**1**

vote

**1**answer

240 views

### Density of Sophie Germain $3\bmod 4$ primes

Is there any reason to expect the density of primes $p$ such that $2p+1$ is also a prime where $p=3\bmod4$ holds would be different from case of $p=1\bmod4$?
What if $2p+1$ is replaced by $2p-1$ and ...

**3**

votes

**1**answer

184 views

### Divisibility of Dirichlet L-functions

Let $k$ be an even integer and $p$ a prime number such that $p-1|k$.
Suppose that $p$ does not divide $L(1-k,\chi)$ and $L(1-k,\psi)$, where $\chi$ and $\psi$ are quadratic characters.
Can we deduce ...

**6**

votes

**1**answer

143 views

### Density of prime divisors of $a^n + b$

Suppose $a > 1, b \neq 0$ be two rational numbers. Is it known in general that the set of prime divisors of (the numerator of) $a^n + b$ has a positive relative density?

**5**

votes

**0**answers

154 views

### Solving polynomial equations modulo $1$

Let $P\in \mathbb{R}\lbrack x\rbrack$ be given. (In practice, the coefficients could be given as, say, decimals to sufficient precision.) Let $M\geq 1$, and let $I$ be an interval in $\mathbb{R}/\...

**6**

votes

**1**answer

223 views

### On property of monic polynomial with integer coefficients

For a monic polynomial with integer coefficients $f$ where $\partial f = 2$, we have
$$
\textrm{inf}(f(x)) > 0 \implies
\textrm{inf}(f(x)) \geq \frac{3}{4} .
$$
Could we generalize this (for ...

**0**

votes

**1**answer

107 views

### Finding out $p$-torsion elements of an elliptic curve $E$ over $\mathbb{Q}_p$

Let $E$ be an elliptic curve over $\mathbb{Q}$.
Then how to compute the $p$-torsion elements of $E$ over the $p$-adic field $\mathbb{Q}_p$ using SAGE or any other means ?
At least can we say whether ...

**0**

votes

**1**answer

96 views

### Number of solutions to a modular congruence

What methods are there for determining the number of solutions to modular congruences of the form $a^m \equiv b^n k \pmod{p}$ with $1 \leq a,b \leq p-1$ where $p$ is a prime?
In the case $m,n$ are ...

**3**

votes

**1**answer

127 views

### The average number of a class of reduced, primitive, positive definite binary quadratic forms

Let $f(x,y) = ax^2 + 2bxy + cy^2 \in \mathbb{Z}[x,y]$ be a positive definite binary quadratic form with even discriminant. We say that $f$ is primitive if $\gcd(a,2b,c) = 1$ and it is reduced if $0 &...

**3**

votes

**1**answer

233 views

### Number of prime divisors of p^2-1 for a prime p

Let $n$ be an integer and $n=p_1^{a_1}\dots p_s^{a_s}$ be its factorization into primes. Denote by $\Omega(n)$ the sum of $a_i$. Does there exist a constant $k$ such that there are infinitely many ...

**3**

votes

**2**answers

353 views

### Relation of these two Dirichlet $L$-functions

Let $\chi$ and $\psi$ be two quadratic Dirichlet characters and let $L(s,\chi)$ and $L(s,\psi)$ their associated Dirichlet $L$-functions.
Is there a realtion between these two Dirichlet $L$-functions:...

**2**

votes

**1**answer

139 views

### Ask for a special function related to the error function

I am wondering whether anyone knows the following integration has a named special function or a reference
$$
F_{a,b}(z) :=\frac{2}{\sqrt{\pi}} \int_0^z \text{erf}(a+b y)\: e^{-y^2} \text{d}y
$$
for ...

**2**

votes

**2**answers

310 views

### How many integer solutions of $a^2+b^2=c^2+d^2+n$ are there?

Are there any references to study the integer solutions (existence and how many) of Diophantine equations like $a^2+b^2=c^2+d^2+2$, $a^2+b^2=c^2+d^2+3$, $a^2+b^2=c^2+d^2+5$...? Actually, I can prove ...

