**3**

votes

**1**answer

174 views

### When are the powers of 2 sum-free mod n?

I've encountered the following question in my research:
Let $A$ be a subset of
$\mathbb{Z}/n\mathbb{Z}$. Let me call $A$ "sum-free" if there is no solution to
$x+y=z$ for $x,y,z \in A$ with distinct ...

**1**

vote

**0**answers

88 views

### Bound on $g(n+1)/g(n)$ for Landau's function

I have read that for the Landau function $g(n)$ (http://mathworld.wolfram.com/LandausFunction.html), one knows that
$\lim_{n \rightarrow \infty} g(n+1)/g(n) = 1$
(stated, but not proved in "On ...

**2**

votes

**2**answers

181 views

### Simultaneous lcms

Suppose that we have some finite number of $k$-tuples then we define the lcm of two of these tuples to be the tuple of lcms of the co-ordinates. E.g. $[(9, 10), (5, 18)] = ([9, 5], [10, 18]) = (45, ...

**0**

votes

**1**answer

155 views

### A conjecture on the prime counting function

I was thinking about the Second Hardy-Littlewood conjecture for quite sometime (some of my posts are related to this). In one of my earlier post I conjectured that the inequality ($\pi(x)$ denotes the ...

**2**

votes

**1**answer

141 views

### isogeny clases of CM abelian varieties

Let $A$ be an abelian variety defined over $\overline{\mathbb{Q}}$ and with complex multiplication by a CM field $K$. Looking at the action of $K$ on $H^0(A, \Omega^1_A)$ one gets a CM type of $K$, ...

**1**

vote

**1**answer

81 views

### Differences of consecutive ordered fractional parts

Let $r$ and $h$ be a real numbers and $n>0$. Write the fractional parts $\{k*r+h\}$, for $k = 1,2, . . . n$, in increasing order as $$ a_1 < a_2 < \cdots < a_n.$$ Let $D_n$ be the set of ...

**5**

votes

**2**answers

191 views

### PSL(2,p) as quotient of triangle groups

As a by-product of some Magma computations, I've observed that, for each prime $p$
such that ${\rm PSL}(2,\mathbb{F}_p)$ can be a quotient of the $(3,3,4)$-triangle group
(i.e. $p \equiv \pm 1 ...

**16**

votes

**0**answers

295 views

### Need explicit formula for certain “$q$-numbers” involving gcd's

The question is motivated by yet another possible approach to a combinatorial problem formulated previously in "Special" meanders. I'm not giving details of the connection as I believe the ...

**7**

votes

**2**answers

276 views

### When is a sequence the sum of two Beatty sequences?

In other words, given a sequence $(s_n)$, how can we tell if there exist irrationals $u>1$ and $v>1$ such that
$$s_n = \lfloor un\rfloor + \lfloor vn\rfloor$$
for every positive integer $n$?
...

**10**

votes

**0**answers

428 views

### One-to-one correspondance between zeta zeros and the prime powers? [closed]

This question is highly speculative, but I would really appreciate some insight into the problem. Previously asked on MSE without answer here.
I have noticed an interesting property related to the ...

**4**

votes

**0**answers

345 views

### Fermat's Last Theorem in $\mathbb{Z}/n\mathbb{Z}$

Let $\mathbb{N}$ denote the set of positive integers.
We define a relation $R\subseteq \mathbb{N}^3$ by
$$ R = \{(x,y,z) \in \mathbb{N}^3: \exists n\in \mathbb{N}: 1< n \leq \max\{x,y,z\} \land ...

**3**

votes

**2**answers

167 views

### Transitivity of discriminant for flat algebras

Sorry if the question doesn't feed this site, I'm reposting it from MSE. Nobody answered it there and I couldn't find the proof in general case(whenever it was mentioned the proof was referred to as a ...

**3**

votes

**0**answers

64 views

### Searching information on a certain function with a fixed point property connecting Moebius $\mu$ and Fibonacci numbers

Let $\mu$ be the Moebius function and define for $1\leq n\in\mathbb{N}$
$$
f(n) =
\left\{
\begin{array}{ll}
\mu\left(\frac{n}{2}\right) + \mu\left(\frac{n}{4}\right), & n\equiv 0, 4, 8\mod 12, \\
...

