**4**

votes

**0**answers

216 views

### The ABC conjecture where A and B are smooth

Mochizuki has already claimed to have proven the ABC-conjecture. But even if his claim turns out to be correct the proof will not be easy to understand. With that in mind I'm asking wether anything is ...

**0**

votes

**2**answers

142 views

### Bounds on the counting function for almost-primes

Let $P_k$ be the set of integers with at most $k$ prime factors (counting with multiplicity, say). There is an almost-prime number theorem which gives asymptotic estimates of the size of $P_k$, and ...

**2**

votes

**0**answers

55 views

### Equidistribution of Brillouin zones

Answering the question about Limiting shape for Brillouin zones Victor Kleptsyn proved that $N$th Brillouin zone is very close to a circle of radius $c\sqrt N$ (you can find all necessary definitions ...

**6**

votes

**1**answer

225 views

### Subsets of [1..N] with no three-term arithmetic progressions and no large gaps

Let S be a subset of [1..N] containing no three-term arithmetic progression, and let h(S) be the size of the largest gap between two consecutive elements of S. By Roth's theorem, h(S) has to grow ...

**2**

votes

**1**answer

281 views

### infinite product of (1-1/(p+1)) over a density 0 set of primes

Let $S$ be a density zero set rational primes, in the concrete situation
$\#\{p<X,p\in S\}=\mathcal{O}(x/(\log x)^{3/2-\delta})$ for all $\delta>0$.
Then can $\prod_{p\in S}(1-\frac{1}{p+1})$ ...

**1**

vote

**0**answers

104 views

### Are these quaternion algebras definite or indefinite?

By investigating a different problem I have ended up looking at Quaternion algebras and have a lot to learn about them. Before I do, however, I want to see if my idea has any hope of being useful. So ...

**2**

votes

**2**answers

340 views

### Is there a nice generating function proof of the following identity?

Consider the Jordan function $J_2(n)$ defined by
$$
J_2(n) = \#\{x \in (\mathbb{Z}/n)^2 \mid ord(x) = n\}
$$
(this is OEIS A007434). One can prove the following identity pretty easily:
$$
\sum_{d \mid ...

**1**

vote

**0**answers

53 views

### Calculating Mahler Coefficients

Assume $p$ be prime number ($p>2$), and let $u$ be any topological generator of the group $1 + p \mathbb{Z}_p$ (an open subgroup of the group of units $\mathbb{Z}_p^\times$ of the ring of $p$-adic ...

**4**

votes

**1**answer

378 views

### About factorization in Zhang's proof of weak Twin Prime conjecture

Why does it need to firstly factorize the number n into two factors q and r( Lemma 4 in the paper,see the following)? What's the motivation. What if it doesn't do this factorization?

**0**

votes

**1**answer

91 views

### Sum of digits of a power [closed]

Are there any explicit formula for a sum of digits for a power in the given base? A problem to be specific: find a sum of digits for a number $2^{100}$ in the system with a base 5. In the system with ...

**5**

votes

**1**answer

166 views

### are the coarse moduli schemes of finite etale covers of $\mathcal{M}_{1,1}$ smooth?

Let$\newcommand{\mM}{\mathcal{M}}$ $\mM_{1,1}$ be the moduli stack of elliptic curves. Let $R$ be a Dedekind domain, say $\mathbb{Z}[1/N]$ for simplicity, and suppose we have a finite etale cover:
...

**0**

votes

**2**answers

274 views

### Does theta(n)<n for all n imply the Riemann Hypothesis and/or vice versa?

I know that better and better bounds of the Chebyshev Theta and Psi functions are implied by knowing that the first (insert large number here) zeta zeroes lie on the Critical Line. These bounds, ...

**2**

votes

**0**answers

117 views

### On sets of coprime numbers

We know that from prime number theorem that the number of primes below $n$ and above $\frac n2$ (denoted by $\pi_{n,\frac n2}$ is approximately $$\pi_{n,\frac n2}\approx\frac{n}{2\ln n}.$$
Denote by ...

