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

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2
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0answers
94 views

Are all zeros of $\xi(a\,s) \pm \xi\left(a\,(1-s)\right)$ on the critical line for $\forall a \in \mathbb{R}/0$?

This question expands on this one and seems to have a stronger result. Take the Riemann $\xi$-function $\xi(s) =\frac12 s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s)$. We ...
4
votes
0answers
283 views

Conjectures on perfect squares

There exist infinitely many sets of three positive integers $(a,b,c)$ where $a\not=b,b\not=c$ and $c\not=a$ such that each of $$ab-1,\ bc-1,\ ca-1,\ ab-a-b+c,\ bc-b-c+a,\ ca-c-a+b$$ is a perfect ...
4
votes
1answer
559 views

How good can we approximate with algebraic curve the egg shaped vanishing of $\Re \zeta(s)$ near the origin?

Related to this question where degree $2$ algebraic curve is good approximation to vanishing of the real part of expression involving zeta. Near the origin, $\Re \zeta(s)$ vanishes in egg shaped ...
6
votes
0answers
100 views

How comes vanishing of the real part of function involving zeta is very well approximated by algebraic curve?

In this question Agno asked about the zeros of $\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$. I fixed $a=2$ and the minus sign and defined: $$ f(s)=\Re \left( ...
0
votes
0answers
134 views

Does the Euler product converge at $s=1$ for the Dirichlet $L$ function?

For the Riemann Zeta function, the Euler product converges on $\{Re(s)=1\}$ except at $s=1$.The zeta series diverges everywhere on $\{Re(s)=1\}$. But the $L$ series converges on $\{Re(s)>0\}$. What ...
14
votes
0answers
230 views

Is the absolute Galois group of the rationals Hopfian?

Is every continuous epimorphism from the absolute Galois group of $\mathbb{Q}$ to itself injective?
7
votes
1answer
232 views

Set of integers having finite intersection with the image of any polynomial of degree $\geq 2$

Is there a set $A$ of positive integers such that $\sum_{n \in A} \frac{1}{n} = \infty$, and there is no polynomial $f \in \mathbb{Z}[x]$ of degree at least $2$ which takes infinitely many values in ...
2
votes
0answers
51 views

Are the complex zeros of $\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$ all on the critical line for $a \lt 0, a \ge 1$?

With $s \in \mathbb{C}, a \in \mathbb{R}$, numerical evidence strongly suggests that the complex zeros in the critical strip of: $$\zeta\left(\frac{s}{a}\right) \pm ...
4
votes
0answers
80 views

When do two lattices have the same stabilizer in the diagonal torus?

This is moved from MSE, where I asked and didn't receive an answer (see http://math.stackexchange.com/questions/1145151/lattices-in-mathbbq-pn-with-the-same-stabilizer) Let $T$ be the diagonal torus ...
16
votes
1answer
722 views

Primes that are sums of two squares with constraints on the squares

It is well known that there are infinitely many primes of the form $a^2+b^2$ (namely all primes congruent to $1$ modulo $4$). On the other hand, Euler raised the problem as to whether there are ...
1
vote
0answers
91 views

Limit of a simple function including a zero of the Riemann Zeta function

Lets consider : $$F(x)= \sum_{n\in\mathbb{N}} n^{-s_0} e^{2i\pi nx}$$ This function is well defined for $x>0$ (Abel summation formula proves it) and I would like to show that if $s_0$ is a zero ...
6
votes
0answers
162 views

P-depletion of Siegel modular forms

Let $F$ be a cuspidal Siegel modular form of genus 2 (of parallel weight $(k, k)$, and level some congruence subgroup $\Gamma \subseteq Sp_4(\mathbf{Z})$ of level $N$). Then $F$ has a series ...
1
vote
1answer
121 views

Infinite sum of asymptotic expansions

I have a question about an infinite sum of asymptotic expansions: Assume that $f_k(x)\sim a_{0k}+\dfrac{a_{1k}}{x}+\dfrac{a_{2k}}{x^2}+\cdots$ with $a_{0k}\leq \dfrac{1}{k^2}$, $a_{1k}\leq ...
0
votes
0answers
63 views

Is it possible the division polynomials evaluated at fixed point to be perfect powers unbounded number of times?

Let $E$ be elliptic curve over the rationals and $P=(X_P,Y_P)$ point on $E$. $\psi_n$ are the division polynomials. Define $a_n=\psi_n(X_P,Y_P)$. Is it possible $a_n$ to be perfect power ...
1
vote
1answer
167 views

Reference request: Research done on whether the Euler prime can be the largest factor of an odd perfect number

(Note: This was cross-posted from MSE.) I posted the following reference request in MSE three (3) days ago, but was unable to elicit any responses. I am cross-posting it to MO, hoping that it is ...
1
vote
1answer
235 views

Constant group scheme and torsors

Let $X$ be a scheme and $G$ a (commutative) constant group scheme. Consider a $G$-torsor $Y$ for $X$, by which I mean that there is a canonical isomorphism: $$g_Y \colon Y \times_X Y \cong Y ...
1
vote
0answers
192 views