**6**

votes

**0**answers

91 views

### Is the analogue of the Simple Continued Fraction in p-adic number fields useful?

Is there an analogue of the Simple Continued Fraction in p-adic number fields? Is it useful and does it have relations to best rational approximation in the p-adic sense? In the analytic case there is ...

**7**

votes

**4**answers

838 views

### Clarification on the weak BSD conjecture

It is usually told that Birch and Swinnerton-Dyer developped their famous conjecture after studying the growth of the function
$$
f_E(x) = \prod_{p \le x}\frac{|E(\mathbb{F}_p)|}{p}
$$
as $x$ tends to ...

**2**

votes

**1**answer

116 views

### Clarification request on sign changes of Hecke eigenvalues

In their paper 'Sign changes of Hecke eigenvalues', Matomaki and Radziwill established in Lemma 6.2 the following result: There exists absolute positive constants $c$ and $\eta$ such that uniformly in ...

**3**

votes

**1**answer

464 views

### On progress towards inverse Galois problem over rationals

I have heard that Progress towards the Inverse Galois problem over $\mathbb{Q}$
is very well documented for sporadic groups ($M_{23}$ is the only case open) and for $PSL_n(q)$.
From where I can read ...

**2**

votes

**1**answer

121 views

### Simultaneous $\pmod{p}$ congruences of two ternary quadratic forms

Let
$$\displaystyle f(x_1,x_2,x_3) = a_1 x_1^2 + a_2 x_2^2 + a_3 x_3^2,$$
$$\displaystyle g(x_1, x_2, x_3) = b_1 x_1^2 + b_2 x_2^2 + b_3 x_3^2$$
be two integral ternary quadratic forms with $f$ ...

**3**

votes

**0**answers

169 views

### Effective prime number theorem

The prime number theorem implies that for every $ϵ>0$, there is $n_\epsilon$ such that for all $n≥n_\epsilon$ the number of primes in $[n,cn]$ is at least $\frac{(c−1−\epsilon)n}{\log n}$ and at ...

**3**

votes

**1**answer

278 views

### Frobenius at ramified primes

Let $E$ be an elliptic curve defined over $\mathbf{Q}$, fix an odd prime $p>3$, let $T_p$ denote the $p$-adic Tate module of $E$, and let $V_p = T_p \otimes \mathbf{Q}_p$.
If the action of $G_\...

**3**

votes

**0**answers

183 views

### Number of solutions to $x_1x_2=x_3x_4\bmod n$

In https://www.math.ksu.edu/~cochrane/research/xyuvmodp.pdf it is shown $x_1x_2=x_3x_4\bmod p$ where $p$ is a prime has $\frac{|\mathcal B|}p+O(\sqrt{|\mathcal B|}\log^2p)$ solutions $(x_1,x_2,x_3,...

**4**

votes

**0**answers

272 views

### Proof for new deterministic primality test possible?

Conjecture:
Let $n \in \mathbb{N}$ and $n$ odd.
Then the number $N=2^2 + n^2$ is prime, if and only if $N$ divides $2^{(N-1)/2} + 1$.
Thanks.

**13**

votes

**3**answers

454 views

### Infiniteness of the Galois cohomology over a number field with coefficients in a finite Galois module

Let $k$ be a number field and $M$ be a nonzero finite discrete $\mathrm{Gal}(\bar k/k)$-module. Is it true that $H^1(k,M)$ is infinite?
This would complete the answer of Daniel Loughran. There is a ...

**5**

votes

**0**answers

111 views

### Primitive element for a number field, and ramification

Let $K=\mathbb Q(\theta)$ be a number field with integral primitive element $\theta$, and let $f(x)$ be the minimal polynomial of $\theta$. Let $p$ be a rational prime. It's well known that if $p$ ...

**3**

votes

**0**answers

101 views

### Globalizing local field extensions with controlled ramification

Let $K_1/k_1, \ldots, K_r/k_r$ be "separable cyclic" extensions of degree $n$ where each $k_i$ is a local field of characteristic 0 (archimedean or not). By separable cyclic of degree $n$, I mean the ...