**1**

vote

**0**answers

57 views

### The genus of specialization of rational surface (compared to genus of parametrization)

Let $f(u,v),g(u,v),h(u,v)$ be polynomials with rational
coefficients or rational functions.
Assume (this happens in general) they parametrize
rational surface $F(x,y,z)=0$.
For rational $a$, assume ...

**2**

votes

**1**answer

205 views

### Reducibility of resultants

This is closely related to this question. Suppose I have the resultant $\mathcal{R}$ of two (or more polynomials) over $\mathbb{Q},$ and suppose $\mathcal{R}$ is not irreducible. What is the ...

**3**

votes

**2**answers

528 views

### What are the necessary conditions for a real number to be a cyclotomic integers？

The motivation of the question is that I try to test when a real number is not an cyclotomic integers. Or more specifically, when a positive real number is not a quantum dimension of a unitary fusion ...

**1**

vote

**1**answer

173 views

### Finding integer representation as difference of two triangular numbers

Since $n = \frac{n(n+1)}{2}-\frac{n(n-1)}{2}$, every natural number can be represented as the difference of two triangular numbers:
$ n = \frac{a(a+1)}{2}-\frac{b(b-1)}{2}$. Finding such a ...

**10**

votes

**2**answers

443 views

### Why going to number fields in number field sieve help beat quadratic sieve?

To factor an $n$ bit integer number field sieve roughly takes $$e^{c{(\ln\ln n)^{\frac23}}({\ln n})^{\frac13}}$$ time while quadratic sieve takes $$e^{c{(\ln\ln n)^{\frac12}}({\ln n})^{\frac12}}$$ ...

**2**

votes

**1**answer

141 views

### Gap between semiprimes

Is there a conjectured gap between semiprimes?
There is a conjectured gap between primes in form of Cramer's conjecture. Using this we have $p_1\leq p_0+c(\log p_0)^2$ for consecutive primes $p_0$ ...

**9**

votes

**1**answer

261 views

### Non-algebraic Hecke characters

Algebraic Hecke characters are ubiquitous in modern number theory. They are in 1-1 correspondence with one dimensional complex Galois representations, and in some precise sense they are the building ...

**10**

votes

**1**answer

444 views

### How much can an Eisenstein series be truncated?

For ease of exposition, I will stick to the simplest case: consider the Eisenstein series for $SL_2(\bf R)$
$$E(z,s)=\sum_{\gamma\in P_{\bf Z}\backslash SL_2(\bf Z)}\text{Im}(\gamma ...

**7**

votes

**4**answers

316 views

### $p$-th Fourier coefficients of newforms of level $\Gamma_1(N)$ with $p|N$

Let $f$ be a newform of level $\Gamma_1(N)$ and character $\chi$ which is not induced by a character mod $N/p$. I learned from these notes by Ribet and Stein that $|a_p|=p^{(k-1)/2}$ where $k$ is the ...

**3**

votes

**1**answer

130 views

### infinite solution of a diophantine quadratic equations

Let $a,b,c,d$ be integers such that $GCD(a,b,c,d)=1$. Assume that the diophantine equation $ax^2+bxy+cxz+dyz-x=0$ has a non-zero solution.Can we assert that it admits infinitely many solutions?
...

**6**

votes

**1**answer

142 views

### power sums and formal divisibility by the Euler totient function

For integers $s \geq 0$ and $n \geq 1$ let $\sigma_s(n) := 1^s + \dots + n^s$ and let
\begin{equation} \sigma_s^*(n) \ : = \ \sum_{\stackrel{\scriptstyle 1 \, \leq \, m \, \leq \, n}{\text{gcd}(m,n) ...

**6**

votes

**1**answer

325 views

### what are the finite etale covers of $\mathbb{Z}_p((x))$?

Let $R$ be the ring of integers of some $p$-adic field $K$ (finite over $\mathbb{Q}_p$) with uniformizer $\pi$ and residue field $k$. I'd like to understand the finite etale extensions of $R((x)) := ...