On a property of Riemann Zeta function zeros

Lets consider the function : $$F(x) = \sum_{n=1} (xn)^{-s_0} e^{-nx} $$ with $s_0$ a zero of the Riemann Zeta function in the critical strip. This sum is well defined for $x \in \mathbb{R}^{+*}$. It ...
3
votes
3answers
344 views

Waring problem for binomial coefficients (generalization of Gauss' Eureka Theorem)

Is there a number k such that every natural number can be written as $\sum_{i=1}^k \binom{a_i}{3}$ for some natural numbers $a_i$'s?
2
votes
2answers
161 views

Relations of eisenstein series with eta quotient

Theorem 1.67 On page 19 of Ken Ono's book The Web of Modularity says: Every modular form on $SL_2(\mathbb{Z})$ may be expressed as a rational function in $\eta(z)$, $\eta(2z)$ and $\eta(4z)$. The ...
5
votes
1answer
339 views

Generating primes with floor of a polynomial $[p(n)]$

Is there a polynomial $p(x)$ with real coefitients and degree at least one that $[p(n)]$ for everey natural number like $n$ be a prime? If yes, what is such a polynomial $p(x)$ and if no, how to ...
3
votes
0answers
64 views

Can we extend the twisted Poisson Summation formula with functions having a singularity in zero?

The following "twisted" Poisson Summation formula for $\chi$ primitive of conductor $q$ : $$ \sum_{n\in\mathbb{Z}}\chi(n)f\left(\frac{nx}{\sqrt{q}}\right) = ...
0
votes
1answer
201 views

Automorphisms of $\mathbb F_q((\frac1T))$

I try to find the automorphisms $\sigma$ of $\mathbb F_q((\frac1T))$ with the following properties: $\sigma$ is an isometry,and $\sigma(\mathbb F_q[T])\subseteq\mathbb F_q[T]$. Almong the ...
2
votes
0answers
119 views

Distribution of Fourier coefficients of Maass forms

In the sense of Maass an automorphic function $\phi$ with Laplace-Beltrami eigenvalue $\frac{(d-1)^2}{4}+t^2$ on $d$-dimensional hyperbolic space which can be thought as ...
2
votes
1answer
162 views

extensions of crystalline representations

Denote by $G_p$ a choice of an absolute Galois group of $Q_p$, the field of $p$-adic numbers. Consider a continuous representations of $G_p$ on a $3$-dimensional $Q_p$ vector space that is a ...
6
votes
0answers
216 views

What is miraculous about the mirabolic subgroup?

I recently asked this question about Euler subgroups and generalizing the automorphic theory of $\mathrm{GL}_n$ to a more general setting. My question here is more specific. As mentioned there, the ...
10
votes
0answers
282 views

Word complexity of primes mod 4

For an infinite binary word $w$, the word complexity $f_w(n)$ is defined as the number of different subwords of length $n$. The asymptotic behavior of this function is an important parameter of the ...
3
votes
1answer
216 views

Non-principal ideals in cyclotomic fields

Let $K=\mathbb Q(\xi_{39})$ be the 39-th cyclotomic field. Pari-GP told me that the prime ideals above $3$ and $13$ are not principal. Is there a way to prove that by hand (no computation made by ...
14
votes
1answer
2k views

Mazur secret Bourbaki report “Analyse p-adique”

Does anyone happen to know if a scan of Mazur's report exists, and, if so, where to find it? It appears in the references for Katz's "Higher congruences" and "Eisenstein measure" papers.
3
votes
2answers
519 views

Automorphisms of $\mathbb C_p$

I am looking for a non-trivial automorphism $\sigma$ of $\mathbb C_p$ such that $\sigma(\mathbb Q_p)\subset\mathbb Q_p$. If $\mathbb C_p$ were spherically complete, then by Hahn-Banach theorem, that ...
5
votes
0answers
135 views

Euler Subgroups and Automorphic L-functions

Recently, I have read about the Whittaker expansion for $\mathrm{GL}_n$ and was struck by the utility of the mirabolic subgroup, $\mathrm{P}_n\subset \mathrm{GL}_n$ of matrices with bottom row $(0\; 0 ...
-1
votes
0answers
42 views

Explicit records for computing multiple of $10^5\#$ (or larger) with additions, subtractions, and multiplications starting from $1$

Related to this question which asks for the complexity of computing multiple of $k!$ with additions, subtractions, and multiplications starting from $1$. There Elkies suggests an interesting way to ...
8
votes
3answers
278 views

Lower bound for spectral radius on $\operatorname{GL}(n,\mathbb{Z})$

Consider the group of matrices $G =\operatorname{GL}(n,\mathbb{Z})$ with integer entries and determinant $\pm 1$. For each matrix $D \in G$, the product of the eigenvalues of $D$ is equal to $\det D ...
1
vote
0answers
105 views

Explicit description/calculation of norm group of ideles of characteristic $p$ global field

I posted the same question earlier in stack exchange, (http://math.stackexchange.com/questions/1130391/algebraic-proof-of-2nd-inequality-of-global-class-field) thinking it is most definitely not a ...
14
votes
1answer
1k views

How important is Weil's decomposition theorem today?

