# Tagged Questions

**3**

votes

**1**answer

254 views

### Primes $p=x^2+27y^2$ and Ramanujan's $x_1^{1/3} + x_2^{1/3} + x_3^{1/3}$

I was trying to generalize,
...

**0**

votes

**0**answers

68 views

### Arithmetic functions associated with Hurwitz Zeta function raised to arbitrary complex powers, $\zeta(s,q)^z$ for $q \in \mathbb{N}$?

If $\zeta(s)$ is the Riemann Zeta function, then $\zeta(n)^z$, with $z \in \mathbb{C}$, $\Re(s)>1$, can be represented as
$$\zeta(s)^z=\sum_{n=1}^\infty \frac{d_z(n)}{n^{-s}}$$
where $d_z(n)$ ...

**7**

votes

**1**answer

536 views

### Application of the Riemann hypothesis and the ABC conjecture to independence results

In Old Home Page of Andreas Weiermann Andreas Weiermann has stated the following:
Quite recently I submitted a preprint about an application of the Riemann hypothesis and the ABC conjecture to ...

**1**

vote

**1**answer

129 views

### Cardinality of the prime divisor set of a k-power sum

Let $a_{1},\dots,a_{n}$ be positive natural numbers ($n>2$) such that $a_{i}\neq a_{j}$ if $i\neq j$. I want to prove that
$$ \left\lvert \left\{ p \text{ prime} \; : \; p \mid \sum_{i=1}^n ...

**2**

votes

**3**answers

782 views

### Where do Set Theory and Number Theory meet together?

As all know, by absoluteness theorems in Set Theory, most of theorems in number theory are $ZFC$-provable if and only if they are consistent with $ZFC$, it's because of absoluteness of essence of ...

**-2**

votes

**1**answer

150 views

### Prove that $\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2})$ [closed]

Let $a,b\in\mathbb{Z}$, and $f\in C^2([a,b])$ such that $|f''(t)|\asymp \lambda$ for $a\le t\le b$. Prove that
$$\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2}).$$
...

**3**

votes

**1**answer

153 views

### Pathological behavior of Lie algebra under a map of abelian schemes

I am trying to understand Example 7.5/9 from the book "Neron models". There one has a discrete valuation ring $R'$ that is the localization of $\mathbb{Z}[\zeta_p]$ at $p$, so that the absolute ...

**5**

votes

**1**answer

162 views

### Which criteria for “uniformly splitting” polynomials?

Let $P(x)$ be an irreducible monic polynomial of degree $\ge4$ with integer coefficients. We all know that over a finite field $\mathbb F_p$, $P$ will often split, and I am interested in polynomials ...

**6**

votes

**1**answer

150 views

### When is $\vartheta(x)>x$? [Skewes number analog]

Let $\vartheta(x)=\sum_{p\le x}\log p$. What is known about the first time $\vartheta(x)>x?$
Bays & Hudson give good upper bounds (slightly improved by Chao & Plymen) on the first crossing ...

**0**

votes

**0**answers

124 views

### Shimura reciprocity law

I have spent quite some time understanding class fields generated by Kummer extensions and class fields generated by modular forms. Now, I am turning the notch of sophistication a bit to study class ...

**1**

vote

**1**answer

365 views

### Numerical evidence and argument against Littlewood conjecture

This is joint work with someone. We got numerical evidence and argument
against Littlewood conjecture, though mistakes are certainly possible.
Littlewood conjecture states that for any two real ...

**6**

votes

**1**answer

263 views

### abelian $\ell$-adic representations in $\widehat{SL(2,Z)}$

In Grothendieck's Esquisse he claims that the action of
$$\text{Gal}(\mathbb Q)\to\text{Out}(\pi_1(M_{1,1})=\text{Out}(\widehat{SL(2,Z)})$$
obtained from the homotopy exact sequence of the étale ...

**5**

votes

**1**answer

459 views

### Modular forms and “too many symmetries”

How do we interpret Barry Mazur's quote of
Modular forms are functions on the complex plane that are inordinately symmetric. They satisfy so many internal symmetries that their mere existence ...