**1**

vote

**0**answers

108 views

### What interesting information can be deduced from knowledge of how deep a geodesic ventures into the cusp

First of all I have to apologise as I am not a geometer and my knowledge of geometry is poor. Let $M$ be the modular surface and $\gamma$ to denote a geodesic in $M$. In the the following paper by ...

**8**

votes

**1**answer

197 views

### Algebraic points of uniformly bounded degree on an algebraic variety

Let $k$ be a perfect field, and let $\bar k$ be a fixed algebraic closure of $k$.
Let $\overline{X}$ be a nonempty smooth algebraic variety over $\bar k$.
Does there exist a natural number $d=d(\...

**4**

votes

**1**answer

139 views

### Irreducible monic polynomials

I am looking for criteria for the irreducibility of monic polynomials with constant term $\pm1$ over $\mathbb Q$. Eisenstein's criterion clearly doesn't apply here.
For instance, for the family of ...

**4**

votes

**1**answer

210 views

### Reference for: Every local field can be realized as the completion of a global field

It is well known that every local field (i.e. nondiscrete topological field locally compact with respect to the topology) is the completion of some global field. I know the argument, a nice ...

**2**

votes

**0**answers

78 views

### Is $\sum_{\rho} \frac{1}{(\rho - s)^{1 + \delta}}$ bounded as Im(s) goes to infinity?

Let $\rho$ denote the zeros of the Riemann zeta-function
and $\delta > 0$.
Is the function
$f(s) = \sum_{\rho} \frac{1}{(\rho - s)^{1 + \delta}}$
bounded as Im(s) goes to infinity?(the real part ...

**17**

votes

**2**answers

899 views

### What is the smallest positive integer for which the congruent number problem is unsolved?

The congruent number problem is the problem of figuring out whether a given positive integer $N$ is the area of a right-angled triangle with all side lengths rational. According to Dickson's "History ...

**4**

votes

**2**answers

195 views

### Group cohomology question, trivial Galois action on discrete Galois module means we can say what about kernel of map

Say we have a number field $K$. Let $G_K = \text{Gal}(\overline{K}/K)$. Let $M$ be a discrete $G_K$-module. We know that $H^1(K, M) := H^1(G_K, M)$, i.e. profinite group cohomology. For each place $v$ ...

**15**

votes

**1**answer

422 views

### Pop's proof that $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})=\mathrm{Aut}\underline\pi^{alg}_{\overline{\mathbb{Q}}}$

I've heard of this result in a paper on which Yves André proves the p-adic analogue (that is, $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)=\mathrm{Aut}\underline\pi^{temp}_{{\mathbb{C}}_p}$), ...

**2**

votes

**0**answers

74 views

### Conditions to check whether a prime ramifies in a certain way

Let $p$ be a prime number and let $f = a_n x^n + \cdots + a_0$ be an irreducible polynomial of degree $n \geq 3$ with a root $\alpha$. We say that $p$ ramifies in a powerful way over $K = \mathbb{Q}(\...

**7**

votes

**1**answer

197 views

### Characters of simply connected semsimple algebraic groups over local fields

Let $G$ be a semisimple algebraic group over $\mathbb{Q}_p$. Then by definition $G$ admits no non-trivial algebraic characters, i.e. homomorphisms $G \to \mathbb{G}_m$.
However, it is quite possible ...

**2**

votes

**0**answers

124 views

### Uniform bound on the Mordell-Weil rank of elliptic curves

I want to know that if there is an uniform upper bound for the rank of elliptic curves over $\mathbb{C}(t)$, the rational function field over complex numbers, and generaly over the function field of ...

**0**

votes

**0**answers

48 views

### Estimating exponential sum of the form $\sum e( \alpha_1 f + \alpha_2 L)$, where $f$ is quadratic and $L$ is linear, on the minor arcs

Let $f(x_1, x_2, x_3)$ be a degree two homogeneous polynomial with coefficients in $\mathbb{Z}$. Let $L(x_1, x_2, x_3)$ be a linear homogeneous polynomial with with coefficients in $\mathbb{Z}$. Let ...

**1**

vote

**0**answers

71 views

### “Algebrazing” canonical subgroups of elliptic curves

I'm puzzled by a part of the construction of the canonical subgroup of a "not too supersingular" elliptic curve. In Katz's paper, one produces a subgroup of the formal group of the elliptic curve but ...

