Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.
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42 views
Galois group of CM fields
I am looking for examples of CM fields whose Galois group is not abelian. By a CM field $K$ I mean a totally imaginary quadratic extension of a totally real field $K_0$. If the extension is not Galois ...
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40 views
Embedding an $n$-simplex in $\mathbb{Z}^n$.
I am trying to understand the proof of embedding an $n$-simplex in $\mathbb{Z}^n$ for specific values of $n$. The proof can be found here. I am stuck on what is meant by "the reflection with axis ...
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45 views
If $x \sim U(Z_n^*)$ then $x^2(mod \; n) \sim U(QR_n)$?
Define:
$Z_n^*=\{x \in Z_n | gcd(x,n)=1\}$
$QR_n=\{x \in Z_n | \exists r \in Z_n \; s.t. \; r^2 =x\}$
How can I show that $x \sim U(Z_n^*) \implies x^2(mod \; n) \sim U(QR_n)$?
Thank you.
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67 views
Efficient way for computation of derivatives of $f(x) = \zeta(1-x) + 1/x $ at integer x?
For some exercises with (divergent) summation of the Stieltjes constants I'm trying a formula, which involves derivatives of the $\zeta()$ -function at negative integers; perhaps better formulated as ...
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50 views
intuitive meaning behind Mertens' theorem
I have just been introduced the topic of distribution of primes, big O notation and aymptotic functions so please correct me if I say something that does not make sense. I am looking to get an ...
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30 views
Mean mean and kth moment of positive integer partitions with constant sum.
I'll start with an example: For a sum of 4, the possible set partitions are {4}, {3,1} and multiset partitions are {4}, {3,1}, {2,2}, {2,1,1}, {1,1,1,1}.
For a given sum $S$, what can be said of the ...
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41 views
Are there (known) bounds to the following arithmetic / number-theoretic expression?
I apologize in advance if this is something that is already well-known in the literature, but I would like to ask nonetheless (for the benefit of those who likewise do not know):
Are there (known) ...
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43 views
sum over primes and its powers
are there examples in number theory where the series
$$ \sum_{p} \sum_{m=-\infty}^{\infty}f(p^{m}) $$ or
$$ \sum_{m=-\infty}^{\infty}f(q^{m}) $$ for fixed prime 'q' appear ??
i believe they may be ...
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0answers
62 views
The zeta-function of Fibonacci sequence?
I have been told that, for a given sequence ${N_r}$, its zeta-function is given by $exp(\Sigma _{r=1}^{\infty} N_rT^r/r)$. Since I have barely any experience with such a sum, I tried to find some ...
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40 views
Any work on properties of $N + \bar \phi (N)$?
I am looking for pointers to any existing materials about the properties of this quantity.
For Euler's cototient, if a number $N$ is written as $2^a \cdot b$ with b odd then the cototient is $$\bar ...
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53 views
$S$ unit equation
From experiments it seems $(1+\sqrt{2})^n+(1-\sqrt{2})^n=2^a-3^b$ has finite solutions $(a,b,n)$, where $a,b,n$ are non-negative integers. From $S$ unit equation we know ...
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40 views
Whether any 3 degree equtition (x,y) can transform to weierstrass equtition?
Whether any $a_1x^3+a_2y^3+a_3x^2y+a_4xy^2+a_5x^2y^2+a_6xy+a_7x^2+a_8y^2+a_9x+a_{10}y+a_{11}$ can transform to weierstrass equaition?
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83 views
Idealclassgroup for quadratic field
I have got a question about an ideal class group, namely the group of $\Bbb{Q}(\sqrt{-185})$.
I can say the following:
I can give a representant system of the group
I can name the class number: ...
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141 views
Ramanujan's sums
Are the series expansions of arithmetic functions in terms of Ramanujan sums computationally useful? I didn't think they would be, but they seem to be good approximations even when summed with few ...
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0answers
52 views
Find total number of sets of integers which satisfy a given equality and inequality
Compute the total number of different sets of integers a1, a2,..,an which satisfy the following equality and constraints:
$$
...
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21 views
Question about the definition of weekly modular of weight k
I'm reading the book A First Course In Modular Forms and it defines the term weakly modular of weight $k$ as following:
Let $K$ be an integer. A meromorphic function $f:H\rightarrow\mathbb{C}$ is ...
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64 views
$a^{N(\mathcal{P})} \equiv a \pmod{\mathcal{P}}$
Let $\mathcal{P}$ a prime ideal in $\mathcal{O}_K$. Show that $a^{N(\mathcal{P})} \equiv a \pmod{\mathcal{P}}$, $a \in \mathcal{O}_K$.
I have this: $N(\mathcal{P})=p^f$, $f$ is the inertia degree of ...