**2**

votes

**0**answers

65 views

### Varieties with few monomials and the n-conjecture

The n-conjecture
is a generalization of abc and basically says that the if
$a_1 + \ldots + a_n=0$, no proper subsum vanishes and $a_i$
are coprime, then the radical of $a_1\cdots a_n$ can't be
too ...

**7**

votes

**1**answer

207 views

### p-adic Stein spaces

The higher cohomology of coherent sheaves vanish on Stein spaces (both complex and p-adic). In the case when the space ($X$) is a curve and we're working in the complex world, this shows that all ...

**2**

votes

**1**answer

91 views

### Does the Lehmer quintic parameterize certain minimal polynomials of the $p$th root of unity for infinitely many $p$?

The solvable Emma Lehmer quintic is given by,
$$F(y) = y^5 + n^2y^4 - (2n^3 + 6n^2 + 10n + 10)y^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)y^2 + (n^3 + 4n^2 + 10n + 10)y + 1 = 0$$
with discriminant $D = (7 + ...

**5**

votes

**0**answers

541 views

### Zeta function double product

Is it possible to write the following double product in terms of the zeta function?
\begin{align}
&\prod_{i=1}^{\infty}\prod_{j=1}^{\infty} \frac{1}{1-(p_i\ p_j)^{-s}}
\end{align}
Extending the ...

**7**

votes

**2**answers

326 views

### Least supersingular prime

Given an elliptic curve over the rationals, what can one say about the size of the smallest supersingular prime?

**12**

votes

**4**answers

614 views

### Zeros of the derivative of Riemann's $\xi$-function

The Riemann xi function $\xi(s)$ is defined as
$$
\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s).
$$
It is an entire function whose zeros are precisely those of $\zeta(s)$.
Since $\xi$ is real ...

**7**

votes

**2**answers

412 views

### Numbers with all N-digit prefixes divisible by N

In base 10, the number 3816547290 contains every digit exactly once. When I take the first N digits, that substring is divisible by N. For example, 381 is divisible by 3, 38165 is divisible by 5, etc. ...

**0**

votes

**1**answer

120 views

### Short arithmetic progressions in quadratic residues

Let p be a prime number of form 4k+1.
I guess that there are c(d) number of 3-terms arithmetic progressions (AP) in the set of quadratic residues modulo p, where c(d) is an integer constant depending ...

**-2**

votes

**2**answers

169 views

### Time estimate to determine if a number is prime [closed]

How long does it take to verify that a given number is a prime number, as a function of its number of digits, in a personal computer, say? How computationally hard is this?

**12**

votes

**1**answer

500 views

### Is there a known primitive recursive upper bound on the nth “Zhang prime”

(This question is pure curiosity. Feel free to close it if you feel it is not appropriate for mathoverflow.)
In 2013 Zhang showed that there are infinitely many pairs of primes which are less that ...

**3**

votes

**1**answer

240 views

### “Weight-mondoromy” for open varieties

Suppose that $X/\mathbb{Q}_p$ is a smooth, projective variety, and choose a prime $\ell\neq p$. Then the weight-monodromy conjecture says that the graded pieces $\mathrm{Gr}_k^M$ of the monodromy ...

**2**

votes

**1**answer

147 views

### Special values of Hecke L-function

The Dedekind zeta function for a number field $K$ is defined as
$\zeta_K(s)=\sum_{I\subset O_K} (N_{K/\mathbb{Q}}(I))^{-s}$.
By attaching a Hecke character $\psi$, we can define ...

**3**

votes

**1**answer

126 views

### How to generate $n$ FP32 rationals s.t. no two distinct k-el. subsets have same sum?

First some
Background: I have lots and lots of integer matrices, whose rows are $k$-combinations (without repetitions and sorted) of numbers from the set $S:=\{1,...,n\}$ and needed to be compared ...