**2**

votes

**0**answers

180 views

### Can estimate upper bound of $|p_{i}|$ or $|q_{i}|?$

when I Find the diophantine-equation rational points $$2y^2=x^6-x^2+2$$
I using Faltings's theorem showed that there are only finitely many solutions,if we assmue that
...

**3**

votes

**1**answer

156 views

### higher dimensional analogue of EGZ theorem

The EGZ theorem states that any multiset of $2n-1$ integers has a subset of size $n$ the sum of whose elements is a multiple of $n$.
Kemnitz-Reiher theorem is a 2-dimensional analogue of EGZ. Here is ...

**0**

votes

**1**answer

291 views

### On a claim of Zagier on extending a map to cocycle

Zagier, in his paper 'Some Surprising Consequences of the Cohomology of SL$_2(\bf{ Z})$' (link, p. 6), studies the action of $\Gamma=PSL_2(\bf Z)$ on a vector space $V$, denoting the action by $v\ |\ ...

**0**

votes

**1**answer

184 views

### Does unique factorization for automorphic L-functions imply a weakened form of Ramanujan conjecture?

Selberg orthonormality conjecture for automorphic L-functions was proven under Ramanujan conjecture, and SOC itself implies unique factorization for those L-functions.
My question is: does the ...

**5**

votes

**1**answer

293 views

### Computing an eigencuspform in $S_2(\Gamma_0(1776))$

Consider
$$\bar{\rho}:G_{\mathbb Q}\longrightarrow\operatorname{GL}_2(\mathbb F_7)$$
the residual 7-adic Galois representation attached to the elliptic curve $y^2=x^3+x^2-4x-4$ of conductor 48. Then ...

**39**

votes

**2**answers

2k views

### Arctangents of odd powers of the golden ratio

While trying to answer this MSE question, I found that arctangents of many odd powers of the golden ratio $\varphi=\frac{1+\sqrt5}2$ are expressible as rational linear combinations of arctangents of ...

**5**

votes

**3**answers

363 views

### Logarithmic integral, $π(x)$ and $x/(\ln x)$

The function $\text{Li}$ (logarithmic integral) is defined for $x>0$
by
$$
\text{Li}(x)=\int_2^{x}\frac{dt}{\ln t}.
$$
The prime number theorem, proven by Hadamard and de la Vallée-Poussin in 1896 ...

**14**

votes

**4**answers

782 views

### Number of $\mathbb F_p$ points constant mod $p$?

I have some affine varieties $X$ defined over $\mathbb Z$, and associated integers $c(X)$, with the property that $\# X_{\mathbb Z/p} \equiv c(X) \bmod p$ for all $p$. (In particular $c(X)$ is usually ...

**4**

votes

**0**answers

107 views

### Number of primes one larger than divisors of a fixed number, which is LCM of 1,2,3,…,L

I am hoping someone can estimate the number of primes that come up this way: take a number $L,$ then let
$$ C = \operatorname{lcm} (1,2,3,\ldots,L). $$
We know that $C$ has quite a lot of divisors; ...

**7**

votes

**2**answers

918 views

### How to prove that this equation has only one solution?

I can't find a way to prove that the following equation has only one solution :
$$
X = \frac{2^Q - 1}{2^{P+Q} - 3^P}
$$
with $X,P,Q$ integers $> 0$.
One trivial solution is $X = 1, P = 1, Q = ...

**1**

vote

**1**answer

96 views

### Line bundle of Half integral weight modular forms

Let $\omega_{X}$ denote the line bundle of cusp forms of weight $\frac{1}{2}$ over $X$, where $X=\Gamma\backslash \mathbb{H}$ and $\Gamma$ is any arbitrary fuchsian subgroup. Is it true that the line ...

**0**

votes

**1**answer

168 views

### Chinese remainder theorem for cyclic subfactor planar algebras

This post was inspired by an exchange with the indian woman mathematician Ajit Iqbal Singh.
The chinese remainder theorem can be stated as follows:
Let $n_1, \dots, n_r \ge 2$ be positive integers ...

