Questions tagged [nt.number-theory]
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
15,905
questions
38
votes
1
answer
2k
views
Degree 17 number fields ramified only at 2
The number $17$ is the smallest odd number that occurs as the degree of a number field $K/\mathbb{Q}$ for which the only finite prime that ramifies is $2$. The non-existence for $n < 17$ follows ...
9
votes
5
answers
2k
views
Optical methods for number theory?
I found a paper: 'A New Method of Finding the Distribution of Prime Number', saying
We stack discs and annuluses with certain rules then turn on the light to illuminate. The projection of ...
12
votes
1
answer
741
views
Overconvergent cohomology and overconvergent modular forms
I've been reading a preprint by David Hansen (with appendix by James Newton) called Universal eigenvarieties, trianguline Galois representations and p-adic Langlands functoriality. In it he talks ...
29
votes
3
answers
2k
views
$\zeta(n)$ as a mixed Tate motive
I am trying to understand why there exists, for each $n \geq 2$, a mixed Tate motive $M$ over $\mathbb{Q}$ such that
$M \in Ext^1_{MT(\mathbb{Q})}(\mathbb{Q}(0), \mathbb{Q}(n))$
and $\zeta(n)$, ...
17
votes
1
answer
670
views
Probability that $n$ is coprime to both $m$ and $m+1$
It is well known that the set $\{(n,m) \in \Bbb N^2 : \gcd(n,m) = 1\}$ of coprime integers has a natural density of $\zeta(2)^{-1}$ in $\Bbb N^2$.
It seems reasonable to think that the density of the ...
25
votes
1
answer
1k
views
Is the following sum irrational?
Is the following sum irrational?
$$S = \displaystyle \sum_{n \text{ squarefree}, n \geq 1} \frac{1}{n^3}$$
The sum clearly converges, so it is bounded above by $\zeta(3) = \displaystyle \sum_{n \geq ...
0
votes
1
answer
328
views
What is a well-known formula of the generalized Hardy Z-function?
Q: What is a well-known formula of the generalized Hardy Z-function??
$\arg_0(z)=\frac{\log(z)-\log(\overline{z})}{2i}$
$a_{k,j}=\frac{1-\chi_{k,j}(-1)}{2}$
$\vartheta_{q,r}(z)=\frac{\log\Gamma(\...
5
votes
1
answer
723
views
Elliptic curve and Galois representation
For an elliptic curve $E$ over ${\Bbb{Q}}$, let us consider Serre's mod $l$ representation by
$\rho_{E,l} \colon {\mathrm{Gal}}({\overline{\Bbb{Q}}}/{\Bbb{Q}}) \to {\mathrm{Aut}}(\phantom{}_lE) = {\...
5
votes
1
answer
323
views
looking for reference on dihedral, tetrahedral, or octahedral forms
I am looking for a reference on dihedral, tetrahedral, or octahedral forms. As far as I read, they are some cuspidal automorphic forms on $GL(2)$ induced from $GL(1)$. Dihedral is from $GL(1)/K$ to $...
19
votes
2
answers
1k
views
Are there any simple, interesting consequences to motivate the local Langlands correspondence?
Let's pretend that we know local Langlands at a fairly high level of generality... i.e. we know something along the lines of:
Let $G=\mathbf{G}(F)$ be the group of $F$-points of a connected reductive ...
2
votes
0
answers
516
views
vanishing of étale cohomology of affine surface
Let $U$ be an affine smooth surface over an algebraic closure of a finite field. Let $\mathscr{A}/U$ be an Abelian scheme and $\ell \neq \mathrm{char}(k)$ be prime.
Are there vanishing results for ...
14
votes
0
answers
1k
views
Is there an infinite field of characteristic 2 whose multiplicative group is torsion free and (direct-sum) indecomposable?
Let $F$ be a infinite field of characteristic 2 whose multiplicative group $F^*$ is torsion free. I would like to conclude that $F^*$ is decomposable or find an example where $F^*$ is indecomposable.
...
5
votes
1
answer
225
views
Rank four quadratic Form with non trivial discriminant in I(k)
Im sure this is a beginners question.