**3**

votes

**1**answer

264 views

### Waring's problem

What is the best known quantitative upper bound for the quantity $G(k)$?
I know that it's due to Trevor Wooley and in simplest form states that
$\limsup_{k \to \infty} \frac{G(k)}{k \log k} \le 1$.
...

**3**

votes

**1**answer

205 views

### Representation of GL(n, F_p) over F_p, for n small

The question is related to this post
Representation theory of the general linear group over a finite prime field
However, I am asking for more detailed references for n small, for example, for n=2, ...

**0**

votes

**0**answers

95 views

### Questions on roots of integral polynomials over $\mathbb{F}_p$

I asked earlier (A question on how polynomials split over $\mathbb{F}_p$ for large primes $p$) on when a polynomial $f(x) \in \mathbb{Z}[x]$ has a small root over $\mathbb{F}_p$, where $p$ is a large ...

**11**

votes

**4**answers

544 views

### How to calculate the infinite sum of this double series?

I'm calculating this double sum:
$$
\sum _{m=1}^{\infty } \sum _{k=0}^{\infty } \frac{(-1)^m}{(2 k+1)^2+m^2}
$$
I know the answer is
$$
\frac{ \pi \log (2)}{16}-\frac{\pi ^2}{16}
$$
which can be ...

**0**

votes

**0**answers

62 views

### Does this quaternary quartic form primitively represent infinitely many sufficiently large powers?

Let $g(x,y,z,t)=(x+y+z+t)^4-a h(x,y)$ where $h(x,y) \in \{x^4,xy^3,x^2y^2\}$
and $a$ is integer.
Does $g$ represent infinitely many powers $r^n$
with $n > 4$, $x+y+z+t,ah(x,y)$ take distinct ...

**28**

votes

**3**answers

922 views

### Why do Pell equations appear in Ramanujan's pi formulas?

While answering this MSE question about the Pell equation $x^2-29y^2=1$, I noticed that certain fundamental solutions appeared in Ramanujan's famous pi formula.
I. Given the fundamental unit,
...

**7**

votes

**1**answer

557 views

### Roots of a polynomial in a finite field related to Fermat's Last Theorem

In my class, we proved the following condition: define the polynomial $P_l(x)$ as
$$P_l(x) = \sum_{r=1}^{l-1}{\frac{1}{r}x^{l-1-r}}$$
Then if for all $a \in \mathbb{Z}/l\mathbb{Z}-\{0,1\},$ ...

**2**

votes

**2**answers

225 views

### Large solutions to Thue equations

Suppose that $f(x,y) \in \mathbb{Z}[x,y]$ is a homogeneous polynomial, or binary form, of degree $d$. The equation
$$f(x,y) = h$$
for a given integer $h$ is known as Thue's equation (so named ...

**23**

votes

**1**answer

937 views

### Which natural numbers are a square minus a sum of two squares?

Question: Which natural numbers are of the form $a^2 - b^2 - c^2$ with $a>b+c$?
This question came up in (Eike Hertel, Christian Richter, Tiling Convex Polygons with Congruent Equilateral ...

**1**

vote

**1**answer

237 views

### A question on how polynomials split over $\mathbb{F}_p$ for large primes $p$

Suppose that $f(x) \in \mathbb{Z}[x]$ is an irreducible polynomial (over $\mathbb{Q}$). Let $p$ be a very large prime with respect to the coefficients of $f$. Then it is possible that $f(x)$ may ...

**0**

votes

**0**answers

124 views

### Is the Jacobi theta function invertible?

Let $\theta$ denote the Jacobi theta function:
$$\theta=\sum_{k=0}^{\infty}{(-1)^kq^{k(k+1)}sin((2k+1)\frac{2\pi}{\omega_1}Re(z))},$$
and we have a complex number $t$. Suppose that we know there ...

**3**

votes

**0**answers

93 views

### Zeta functions with Brauer class

In algebraic geometry there are examples when a variety $X$ is somehow related (I call it double-mirror) to another variety $Y$, together with
a 2-torsion Brauer class. By "related" I mean statements ...