**2**

votes

**0**answers

91 views

### Residual Representation of a Motive

Suppose we have $M$ a hypergeometric motive, and $\rho$ its associated Galois rep over $\mathbb{Q}_{l}$. Is there any easy/concrete way to find $\bar{\rho}$, the residual representation at a prime (in ...

**28**

votes

**3**answers

1k views

### irreducibility of discriminant

This must be well-known to everyone but me, but here goes: take a general (monic) polynomial $p(x) = x^d + a_{d-1} x^{d-1} + \dotsc + a_0.$ The discriminant is a polynomial $D(a_0, \dotsc, a_{d-1}).$ ...

**2**

votes

**0**answers

214 views

### Generalized Ramanujan's identity with hyperbolic cotangent

Three weeks ago I derived an identity, which generalizes Ramanujan's identity with hyperbolic cotangent. Don't you know is it original or not?
$$
\sum_{n=1}^\infty \frac{\coth(\pi ...

**10**

votes

**2**answers

467 views

### Computing millions of coefficients of non self-dual modular forms

To test some conjectures made by some colleagues, I need to compute millions of coefficients of non self-dual modular forms, preferably in low weight (say 2 or 3). A form such as this.
For elliptic ...

**3**

votes

**2**answers

258 views

### multiplicative functions of powers

Suppose I have a multiplicative function $f(n),$ and I want to understand the behavior of
$$
\sum_{n<x} f(n^k),
$$ for some integer $k.$ This seems like it should be easy (since the Dirichlet ...

**4**

votes

**2**answers

332 views

### Dirichlet's approximation only using prime power as denominator

I am not sure whether this is a suitable question for MO. We know the classical version of Dirichlet's approximation theorem that if $x$ is a real number and $Q>0$ there exist $p,q\in \mathbb{Z}$ ...

**4**

votes

**1**answer

118 views

### Asymptotic for binomial sums

Let $S(n, t) = \sum_{k = 0}^n {n \choose k} ^t$.
The task is to find asymptotic behavior of $S(n,5)$, $n \to \infty$.
Asymptotic for $S(n,0)$ and $S(n,1)$ is very simple.
For $S(n,2)$ we can use ...

**8**

votes

**1**answer

465 views

### Plot of Ramanujan tau function

There is a picture on wikipedia of Ramanujan tau function. At first I noticed that there are exceptional red point (where the red points are sparse in the lower part), this should be due to Sato-Tate ...

**4**

votes

**1**answer

182 views

### Summation of an infinite q-series

When calculating a Partition function, I encounter the following summation
$$\sum_{n=0}^{\infty} x^n q^{n^2}.$$
I know that the sum $\sum_{n=-\infty}^{\infty} x^n q^{n^2}$ is a Theta function, but I ...

**4**

votes

**1**answer

238 views

### Fourier expansion of automorphic forms

we know that for $r \in \{1,2,3,4\},$ $\lambda_{Sym^rf}$ is an automorphic form (here $f$ is a modular form for the full modular group) and this fact is conjectured for $r\geq 5$ by Langlands and ...

**22**

votes

**1**answer

638 views

### How big can a set of integers be if all pairs have small gcd?

Suppose $A\subset[1,N]$ is a set of integers. If for any distinct $a,b\in A$ we have $(a,b)\leq M$ then how big can $|A|$ be?
If $M=1$ then $|A|$ is at most $\pi(N)$ since the map $a\mapsto P_+(a)$ ...

**8**

votes

**2**answers

272 views

### What is the value of $p$-adic $\zeta$-function at positive integer point?

$p$-adic zeta function is a $p$-adic interpolation of the Riemann $\zeta$-function for the values $\zeta(1−k)$, $k\ge 1$ (see $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions by Neal Koblitz) ...

**16**

votes

**1**answer

710 views

### Reference Request: Conductors of Twists of Hyperelliptic Curves

It is my understanding that if I twist a hyperelliptic curve of genus 2 whose Jacobian has conductor $N$ by a prime $p$ with $p\nmid N$, that the conductor of the Jacobian of the twist is expected to ...