**7**

votes

**0**answers

74 views

### Approximation to a certain Weyl-sum

Let $ S(\alpha)=\sum_{x\leq X}\sum_{y\leq Y}e \left(\alpha x y^3 \right)$, for some $X,Y \geq 1$ and write $\alpha = a/q + \beta$ for $(a,q)=1$, as usual.
For the 'classical' cubic Weyl-sum ...

**9**

votes

**0**answers

208 views

### Hodge–Tate structures of modular forms

The title refers to the paper of Faltings:
Hodge-Tate structures and modular forms.
Math. Ann. 278 (1987), no. 1-4, 133–149.
The main theorem in the paper says that the associated Galois rep to a ...

**11**

votes

**1**answer

622 views

### Subsequence and integers as a sum of $\frac{1}{n}$

For all $M \in \mathbb{Z}$, is there a finite sequence of positive integers (not necessarily distinct) $(n_i)_{i \in I}$, s.t. $\sum_{i \in I} \frac{1}{n_i} = M$, and there is no subsequence $(n_i)_{i ...

**3**

votes

**2**answers

267 views

### Representations of complex semi-simple algebraic group “defined over $\mathbf{Z}$”?

If $G$ is a split semisimple linear algebraic group over $\mathrm{Spec}(\mathbf{Z})$ then does every (algebraic) irrep of $G_{\mathbf{C}}$ extend to a morphism $G\to\mathrm{GL}_n$ over ...

**1**

vote

**0**answers

139 views

### Conway's box function iterated to produce a hierarchy of nested sets of real numbers

Conway's box function is the inverse of Minkowski's question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence). When the ...

**3**

votes

**1**answer

148 views

### Products of relative prime numbers with least sum

Let $P(n)$ be the set of subsets $P$ of $\mathbb{N}$ with the properties
All elements of $P$ are relative prime to each other.
The product of all $k \in P$ is greater or equal to $n$.
Now let ...

**5**

votes

**2**answers

322 views

### Order of vanishing of an integer polynomial at a point

Let $f(x,y)$ be a polynomial with integer coefficients, and let $\alpha=(\alpha_1,\alpha_2)\in \mathbb{C}^2$ be a complex point. I want to show that $f$ cannot vanish at $\alpha$ to high order unless ...

**2**

votes

**0**answers

57 views

### Is this similarity to the Fourier transform of the von Mangoldt function real? [duplicate]

This question proposed in SEM but no answer and it's interesting to know connection
between Fourier analysis and number theory .
Mathematica knows that the logarithm of $n$ is:
$$\log(n) = ...

**2**

votes

**2**answers

155 views

### Pseudo-decision procedures for first order arithmetic

I was reading this paper http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.117.2911&rep=rep1&type=pdf in which the author describes an algorithm, based on Groebner basis, ...

**2**

votes

**1**answer

165 views

### Does $\prod_{n=2}^{\infty} \left(\frac {1}{1-\frac{\chi_k(n)}{n^s}} \right)$ converge for non-principal characters for all $\Re(s) > \frac12$?

This question loosely builds on this one.
Take the following infinite product:
$$N(s,\chi_k)=\prod_{n=2}^{\infty} \left(\frac {1}{1-\dfrac{\chi_k(n)}{n^s}} \right)$$
with $\chi_k$ a Dirichlet ...

**3**

votes

**1**answer

1k views

### Is $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ a rational number?

How can we determine whether $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ is rational or not?
Is it transcendental or algebraic?

**6**

votes

**1**answer

101 views

### Lattice parallelogram of minimal area containing convex lattice polygon

What is the minimal constant $\alpha$ so that for any convex lattice polygon $F$ there exists a lattice parallelogram $P\supseteq F$ of area $A(P)\leq \alpha\cdot A(F)$?
It is not hard to show that ...

**10**

votes

**0**answers

244 views

### Reference request: a combinatoric result [closed]

When I tried to construct a counterexample in my research, I encountered the following result, which should be true.
Let $m=m(n)$ be a function that grows faster than $\sqrt n$, so $m(n) = ...