**7**

votes

**1**answer

255 views

### Euler's Triangular Number closure properties

Burton, in "Elementary Number Theory", states that the following problems are due to Euler 1775:
If $n$ is a triangular number, then so are $9n+1$, $25n+3$ and $49n + 6$.
R. F. Jordan in the J. ...

**1**

vote

**1**answer

307 views

### Improving the bound $q < n\sqrt{3}$ for an odd perfect number $N = {q^k}{n^2}$ given in Eulerian form

(Note: This was cross-posted from MSE.)
Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q, n) = 1$).
Therefore, ...

**4**

votes

**0**answers

130 views

### The divisors of $p-1$ and high-degree residues modulo $p$

Here is a somewhat more explicit version of a question that I asked a while ago.
Suppose that $p$ is a prime of the form $p=2n(n+1)+1$, with a positive integer $n$. Can every odd prime divisor of ...

**8**

votes

**1**answer

281 views

### minimizing an integral over integer-coefficient polynomials $\displaystyle \inf_{f \in \mathbb{Z}[x]} \int_a^b f(x)^2 \, dx $ [duplicate]

Let's consider the space $L^2[a,b]$ of functions on the interval and the norm:
$$ ||f(x)||^2 = \int_a^b |f(x)|^2 \, dx $$
Now what if we consider only polynomials with integer coefficients: $f(x) ...

**0**

votes

**0**answers

72 views

### Lucasian Primality Criterion for Specific Class of $k \cdot 2^n-1$

Definition
Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$ , where $m$ and $x$ are nonnegative integers .
Conjecture
Let $N=k\cdot 2^n-1$ such ...

**2**

votes

**2**answers

128 views

### Number of *distinct* dot products of an integer vector by elements of a hyper-rectangle

Imagine a vector $\boldsymbol{v}$ composed of integers, and the set $S$ of all integer vectors within a hyper-rectange, with one corner at the origin and other at $\boldsymbol{m}$. In other words: $S ...

**6**

votes

**0**answers

165 views

### Can these two proofs of the parametrization of pythagorean triples be unified?

I am interested in the classical parametrization of rational solutions to $x^2 + y^2 = 1$.
One proof is the classical stereographic projection technique (see, e.g., here): Choose a rational point $P$ ...

**8**

votes

**2**answers

497 views

### Adjoining torsion points from abelian varieties

Let $L/\mathbb{Q}$ be the field generated over $\mathbb{Q}$ by all of the (projective) coordinates of all of the torsion points of all abelian varieties defined over $\mathbb{Q}$. Is $L$ algebraically ...

**7**

votes

**1**answer

298 views

### rational points of a hyperelliptic curve

I have the following hyperelliptic curve of genus $2$:
$$
y^2 = 561 x^6 - 41904 x^5 + 627264 x^4 + 11860992 x^3 - 197074944 x^2 + 124416^2
$$
I need to find all the rational points on this curve. ...

**1**

vote

**1**answer

192 views

### computing height on elliptic curve of the form $y^2=x^3-nx$

Let $E$ be the elliptic curve
$$y^2 =x^3 - 19*67 x$$
and $P=[26011/625,2159616/15625]$, I want to compute $\hat{h}(P)$ using formula given in
Fujita, Y., & Terai, N. (2011). Generators for the ...

**2**

votes

**1**answer

187 views

### Kloosterman sum zeroes

Can we prove the following statement about Kloosterman sums? Recall that a Kloosterman sum is given by:
$$K(a,b,m)=\sum_{0\leq x\leq m-1,\,\gcd(m,x)=1}e^{\frac{2\pi i}{m}(ax+bx^*)}$$
Where $x^*$ is ...

**10**

votes

**1**answer

1k views

### When does a Catalan number equal a Fibonacci number?

The $n=3$'rd Catalan number (A000108) is $1,1,2,5$ : $\frac{\binom{2n}{n}}{n+1}=\frac{\binom{6}{3}}{4}=\frac{20}{4}=$ 5.
The $n=4$'th Fibonacci number (A000045) is $1,1,2,3,5,...$ : 5.
Q. Which ...