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

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1 answer
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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 ...
Jeremy Rouse's user avatar
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 ...
Chris Birkbeck's user avatar
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 $\zeta(n)$, ...
mtm93's user avatar
  • 291
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 ...
Siméon's user avatar
  • 635
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 ...
Stanley Yao Xiao's user avatar
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(\...
Sangkyu Kim's user avatar
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) = {\...
Pierre's user avatar
  • 101
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 $...
user52959's user avatar
  • 131
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 ...
user19918273's user avatar
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 ...
user avatar
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. ...
Chebolu's user avatar
  • 575
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(...
nxir's user avatar
  • 1,409
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/\...
user52959's user avatar
  • 131
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 $\...
john mangual's user avatar
  • 22.6k
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 ...
Angsuman's user avatar
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)}{...
B . O's user avatar
  • 139
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 ...
Simon Rose's user avatar
  • 6,240
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 ...
Włodzimierz Holsztyński's user avatar
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 ...
Mostafa's user avatar
  • 4,454
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-...
David S. Newman's user avatar
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 $S \subset \{0,\dots,n\}$. ...
Joël's user avatar
  • 25.7k
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 $p>g^{1\over 3}$ ...
BHZ's user avatar
  • 1,168
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 ...
Joni Teräväinen's user avatar
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 ...
Stanley Yao Xiao's user avatar
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 $|\...
Steven Stadnicki's user avatar
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 ...
GFS's user avatar
  • 253
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 ...
Shivam Patel's user avatar
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 ...
Daniel Loughran's user avatar
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 ...
Suman's user avatar
  • 1,211
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 $\Xi(z)$ ...
mike's user avatar
  • 603
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 ...
Andreas Holmstrom's user avatar
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 $...
Paul McKenney's user avatar
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')...
GFS's user avatar
  • 253
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 ...
Pgatti's user avatar
  • 147
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 $a=1$ ...
Martin's user avatar
  • 1,101
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 \...
Shivam Patel's user avatar
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 ...
IBazhov's user avatar
  • 600
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 ...
ellCurves's user avatar
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 $\Gamma=PGL_2(\mathcal{O})$...
Subhajit Jana's user avatar
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 ...
Kevin Buzzard's user avatar
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 ...
Daniel Loughran's user avatar
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 ...
joaopa's user avatar
  • 3,655
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 ...
7-adic's user avatar
  • 3,764
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 ...
Shivam Patel's user avatar
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.
user avatar
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 ...
George Shakan's user avatar
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 ...
aglearner's user avatar
  • 14k
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 $...
SJY's user avatar
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