x_n + 1)}\over2}.$$What can we say about the behavior of $x_n ...

**-1**

votes

**1**answer

157 views

### When the Ratio of Two Factors is a power of $2$ (i.e. $\lfloor{\frac{a}{b}}\rfloor = 2^{s-2}$)

We can write, $n!= 2^s \times a \times b$ where $gcd(a,b)=1$ and $2^{s+1} \nmid n!$.
Problem: Is there infinite number of $n$ when $\lfloor{\frac{a}{b}}\rfloor = 2^{s-2}$?
Question:
1.How ...

**2**

votes

**1**answer

173 views

### Kummer extension of Galois modules

Let $k$ be a field of characteristic $p \geq 0$, $n$ an integer prime to $p$, and $x$ an element of $k \setminus \{0, 1\}$. I have read that the $n^{th}$ root of $1-x$ gives rise to a Galois module $E$...

**6**

votes

**2**answers

250 views

### Is there any formula to find number of Pythagorean triplets between two integers 2 and j, j>2?

Given $j \geq 5$, is there a formula for the number of Pythagorean triplets $(a, b, c)$ satisfying the constraint that $a, b, c \leq j$?
There exists at least one Pythagorean triplet for $j\geq5$; ...

**2**

votes

**3**answers

333 views

### Find a distinct postive integer solution to this $xyzw=504(x^2+y^2+z^2+w^2)$ diophantine equation

Following problem though not a research problem
if $x,y,z,w$ are postive integers,and such
$$xyzw=504(x^2+y^2+z^2+w^2)$$
such example $(x,y,z,w)=(21,63,84,84)$ hold,
Now My problem there exist ...

**2**

votes

**1**answer

109 views

### An upper bound for the number of prime numbers in non-linear progressions

Let $f(x)$ be a non-linear polynomial over $\mathbb{Z}$.
Consider the following sum $\pi_f(x):= \#\{y: y< x \text{ and } f(y) \text{ is prime} \}$.
Can we get an upper bound for $\pi_f(x)$?

**9**

votes

**2**answers

523 views

### congruent number problem [closed]

I am studying the congruent number problem
and I heard that there is a paper by Kazuma Morita
which claims to solve this problem from my colleague.
I saw the paper on his homepage but it is very ...

**1**

vote

**0**answers

42 views

### How many rational points on $F(x,y)=m$ for homogeneous $F$?

Let $F \in \mathbb{Q}[x,y]$ be homogeneous of degree $d$ and $m$ is rational.
Assume the curve $C : F(x,y)=m$ is irreducible and of genus greater
than one.
Currently, how many rational points $C$ ...

**0**

votes

**1**answer

171 views

### Prime quadratic non-residue

NC Ankeny showed assuming Riemann Hypothesis the least quadratic non residue( let it be '$r$') modulo some prime $p$ to be $O(\log^2 p)$. It is easy to see that $r$ is a prime.
I have following ...

**5**

votes

**3**answers

192 views

### 2-adic valuation of odd harmonic sums

(This question is cross-posted on math.stackexchange)
I'm playing with p-adic valuations, and find that the odd harmonic sums, $\tilde{H}_k=\sum_{i=1}^{k}\frac{1}{2i-1}$, has 2-adic valuation $||k^2||...

**2**

votes

**0**answers

64 views

### Rank growth in ray class fields of primes that are inert in an imaginary quadratic extension

If $K=\mathbb{Q}[\sqrt{-d}]$, $E$ is an elliptic curve and $\operatorname{rank}(E(K))=\operatorname{rank}(E(\mathbb{Q}))$, are there infinitely many primes $l$ of $\mathbb{Q}$ so that:
(1.) $\...

**4**

votes

**1**answer

288 views

### Coefficients of factors of $x^n-1\in\mathbb{Q}[x]$

If you factor $x^n-1\in\mathbb{Q}[x]$, then for $n\leq 104$ the coefficients of the factors are in $\{-1, 0, 1\}$. (This is not true for $n=105$, however). Let $U$ be the set of positive integers $n$ ...