**0**

votes

**0**answers

87 views

### A sum on a minimums conjecture

Prove or disprove:
$$m(n,k,s)=\sum_{a_1=1}^n \sum_{a_2=1}^n \cdots \sum_{a_k=1}^n \min(a_1, a_2,\cdots, a_k)^s =$$
$$ \sum _{i=0}^{k-1} \frac{(-1)^i}{i!} F(n,i+s) \sum _{j=0}^{k-1} \frac{\partial ...

**1**

vote

**0**answers

95 views

### Twists of Hecke character

Let $\chi$ be a self-dual Hecke character over a CM field $E$ with root number $-1$. Then, how to show the existence of a finite order Hecke character $\eta$ over $E$ such that the twist $\chi\eta$ is ...

**4**

votes

**1**answer

155 views

### Unicity of additive, $(-1)$-homogeneous, and shift invariant probability measures on $\mathbf N^+$

Let $\mathcal D$ be the set of all (finitely) additive probability measures $\mu^\ast: \mathcal P(\mathbf N^+) \to [0,\infty[$ such that $\mu^\ast(k \cdot X + h) = \frac{1}{k} \mu^\ast(X)$ for all $X ...

**1**

vote

**0**answers

54 views

### Does there exist $k\ge2$ s.t. $X \subseteq \mathbf N^+$ has positive upper Banach density if the counting function of $X$ is $\gg n/\log^{[k]}(n)$?

Does there exist an integer $k \ge 1$ such that ${\sf bd}^\ast(X) > 0$ whenever $X \subseteq \mathbf N^+$ and $\pi_X(n) \gg \frac{n}{\log^{[k]}(n)}$ as $n \to \infty$? Here, ${\sf bd}^\ast$ is the ...

**10**

votes

**1**answer

443 views

### Which degree does a motivic Galois representation show up in?

Consider a representation $\rho: \operatorname{Gal} (\overline{\mathbb Q} | \mathbb Q ) \to GL_n ( \overline{\mathbb Q}_\ell)$ that is a subrepresentation of $H^i(X, \overline{\mathbb Q}_\ell (j))$ ...

**2**

votes

**0**answers

63 views

### Bitwise operation of two square roots

Let $\sqrt 2 = 1.a_1a_2\dots _2$, and $\sqrt 3 = 1.b_1b_2\dots _2$. What can one say about the number $n = 0.c_1c_2\dots$ where $c_i = 1$ if $a_i = b_i$ and $0$ otherwise? There is no reason to ...

**3**

votes

**1**answer

163 views

### A question about “small” sets of algebraic numbers

For the sake of brevity let me use the following terms. A subset $X$ of $\bar{\mathbb{Q}}$ will be called "small" if for any number field $K$, the intersection
$X\cap K$ is finite. Similarly, a set ...

**9**

votes

**1**answer

458 views

### A weakening of the Littlewood conjecture

For real numbers $x$, let $\|x\|$ denote the distance from $x$ to the nearest integer. Define a function $\ell:\mathbb{R}^2\rightarrow\mathbb{R}$ by
...

**4**

votes

**1**answer

296 views

### Does the Euler product for $L(s,\chi_4)$ also converge in the right half of the critical strip?

This question expands on this one from MSE.
In the literature about Dirichlet $L$-series, I found that their Euler products:
$$L(s, \chi) =\prod_p \bigg(\frac {1}{1-\frac{\chi(p)}{p^s}} \bigg)$$
...

**6**

votes

**1**answer

121 views

### Monte-Carlo computation of the Smith normal form

Quite some time ago I saw an article where a Monte-Carlo algorithm for computing the Smith normal form of an integer matrix was described. In this article the following problem was posed:
Suppose $P, ...

**8**

votes

**2**answers

653 views

### Is $f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)}$ essentially bounded?

Let $$f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)}
$$
Is it true that $\|f\|_{L^{\infty}(\mathbb{R}^2)}<\infty$? i.e. is $f$ essentially bounded?

**4**

votes

**0**answers

369 views

### Best known bounds on certain exponential sums

What are the best bounds currently known for the following exponential sum:
$$\sum_{x < p \le 2x} e(\alpha p^k)$$
for values of $\alpha$ far from a rational with small denominator. ($p$ refers ...