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77 views
Estimation on Primorial Influence
As you know, Primorial ($\#$) notion is defined as the product of first $n$ prime numbers. That is,
$$
n\# = \prod_{i=1}^nP_i
$$
An there is some effect named Primorial Influence (explained ...
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36 views
Lattice Reduction Problem: Minimizing the Longest Vector
Suppose we have a basis for an integer lattice formed by the vectors $\vec v_1, \vec v_2, \ldots,\vec v_n$. Then let $A$ be the augmented matrix $( \vec v_1| \space \vec v_2| \cdots |\space \vec ...
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86 views
Number of solutions to polynomials over finite fields
Let $m = p_{1}p_{2}\cdots p_{s}$. Let $N_{f}(n)$ denote the number of solutions to $f \equiv 0 \pmod{n}$. If $f = f(x)$, a polynomial in the single variable $x$, then from the Chinese Remainder ...
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69 views
solution count of quadratic form congruences
Let $(L,q)$ be an even Lorentzian lattice of signature $(1,l-1)$, i.e., $q(\lambda) \in \Bbb Z$ for all $\lambda \in L$ and the (non-degenerate) quadratic form $q$ is of type $(1,l-1)$ (the ...
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43 views
Upper bound for non-square-free sum
Given a multiplicative function $f$, is there any general method of getting a upper bound of $\sum^{'}_{n<x} f(n)$, where the sum is restricted to all those non-square-free $n$?
For example, when ...
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17 views
group law on weil-chatelet group
Is there a reference for the gemetric definition of the group law on the Weil-Châtelet group of an Abelian variety more recent than the original Weil's paper ("On algebraic groups and homogeneous ...
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74 views
Elementary divisors of a finite abelian group
Suppose $A$ is a finite abelian group.
(a) Extract from the function $h_A(n) = |\{x \in A : x^n = e\}|$ ($n \in \mathbb{Z}$) the elementary divisors of A using the fact that for a cyclic group $C$ of ...
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44 views
absolute galois group of Puiseux series with coefficients in $\bar{\mathbb{F}_p}$
Let $K$ be the field of Puiseux series with coefficients in $\bar{\mathbb{F}_p}$ (the algebraic closure of the field with elements). What is the absolute Galois group of $K$ ?
Thank you to anyone ...
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37 views
What's the probability to find a value of $t<T$ where $|P_k(it)|<\epsilon$?
Given $P_k$, the truncated Prime $\zeta$ function, defined like
$$
P_k(it):=\sum_{n=1}^k p_n^{it},
$$
where $p_n$ is the $n$th prime. What's the probability to find a value or range of $t$ less than ...
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57 views
A natural way of thinking of the definition of an Artin $L$-function?
Emil Artin knew that given a finite extension of $L/\mathbb{Q}$, the local factor of the zeta function $\zeta_{L/\mathbb{Q}}$ at the prime $p$ should be $\displaystyle\prod_{\mathfrak{p}|p}\frac{1}{1 ...
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84 views
Question about the elementary divisors of a special matrix
I have the following question:
Is there a closed formula for the elementary divisors of the Matrix $M={(m_{ij})}_{i=1,...,n,\ j=1,...,k}$, where ${m}_{ij}$ is the greates common divisor of $i$ and ...
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23 views
Series function help
I want to find a function such that $$ \sum_{0<j<n/k
} f(kj)=1 $$
Where the sum j is taken over the natural numbers,
And the series is satisfied for all integers k and n, I was thinking of ...
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35 views
Does an irreducible polynomial in K(t)[x] give an irreducible polynomial in K[t][x]
Let $K[t]$ be the ring of polynomials over a field $K$. Let $K(t)$ be its fraction field. Let $f$ be an irreducible polynomial in $K(t)[x]$. There exists an element $a\in K[t] $ such that $af$ is in ...
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28 views
Dilation mod p, small gaps
If I have a set $A \subset \{1,2,3,...,\epsilon p \} \subset \mathbf{Z}/p \mathbf{Z}$ does there exist a dilation $ \lambda $ such that $ \lambda A$ has no gap larger than $s$ (where $\epsilon = ...
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79 views
Permutations within a specific boundary
Let's have the following sequence of natural numbers: 1, 2, 3, 4, 5, 6, 7, 8. The permutations of these 8 numbers are equal to 8!. We can obtain some of these permutations by adding and subtracting ...
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49 views
Inequality help
Can someone help me prove the inequality,
$$ \frac{\sum\limits_{k=1}^n \Lambda(k) \frac{k}{n}\lceil\frac{n}{k}\rceil\{ \frac{n}{k} \}}{\sum\limits_{k=1}^n \Lambda(k)}<\ \frac{\sum\limits_{k=1}^n ...
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44 views
How good might $|\vartheta(x)-x|$ be?
This question is about Chebyshev's first function, $\vartheta(x) = \sum_{p\leq x}\log p.$
Assuming the truth of the Riemann hypothesis, $|\vartheta(x) -x|= O(x^{1/2+\epsilon})$ for $\epsilon > 0.$
...