Let $k$ be a field and $I(k)$ the fundamental ideal in the Witt-ring W(k).
The Arason-Pfister-Hauptsatz states:
"If $\varphi$ is any anisotropic class in $I^n(...
5
votes
1
answer
224
views
Log weight removal in general (weaker) prime number theorem
Let $a_n$ be a sequence of non-negative numbers.
Assume that $$\limsup _{X\to \infty}\frac{\sum_{p\leq X} a_p\log p}{X}\leq 1.$$
Can we prove that $$\limsup _{X\to \infty}\frac{\sum_{p\leq X} a_p}{X/\...
4
votes
1
answer
230
views
How does associativity get twisted by elements of $H^3(G)$?
In Braided Monoidal Categories by Joyal and Street, §6 a monoidal category is $V =T(G,M,h)$ built using a recipe:
objects are are elements of $G$ ✓
$V_0(x,y) = M$ if $( x=y)$ or else $\...
4
votes
2
answers
579
views
quadratic residue difference set
Let $N=pq$ where $p$ and $q$ are primes of the form $4k+1$. Let $\mathbb{Z}_N$ be the set of integers modulo $N$ and $\mathbb{Z}_N^*$ be the units in $\mathbb{Z}_N$. Let $QR$ be the quadratic ...
3
votes
1
answer
287
views
A question on the big-O value of the complex integral especially in the number theory
My question is quite simple and elementary.
Let $A(x)=\sum_{1}^{x}a(n)$ and $\alpha(s)=\sum_{1}^{\infty}a(n)n^{-s}$. Then, as we know,
$$ A(x)= \int_{\gamma-i\infty}^{\gamma+i\infty}\frac{\alpha(s)}{...
3
votes
1
answer
314
views
Is there a nice way to write the generating function obtained by taking the quadratic coefficients of another one?
Suppose that you have a generating function
$$
f(q) = \sum_{k=0}^\infty a_k q^k
$$
It's not too hard to obtain the generating function
$$
f_{n,m}(q) = \sum_{k=0}^\infty a_{nk + m}q^k
$$
by taking a ...
4
votes
1
answer
210
views
Prime residua races and two views on primes
Let $\ a>1\ \ r\ \ k\ $ be arbitrary natural numbers such that $\ a\ r\ $ are relatively prime. The natural conjecture below, is it known?, is probably true in full generality:
Q1. There exists a ...
4
votes
1
answer
524
views
Hyperrectangles with integer diagonals
What is the largest value of $n$ for which there exists $n$ (not necessarily distinct) complete squares of natural numbers such that the sum of every subset of it is also a complete square? ( For ...
2
votes
1
answer
270
views
How does this sequence grow
Let $a(n)$ be the number of solutions of the equation $a^2+b^2\equiv -1 \pmod {p_n}$, where $p_n$ is the n-th prime and $0\le a \le b \le \frac{p_n-1}2$. Is the sequence $a(1),a(2),a(3),\dots$ non-...
23
votes
1
answer
796
views
integers which are sums of binomial coefficients: $\sum_i {n \choose k_i}$
Let $n$ be an integer. For $S$ a subset of $\{0,\dots,n\}$, define
$$m(S) = \sum_{k \in S} {n \choose k}.$$
Let $M_n$ be the set of integers of the form $m(S)$ for all sets $S \subset \{0,\dots,n\}$. ...
11
votes
1
answer
674
views
On the order of finite simple groups
About the order of finite simple groups there exists a very interesting result which stated as follows:
Let $G$ be an non-solvable simple group of order $g$. If $p\mid g$, where $p>g^{1\over 3}$ ...
9
votes
0
answers
391
views
In which orders can the numbers of prime factors of consecutive integers be?
Let $\omega(m)$ be the number of distinct prime divisors of a positive integer $m>1$. I am interested in the relative orders in which the numbers $\omega(n+1),...,\omega(n+k)$ can occur.
Given a ...
2
votes
0
answers
305
views
Can the affine sieve be used to sieve for $k$-free values?