**0**

votes

**0**answers

68 views

### A question on indefinite binary quadratic forms

Let $f(x,y) = ax^2 + 2bxy + cy^2$ be an irreducible, indefinite (so that $b^2 - ac > 0$) binary quadratic form. Put $d = b^2 - ac$. We say that two pairs of integers $(x_1, y_1)$ and $(x_2, y_2)$ ...

**1**

vote

**0**answers

93 views

### Analog of Baker's theorem on linear combination of $\log a \log b$

Baker's theorem basically says that, given algebraic numbers $a_1,\ldots,a_n$ and $m_1,\ldots,m_n$, if there is no good reason for a linear combination
$$\sum m_i\log a_i$$
to cancel, then it is ...

**0**

votes

**0**answers

153 views

### Deduction formula for Goldbach counting function

Assume $N\geq 1$ is integer and $P\geq 1$ is square-free integer. Goldbach counting function, $S_P(N,x)$, is defined to be the number of $n$ between 1 and $x$ such that $(N-n)(N+n)$ is co-prime to $P$....

**5**

votes

**0**answers

121 views

### Chowla's Construction of prime having least quadratic non-residue $\gg \log p$

This paper by NC Ankeny mentions that " S. Chowla has proved that there exist infinitely many primes $k$ where the
first $c_1\log k$ residues $(\bmod k)$ are all quadratic residues".
I recently ...

**4**

votes

**1**answer

313 views

### Is the set $S:=\{n\mid n\in\mathbb{N}\text{ and $n$, $n+1$ and $n+2$ have the same number of divisors}\}$ infinite?

Is the set $S:=\{n\mid n\in\mathbb{N}\text{ and $n$, $n+1$ and $n+2$ have the same number of divisors}\}$ infinite?
Example: $33\in S$.

**4**

votes

**0**answers

105 views

### Affine group, $[L : \mathbb{Q}] = n\varphi(n)$ or ${1\over2}n\varphi(n)$ [closed]

Let $L$ be the Galois closure of $K = \mathbb{Q}(\sqrt[n]{a})$, where $a \in \mathbb{Q}$, $a > 0$ and suppose $[K : \mathbb{Q}] = n$. How do I see that $[L: \mathbb{Q}] = n\varphi(n)$ or ${1\over2}...

**5**

votes

**1**answer

195 views

### Numbers divisible only by primes of the form 4k+1

Let $A(N)$ denote the number of positive integers $n\le N$ composed of prime numbers $p\equiv 1\pmod 4$ only. Is there an asymptotic formula for $A(N)$ (as $N$ tends to infinity)?

**1**

vote

**0**answers

78 views

### Counting points in a certain 4-dimensional region

Let $(a,b,c)$ be a fixed tuple of co-prime integers, with $a \ne 0$ and at least one of $b,c$ non-zero. Define
$$L = -\frac{a p_1 q_1 - b p_2 q_1 - b p_1 q_2 + 4 c p_2q_2}{a},$$
$$Q = \frac{(b^2 - ...

**7**

votes

**0**answers

127 views

### Efficient Dirichlet approximation (continued fractions?) over a number field

Is there an efficient algorithm for Dirichlet approximation for a given (high-degree) number field and its ring of integers, perhaps analogous to the Euclidean/continued fractions algorithm for the ...

**3**

votes

**1**answer

186 views

### Is there an infinite family of primes $q_{1},q_{2},…$ so that the rank of $E(\mathbb{Q}(\sqrt{-q_{i}}))$ equals that of $E(\mathbb{Q})$?

It is known that the group of $K$-rational points of an elliptic curve $E$ is finitely generated if $K$ is a number field of finite degree over $\mathbb{Q}$.
Much less is known if $K$ is infinite-...

**4**

votes

**2**answers

283 views

### Mordell-Weil rank of an elliptic curve over $\mathbb{Q}(\sqrt{-1},\sqrt{2},\sqrt{3},\sqrt{5},…)$?

It is known that the group of $K$-rational points of an elliptic curve $E$ is finitely generated if $K$ is a number field of finite degree over $\mathbb{Q}$.
The picture is less clear if $K$ is ...