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0answers
50 views
Find the orders..
$\newcommand{\ord}{\operatorname{ord}}$
Find the orders below:
\begin{align}
& (a) \quad \ord_{11}5 \\
& (b) \quad \ord_{7}4 \\
& (c) \quad \ord_{23}22!
\end{align}
For the most part, I ...
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0answers
85 views
Integral basis of $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$
how i can find a integral basis $\mathbb{Z}_K$, if $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$?
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63 views
$\int_{\gamma}\frac{dz}{\sqrt{1-z^2}}=2\pi$ Along the path $\gamma(t)=2e^{it}$ for $0\leq t\leq 2\pi$ Implies Sine is $2\pi$ periodic
Ok so to back up a bit, by a trig substitution we have for $f:(-1,1)\rightarrow\mathbb{R}$:
$$f(x)=\int_0^x\frac{dt}{\sqrt{1-t^2}} = \arcsin(x)$$
Now according to the notes here: Elliptic Functions ...
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0answers
90 views
Matiyasevich polynomial proof
Can someone provide a proof, or a link to a proof, of why does the Matiyasevich polynomial always generate primes for the nonnegative results? Any help will be appreciated.
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47 views
Rationality and convergences
I observed that convergence of partial fractions and I am writing this samll story for seeking more clarity and justifications etc.
If f(n) = $\frac{p(n)}{q(n)}$, where p and q are polynomials of n ...
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68 views
A question on algorithm complexity
It is well-known that the evaluating the Discrete Fourier Transform definition directly has a complexity $O(N^{2})$
for a signal with bandwidth $N$. How to see or show that the fast
Fourier transform ...
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66 views
the 2-rank of field
Let the field $K=\mathbb{Q}(\sqrt{p_1}, \sqrt{p_2 q}, i)$ where $p_1, p_2 \equiv 1 \mod{4}$ and $q \equiv 3 \mod{4}$, kronecker(2,$p_1$)=1 and kronecker(2,$p_2$)=kronecker($p_1$,$p_2$) = ...
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118 views
Fibonacci and Lucas relations
We know that $H_n$ = $L_n + mF_n$, where $n = 0$ or $n > 0$ is simply relation between Fibonacci sequence and generalized Fibonacci-Lucas sequence. Are there any methods to prove the following ...
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176 views
Legendre and Jacobi symbols
I have a problem with Tonelli-Shanks algorithm with numbers $n = 87463$ and $p = 17$. Solutions are supposed to be $x_1 = 7$, $x_2 = 10$, but I get $11$ and $6$.
First with sieving I get a list of ...
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180 views
The Lucas Theorem and facts
I have studied the Lucas theorem and I encountered the following facts.
How to deduce the following facts from The Lucas theorem?
(1) If d, q > 1 are integers such that , $$\binom{nd}{md}$$ ...
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56 views
series: can the result be zero for a continuous interval of its argument?
I'm considering the series
$$ f_c(x) = \sum_{k=c}^\infty \left( c^{k-1} \binom{k}{c} \cdot \prod_{j=1}^{k-1} (x-1/j) \right) $$
where the parameter $c \in \mathbb N ,c \gt 0$ and fixed for a certain ...
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123 views
Lucas' theorem Consequence
Lucas' theorem consequence
$$\binom {m}{n}=\ \prod_{i=0}^k\;\ \binom {m_i}{n_i} \pmod{p},$$
$$m=m_k\;p^k+m_{k-1}\;p^{k-1}+\cdots +m_1\; p+m_0,$$
$$n=n_k\;p^k+n_{k-1}\;p^{k-1}+\cdots +n_1\;p+n_0$$
...
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64 views
Infinitesimals and infinite elements among the transseries
In the quest for extensions of $\mathbb{R}$ and $\mathbb{C}$ that contains infinitesimals, infinities (and even more exotic beasts like $\omega - 1$ and $\sqrt{\omega}$) I came across the theory of ...
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98 views
Is there any new improvement in the proof or disproof of the twin prime conjucture?
I think this is not the first question about twin primes here, but my own is the latest one!
I am a postgraduate student in Mathematics interested in the field of number theory. While searching on ...
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39 views
A question regarding the method followed in Cohen & Selfridge's paper on covering systems.
Note: I have posted this question on MO before. No one replied, so I am reposting it here.
I am reading this paper by Cohen and Selfridge that deals with covering systems. Its link is
...
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0answers
129 views
A special factorization
Suppose that monic polynomial $f(x)\in\Bbb Z[x]$ such that for all $m\in\Bbb Z$, $m>1$, there's no integers $\langle r,r_1,\ldots,r_m\rangle$ such that $f(r)=f(r_1)\cdots f(r_m)$. Is there any ...
