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

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0
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10 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
0answers
215 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>...
0
votes
0answers
74 views

Summing a series of integrals [on hold]

EDIT: This IS related to my research (investigating representations of the harmonic mean)and I gave the wrong formula for the sum the first time around. The sum formula has been amended below. I ...
3
votes
1answer
123 views

Intuition behind the definition of $E_{\omega}$ in Wall's paper

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 ...
13
votes
2answers
246 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 ...
2
votes
1answer
82 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{...
20
votes
1answer
892 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 ...
3
votes
1answer
126 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
0answers
94 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
0answers
157 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
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0answers
79 views

What is the sharpest bound of this sum?

Fix $y\geq 1$ and let $\delta$ be a small enough positive real number. Put $$\mathcal{D}^{+}=\left\{d=p_{1}...p_{l}: p_{l}<...<p_{1},\ p_{m} \leq y_{m} \ \textrm{for all odd} \ m \right\},$$ ...
-4
votes
0answers
70 views

How do I go about solving the following problem? [on hold]

Given $(m_1,m_2, ...,m_r)\in Z^r_{\geq 0}$, and $a_1, a_2, · · · , a_r \in \mathbb{N}$ such that: $\sum_{i=1}^r a_im_i=qL$ where $L$ denotes the least common multiple of $a_1, a_2, · · · , a_r$ and $...
1
vote
0answers
64 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
3answers
357 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
1answer
153 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 $\chi^{\sigma} \in S$ ...
28
votes
2answers
516 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
1answer
154 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
1answer
169 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 $E$...
5
votes
2answers
244 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
3answers
318 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
1answer
104 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
2answers
493 views

congruent number problem [on hold]

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
0answers
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
1answer
169 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
3answers
190 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
0answers
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
1answer
275 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 $n$ ...
-3
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0answers
33 views

problem related to conformal map of doubly connected region [closed]

Is the explicit result known? Product[(1 - Cos[x]/Cosh[n*h]), {n, 1, Infinity}]
0
votes
0answers
65 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
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0answers
92 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
0answers
144 views
+50

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
0answers
120 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
1answer
309 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
0answers
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
1answer
190 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
0answers
76 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
0answers
119 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
1answer
182 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
2answers
281 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 ...
2
votes
0answers
118 views

Spectral decomposition on GL(n)

If $\Delta_1, \ldots, \Delta_{n-1}$ are a basis of the ring of commuting bi-$SL(n,R)$-invariant differential operators, $L_0^2=L_0^2(SL(n,Z)\backslash SL(n,R))$ is the space of cuspidal automorphic ...
0
votes
1answer
215 views

S. Chowla real quadratic fields

Let $K=\mathbb{Q}(\sqrt{d})$ be a real quadratic fields. S. Chowla conjectured that if $d=4m^2+1$ for $m \in \mathbb{Z}$ then there is exactly 6 real quadratic fields which has class number one. Hideo ...
2
votes
0answers
84 views

characteristic ideal of the Iwasawa module

Let $H$ be a complex biquadratic Galois extension of $\mathbb{Q}$ such that the galois group of $H$ is isomorphic to the Klein Group. Let $H_{\infty}$ be an anticyclotomic $\mathbb{Z}_p$-extension of $...
3
votes
0answers
144 views

Can someone explain this appearance of the Fibonacci series in the formula of the Fibonacci series? [closed]

I have found the Fibonacci series as a function. The function is as follows :- $$F(x) = 1 - 0×f_1(x) + 1×f_2(x) - 1×f_3(x) + 2×f_4(x) - 3×f_5(x) + 5×f_6(x) - 8×f_7(x) + 13×f_8(x) - 21×f_9(x) + 34×f_{...
2
votes
2answers
266 views

Asymptotic estimate of the probability of $(n, P(\sqrt{x})) \leq x$?

Let $P(x)$ be the product of all primes less or equal to $x$. The probability of $(n, P(\sqrt{x})) \leq x$ for an arbitrary $n$ is then given exactly by $$ \prod_{p\mid P(\sqrt{x})}{\left(1-\frac{1}{p}...
4
votes
1answer
222 views

How to construct an abelian variety with CM by a given CM field?

Let $F$ be a totally real number field, and let $K$ be a quadratic extension of $F$ which cannot be embedded into $\mathbb{R}$. Then $K$ is a so called CM field. For instance, take $F = \mathbb{Q}(\...
1
vote
0answers
106 views

Probability that two integers selected from a fixed interval are relatively prime [closed]

I found the answer to a very similar question already asked here on mathoverflow: what is the probability that two natural numbers are relatively prime? The answer given in the link below was $\frac{6}...
10
votes
1answer
238 views

Why is there a factor $p$ in the definition of $T_p$ via Hecke correspondences on modular curves?

Fix $N\ge4$. Let $Y_1(N)$ and $X_1(N)$ be the usual modular curves. I want to view them as schemes over $\mathbb Z$ representing the moduli functors of (usual or generalized) elliptic curves with (...
1
vote
0answers
148 views

Optimal Diophantine approximation of $\pi$

If the 'optimal' Diophantine approximation of $\pi$ is given by the maximum value of $M=-\log_q(\min_{\forall p \in \mathbb{N}} |\frac{p}{q}-\pi|)$ for $q \geq 2$, what is the maximum value of $M$?
1
vote
1answer
194 views

Approximation of sets

Is the following true? For every $\varepsilon>0$ there is a finite subset $W$ of $\mathbb{N}\times \mathbb{N}\times \mathbb{N}$, such that $$|p_1(W)\cap p_2(W)\cap \{p_1(x)+p_2(x):x\in W\}\cap \{...
2
votes
1answer
156 views

Restriction to the diagonal of Hilbert eigenforms

Do you know of any reference that discusses whether the restriction to the diagonal of a Hilbert eigenform is an (elliptic) eigenform?