The affine sieve, developed initially by Bourgain-Gamburd-Sarnak in the paper "Affine linear sieve, expanders, and sum-product" published in Inventiones Mathematicae in 2010, deals generally with the ...
15
votes
4
answers
988
views
Can we get good rational approximations in all residue classes?
The classic Hurwitz theorem for rational approximations (in simplest form; the constant can of course be improved) gives infinitely many approximations $\frac mn$ to an irrational $\alpha$ with $|\...
14
votes
1
answer
1k
views
Is the adjoint L-function on GL(m) holomorphic?
Let $\pi$ be an automorphic representation on $\mathrm{GL}(m)/\mathbb{Q}$.
Define $$L(s,\pi,\mathrm{Ad}):=\frac{L(s,\pi\times\overline{\pi})}{\zeta(s)}.$$ This is an $L$-function with Euler product of ...
8
votes
0
answers
986
views
On the sum of consecutive primes and product of first and last
Lets consider the sequence of consecutive prime numbers $p_1=2 , p_2=3 ,p_4=5 , \cdots$
. $(p_n,p_{j})$ is to be called good prime pair if $$\sum_{i =n }^{j}p_i= p_n p_{j}$$
Meaning the sum of set of ...
6
votes
1
answer
1k
views
A weak version of Bass' conjecture
Let $A$ be a finitely generated $\mathbb{Z}$-algebra which is a UFD. Then (a special case of) the Bass conjecture states that $K_0(A)$ is a finitely generated abelian group. As far as I am aware, this ...
2
votes
1
answer
403
views
Some questions related to Iwasawa invariants of elliptic curves
Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$.
Let $\mathbb{Z}_{p}$ denote the ring of $p$-adic integers, and $\mathbb{Q}^{cyc}$ be the ...
6
votes
2
answers
761
views
Are there any new results on approximating Riemann $\Xi$ function by Polya-like Fourier transforms?
I posted [this question][1] at math.stackexchange.com and was told that it is more appropriate to post this research related question here at mathoverflow.
So I re-post it below.
Riemann $\Xi(z)$ ...
5
votes
2
answers
1k
views
Local factors of Hasse-Weil zeta function - what do they have in common?
Let $X$ be a regular scheme, flat and of finite type over $Spec(\mathbb{Z})$ (add "projective" if you want). Then the Hasse-Weil zeta function of $X$ is defined as a product over all prime numbers of ...
3
votes
0
answers
235
views
When are two groups $\mathbb{Z} + \alpha \mathbb{Z} + \alpha^2\mathbb{Z}$ equal? [closed]
If $\alpha$ and $\beta$ are irrational, and $$\mathbb{Z} + \alpha \mathbb{Z} + \alpha^2\mathbb{Z} = \mathbb{Z} + \beta \mathbb{Z} + \beta^2 \mathbb{Z}$$ does it follow that one of $\alpha + \beta$ or $...
8
votes
1
answer
625
views
Absolute convergence of Rankin–Selberg series
Let $\pi$ and $\pi'$ be two general automorphic representations on $\operatorname{GL}(n)$ and $\operatorname{GL}(n')$ over $\mathbb{Q}$.
I heard that the Rankin-Selberg $L$-function $L(s,\pi\times\pi')...
2
votes
1
answer
1k
views
Maximal unramified extension and inertia group for separable closure
I have a problem in understanding the inertia group of an infinite extension. I am studying it in this context.
Let $K$ be a field, $v$ a discrete valuation on $K$, and $\mathcal{O}_v$ the discrete ...
2
votes
0
answers
189
views
Equation in the Gaussian Integers
Let $a,b \in \mathbb{N}$. Is there a possibility to characterize the solutions of $a N(\alpha) - b N(\beta)=1$ where $\alpha,\beta \in \mathbb{Z}[i]$? In particular I am interested in the case $a=1$ ...
5
votes
0
answers
136
views
On the comparison of Egyptian fractions of two kinds
I posted the question on MSE here but it did not get any answer.