Andre Weil's Apprenticeship of a Mathematician (p. 46) tells how he as a student realized that all of Fermat's uses of descent are unified in one principle: "If $P(x,y)$ and $Q(x,y)$ are homogeneous ...
1
vote
1answer
168 views

Heegner points on elliptic curves

I want to know about Heegner point computations for a CM elliptic curve. What is the best reference book/paper for reading?
3
votes
0answers
92 views

ramified principal prime ideals in Artin-Schreier extension

Let $q$ be a power of $2$ and $K$ be the quadratic extension of $\mathbb F_q(T)$ defined by $K:=\mathbb F_q(T)[y]$ where $y^2+y=f(T)\in\mathbb F_q(T)$. Put $\mathcal O_K$ the integral closure of ...
1
vote
1answer
131 views

Is there a Poisson Summation formula for imprimitive Dirichlet characters?

I was wondering if there exists a Poisson Summation formula (like the one existing with primitive character) for imprimitive Dirichlet characters ? For a primitive Dirichlet character $\chi$ we have: ...
5
votes
0answers
191 views

Legendre polynomials and formal groups

Let $P_n(x)$ be Legendre polynomials: $$\frac{1}{\sqrt{1-2tx+t^2}}=\sum\limits_{n=0}^{\infty}P_n(x)t^n.$$ Usual arguments from the theory of formal groups allow to prove that for any $n$ ...
1
vote
0answers
138 views

Fourier expansions at the cusps of $\Gamma_0(N)$

My question may be basic but I can't find any answer. Let $N$ be a positive integer. I need to find the constant term (of the Fourier series) at each cusps of a modular form ...
14
votes
0answers
369 views

Can there be arbitrarily many cubic fields unramified outside $\{p,\infty\}$?

Observe, trivially, that since quadratic fields correspond to rational integers modulo squares (viz. discriminants), there are (roughly about, but certainly at most) $2^{|S|+1}$ quadratic fields ...
2
votes
0answers
86 views

Linear forms with best approximation vectors lying in a subspace

Setup: For $u \in \mathbb{R}^n$, let $\rho(u)$ be the Euclidean length, $\sqrt{u_1^2 + \ldots + u_n^2}$. For $x \in \mathbb{R}$ let $\|x\| = \min_{k \in \mathbb{Z}} |x - k|$, and for $x \in ...
2
votes
2answers
116 views

asymptotic for a sequence coming from multiplicative sets

Let $a,b$ two multiplicatively independant positive integers with $1<a<b$ that is $a^mb^n\ne1$ for all $m,n\in\mathbb N\setminus\{0\}$ We sort out the set $E=\{a^mb^n\mid m,n\in \mathbb ...
1
vote
0answers
129 views

Consequences failure of $\tau$ conjecture

A question Bounds for constructing $n!$ with additions, subtractions, and multiplications starting from $1$ was asked on constructing $ak!$ with ring operations. $\tau$ conjecture states if $\exists$ ...
0
votes
0answers
129 views

Ring of integers in Artin-Schreier extension

Question put in mathstackexchange but received no answer. It is well-known( see Goldschmidt book: Algebraic Functions and Projective Curves) that for $q$ a power of $2$ a quadratic separable ...
94
votes
1answer
4k views

What is a Frobenioid?

Since there will be a long digression in a moment, let me start by reassuring you that my intention really is to ask the question in the title. Recently, there has been a flurry of new discussion ...
11
votes
2answers
550 views

Formal group law over $\mathbb{F}_p$

Let $p$ be a prime. For each $n > 0$ there is a unique 1-dimensional commutative formal group law $F$ over $\mathbf{Z}$, $F(X, Y) = X + Y + \dots \in \mathbf{Z}[[X, Y]]$, whose logarithm function ...
4
votes
2answers
450 views

How to calculate the sum of remainders of N?

I'm trying to sum the remainders when dividing N by numbers from 1 up to N $$\sum_{i = 1}^{N} N \bmod i$$ It's easy to write a program to evaluate the sum if N is small in O(N) but what if N is large ...
1
vote
0answers
52 views

Question regarding a theorem of Erdos and Renyi on $B_2(g)$ sequence

Let $S \subset \mathbb{N}$. We say $S$ is of type $B_2(g)$ if the number of representation of the form $n = s_1 + s_2 \ (s_1 \leq s_2)$ is bounded by $g$ for every $n \in \mathbb{N}$. Let $S(n)$ be ...
1
vote
1answer
202 views

Sum of two squares - Number of steps in Fermat descent

If a prime $p$ can be written as the sum of two squares, then one can construct this representation via Fermat descent if we know an $x$ such that $x^2 \equiv −1 \mod p$. Is there a possibility to say ...
10
votes
0answers
698 views

Primes and Parity

This problem is motivated by the polymath4 project. There, the aim was to find an efficient deterministic algorithm for finding a prime larger than $N$. The hope was to find a polynomial algorithm in ...