**18**

votes

**2**answers

559 views

### Distribution of the Error term in GH Hardy's “curious result” $\sum_{\nu \leq n } \{ \nu \theta \}^2 = \tfrac{1}{12} n + O(1)$

In an early paper, GH Hardy talks about the distribution of "curious" sum:
$$ \sum_{\nu \leq n } \{ \nu \theta \}^2 = \tfrac{1}{12} n + O(1)$$
where $\{x\}:=x-\left \lfloor x \right \rfloor -1/2$. ...

**6**

votes

**2**answers

225 views

### Additive energy of Piatetski-Shapiro sequences

Let $c>1$, and let $A$ denote the set
$$
\Big\{ \lfloor n^c \rfloor, \quad 1 \leq n \leq N \Big\}.
$$
Thus $A$ consists of the first $N$ elements of a so-called Piatetski-Shapiro sequence.
The ...

**14**

votes

**3**answers

674 views

### Cyclotomic polynomials: $\Phi_n(p)$ is like $p^{\phi(n)}$ for big enough $p$, right?

Apologies in advance if this turns out to be simple. So far I haven't found a proof or a reference.
Although I like $p$ to be a prime, I can ask the following for positive integers $n$ and $p$, ...

**15**

votes

**3**answers

798 views

### Reference for a linear algebra result

I asked the following question (http://math.stackexchange.com/questions/1487961/reference-for-every-finite-subgroup-of-operatornamegl-n-mathbbq-is-con) on math.stackexchange.com and received no ...

**5**

votes

**1**answer

166 views

### Is this subset-sum-type problem discussed in the literature?

Let $y \in \mathbb{Z}_+^n$, with $y_1 < \dots < y_n$. I am interested in finding a 0-1 matrix $A$ and $x \in \mathbb{Z}_+^m$ s.t. $m$ is minimal and $Ax = y$, where I am guaranteed that at least ...

**1**

vote

**1**answer

194 views

### How often is $2^n-1$ a number with few divisors?

It is unknown whether $2^n-1$ is (a Mersenne) prime for infinitely many values of $n$. Such $n$ must necessarily be prime itself due to the factorization below ...

**5**

votes

**0**answers

153 views

### Effective bound of $L(1,\chi)$

Let $d$ be a fundamental discriminant and let $\chi$ be the associated primitive real character of modulus $\vert d \vert$. Assuming GRH, Littlewood proved that as $\vert d \vert$ grows large,
$$L(1, ...

**7**

votes

**2**answers

183 views

### How to formalize the *loci of equal arg($\zeta(s)$)* (“isogones”) in the near of a nontrivial root

(This is an extension and specification of a question which I initially asked in MSE having now one comment (which I could not yet digest completely) and which I also detailed further (after working ...

**11**

votes

**2**answers

537 views

### First formulation of the Dedekind and Hasse-Weil conjectures

I'm looking for the original statement of two important conjectures in number theory concerning L-functions. I'm particularly interested in pinning down the year in which they were first formulated:
...

**8**

votes

**0**answers

176 views

### Attractors of arithmetically small points

Consider the "points" to be in $\mathbb{G}_m(\mathbb{C}) = \mathbb{C}^{\times}$. By a sequence of "arithmetically small points" is meant a sequence of pairwise different algebraic points $\beta$ that, ...

**6**

votes

**1**answer

184 views

### $N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}?$

Assume that $ab \neq 0$. What is $$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}?$$I need this result, but unfortunately I am not a number theorist. Could ...

**6**

votes

**2**answers

264 views

### Motivation for Hirzebruch-Jung Modified Euclidean Algorithm

Let $a,b \in \mathbb{N} \ \ s.t. \ \ a > b$ have $\gcd(a,b) =1$. We can define the Hirzebruch-Jung modified euclidean algorithm as follows:
Let $e_i \in \mathbb{N} >2$, and $ r_k \in ...