**2**

votes

**0**answers

365 views

### Analytically continuing the limit of this series?

Main Question
I believe the following formula gives the right answer:
$$ \lim_{k \to \infty} \lim_{n h \to k} \left( \sum_{r=1}^n c_r f(hr) h\right) = \int_0^\infty f(x) \, dx \times \sum_{r=1}^\...

**1**

vote

**2**answers

311 views

### Expository articles on Algebraic Number Theory

I am about to start learning Algebraic Number Theory and thus was looking for some expository articles on this subject. So far I have found two such articles:
Dickson, L. E.. (1917). Fermat's Last ...

**2**

votes

**1**answer

163 views

### Fundamental Units in Totally Real Cubic Fields

How much is known about the fundamental units in totally real cubic fields? For example, Daniel Shanks has a family of totally real cubic fields for which the fundamental units are known; those with ...

**1**

vote

**1**answer

182 views

### Repdigit numbers, which are sum of consecutive squares

Following up on this question,
http://math.stackexchange.com/questions/1788015/is-112122132142152162-1111-special/1788102?noredirect=1#comment3649733_1788102
is anything known about the sequence of ...

**2**

votes

**1**answer

148 views

### Character group of the multiplicative rationals

I was reading some stuff on Hecke characters and came across an issue I have not been able to resolve. I posted it here on math stack exchange first.
Let $\mathbb{Q}^{\times}$ be the multiplicative ...

**6**

votes

**2**answers

270 views

### Noncoprime polynomial values

Let $p_1, \ldots, p_n$ be a finite sequence of nonconstant polynomials with integer coefficients. Does there exist a finite sequence of integers $x_1, \ldots, x_n$ such that the integers $p_1(x_1), \...

**4**

votes

**2**answers

233 views

### Adjoint semi-simple algebraic groups over non-algebraically closed fields

Let $k$ be a field of characteristic zero and let $G$ be an adjoint semi-simple algebraic group over $k$.
On p34 of the paper "Sansuc - Groupe de Brauer et arithmétique des groupes
algébriques ...

**12**

votes

**1**answer

461 views

### A combinatorial identity involving harmonic numbers

The harmonic numbers are given by $$H_n=\sum_{k=1}^n\frac{1}{k}.$$
Numerical calculation suggests
$$
\sum_{k=1}^{n}(-1)^k{n\choose k}{n+k\choose k}\sum_{i=1}^{k}\frac{1}{n+i}=(-1)^nH_n.
$$
I can not ...

**2**

votes

**1**answer

137 views

### Want more details about the image of a Maass form in the AIM press release concerning LMFDB

Actually I came upon this through MO a couple of days ago: in here
(http://aimath.org/aimnews/lmfdb/) there is a mesmerizing image
The caption reads
A Maass form, one of the 20 different types ...

**1**

vote

**1**answer

105 views

### Linear functions

Let $(f_1, f_2, \ldots, f_n)$ be an $n$-tuple of functions mapping non-negative integers to non-negative integers. Let $m$ be a positive integer.Suppose there exists a function $f$ apping non-negative ...

**1**

vote

**2**answers

189 views

### Implicit constant in Tenenbaum's result

In his famous book 'Introduction to Analytic and Probabilistic Number Theory', Gérald Tenenbaum established the following result (Theorem III.3.5):
Let $g$ be a positive multiplicative function and ...

**15**

votes

**1**answer

332 views

### Combinatorics problem about sum of natural numbers

Following combinatorics problem is claimed to be an open problem in "The Princeton Companion to Mathematics" (pp. 6)
Let $a_1,a_2,a_3,...$ be a sequence of positive integers, and suppose that each ...

**1**

vote

**0**answers

103 views

### Ramanujan sum type

I try to show
$$\sum _{k=1}^{\infty } \frac{e^{-2 k} k}{e^{-2 k}+1}=\frac{\pi ^2}{48}-\frac{\pi ^2-6 \left(\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}(1)+\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}\left(\frac{-2 i+\pi }...