Consider $$S(n)=\left\{(a_1 ,a_2,a_3, \dots, a_n)\mid a_1\le a_2\le\cdots\le a_n, \; \sum_{r=1}^{n}\frac{1}{a_r} = 1\right\} \subset \...
15
votes
1
answer
939
views
Curves on K3 and modular forms
The paper of Bryan and Leung "The enumerative geometry of $K3$ surfaces and modular forms" provides the following formula. Let $S$ be a $K3$ surface and $C$ be a holomorphic curve in $S$ representing ...
8
votes
2
answers
880
views
Elliptic Curves with equal trace of Frobenius Values
Suppose we have two elliptic curves over $\mathbb{Q}$ with trivial rational torsion. Is there some density $\delta$ such that if the trace of Frobenius values of the two elliptic curves are equal on a ...
8
votes
1
answer
366
views
Eisenstein series over a definite division algebra
Let $D$ be the definite quaternion division algebra over $\mathbb{Q}$. $\mathcal{O}$ is a maximal order inside $D$, let's fix $\mathcal{O}$ to be the Hurwitz quaternion. Let $\Gamma=PGL_2(\mathcal{O})$...
10
votes
1
answer
527
views
Example of a non-smooth irreducible component of the generic fibre of a Hida family?
Is there a known example of a non-smooth irreducible component of the rigid generic fibre of a Hida family?
Let me explain some of the context around this question (but I'm not going to explain Hida ...
21
votes
0
answers
598
views
Bounding failures of the integral Hodge and Tate conjectures
It is well know that the integral versions of the Hodge and Tate conjectures can fail. I once heard an off hand comment however that they should only fail by a "bounded amount". My question is what ...
5
votes
1
answer
261
views
$H^1$ and fractional ideals group
Let $L/K$ be a Galois extension with Galois group and $\mathfrak p$ be a prime of the ring of integers $\mathcal O_K$.
I would like to prove that $H^1(G, I_{\mathfrak p})=1$ where $I_{\mathfrak p}$ is ...
13
votes
5
answers
4k
views
Brief Introduction to Modular Forms
What are the best introductory texts on modular forms that are suited for a brief six week course intended for advanced undergraduates? The students will be quite sharp and as far as prerequisites go, ...
9
votes
1
answer
513
views
standard zero free region of automorphic L-function on GL(N)
Let $L(s,\pi)$ be the standard(Godement-Jacquet) $L$-function of $\pi$, where $\pi$ is a cuspidal automorphic represetation of $GL(m,A_Q)$.
What's the standard zero-free region for $L(s,\pi)$? any ...
5
votes
2
answers
566
views
How many integers are of the form $n/d(n)$, where $d(n)$ is the number of divisors of $n$?
This was post by me on Maths SE: but it did not get any solution.
Some months ago I made the following conjecture -
Let $d(n)$ denote the number of divisors of $n $.
Then let $N$ be a number such ...
1
vote
0
answers
320
views
Where can I find the article of A. Borel: "Values of zeta-functions at integers, cohomology and polylogarithms"? [closed]
Where on the internet can I find this article?
I know that it is in this book: Current trends in mathematics and physics, Narosa, New Delhi, 1995.
17
votes
1
answer
1k
views
Sum and product estimate over integers, rationals, and reals
My question is the following: is finding a lower bound for $|A+A\cdot A|$ (as a function of $|A|$) where $A$ is any finite subset of the positive integers equivalent to finding the same lower bound ...
13
votes
1
answer
856
views
Colourings of $\mathbb Q\times \mathbb Q$ in three colours
Using two-adic valuation Monsky coloured $\mathbb Q\times \mathbb Q$ in red, blue, and green, so that on each line points of at most two colours are present.
Question. I would like to know if there ...
3
votes
0
answers
181
views
Finding an integral basis for the lattice of the form $\mathbb{Z}^J \cap \mathbf{p}^{\perp}$
Let $\mathbf{p}$ be a primitive point in the lattice $\mathbb{Z}^J$
and denote the $J-1$ dimensional vector space $V = \mathbf{p}^{\perp} \subseteq \mathbb{R}^J$.
Let $\Lambda' = \mathbb{Z}^J \cap V $...