**0**

votes

**0**answers

7 views

### Minimal number of different values in the sequence $(\mu(d_i)\varphi(d_i))_{i=\overline{{1,\tau(m)}}}$

Let $m=p_1\ldots p_k$ be the prime factorization of some positive integer $m$ and $k\geq 2$.
Let $d_1,\ldots,d_{\tau(m)}$ be all divisors of $m$, where $\tau(m)$ counts the number of divisors of $m$....

**0**

votes

**0**answers

64 views

### How do these primes jump?

I have several questions regarding the analysis, behaviour, and expression of a simple sieving algorithm which uses associative arrays. The pseudocode below assumes integer addition, string ...

**0**

votes

**0**answers

68 views

### The number of fixed points of an automorphism of $\mathbb{Z}_m\times\mathbb{Z}_n$

Let $m$ and $n$ be two positive integers such that the groups $\mathbb{Z}_m$ and $\mathbb{Z}_n$ have no common direct factor. Then an automorphism $f$ of $\mathbb{Z}_m\times\mathbb{Z}_n$ is of type
$$\...

**5**

votes

**0**answers

166 views

### Semisimplicity of Frobenius on *integral* Tate module

Let $K$ be a number field and $A/K$ an Abelian variety; let $l$ be a (rational) prime. Do there exist infinitely many primes $\mathfrak{p}$ of $K$ such that the Frobenius at $\mathfrak{p}$ acts ...

**9**

votes

**1**answer

242 views

### Rational curves on the Fermat quartic surface

Let $X$ be the Fermat quartic $x^4+y^4+z^4+w^4=0$ in $\mathbb P^3$. It is known that $X$ contains infinitely many $(-2)$-curves, that is, smooth rational curves. (One way to obtain in infinitely many ...

**5**

votes

**2**answers

181 views

### Definition of Hecke operators on orthogonal modular forms

In his paper Automorphic forms with singularities on Grassmannians, Borcherds poses Problem 16.5:
"Describe how the correspondence in this paper behaves under
the
action of Hecke operators."
Since ...

**2**

votes

**0**answers

104 views

### When two non-equivalent binary forms primitively represent the same infinite subset of the integers

Let $F(x,y)$ be an irreducible binary form with integer coefficients and degree $d \geq 3$. We say that an integer $n$ is primitively represented by $F$ if there exist coprime integers $x$ and $y$ ...

**-3**

votes

**0**answers

44 views

### How to figure a complentary set of a Diophantine equation [on hold]

Given a subset of the real numbers defined as 2xy-x-y+1 for x >1, y > 1 how can I determine the complementary set?

**2**

votes

**0**answers

57 views

### If $N = q^k n^2$ is an odd perfect number, if $n < q^{k+1}$, does it follow that $k > 1$?

Let $N = q^k n^2$ be an odd perfect number with Euler prime $q$. According to Dickson (as pointed out recently by Beasley), Descartes conjectured $k=1$ in a letter to Mersenne in 1638, with Frenicle'...

**1**

vote

**0**answers

42 views

### Higher dimensional analogs of logarithmic density

For a set $A\subseteq \mathbb{N}$ its lower/upper asymptotic/logarithmic densities are given by
\begin{align*}
\underline{d}(A)=\liminf_{N\to\infty} \frac{|A\cap [1,N]|}{N},\\
\bar{d}(A)=\limsup_{N\to\...

**2**

votes

**1**answer

141 views

### Katz $p$-adic L function and ordinary condition

Let $H$ be a CM field and $F$ be the maximal totally real subfield of $H$. Can we construct a Katz $p$-adic L-functions of Hecke characters without the ordinary condition (i.e every prime of $F$ above ...

**0**

votes

**0**answers

117 views

### Localisation of the formal power series ring

Let $A \colon= K[[X_1,...,X_d]]$ be a formal power series ring of $d$-variables over a field $K$.
Let ${\frak a}$ be a height $r$ prime of $A$ given by ${\frak a} \colon= (f_1,...,f_r)$, where
$f_1 ...

**0**

votes

**0**answers

47 views

### Image of composition of integral upper triangular matrices

For $A,B$ integral upper triangular matrices on $\mathbb{Z}^k$, do we know something about the image $\text{im}(AB)$ in terms of $\text{im}(A)$, $\text{im}(B)$, unions, intersections, determinants, ...

**0**

votes

**1**answer

70 views

### Application of the EGZ theorem

Given $r$ numbers $a_1,a_2,...,a_r$ and $n=qP$ where $P$ is the product of these $r$ numbers. $q$ is a natural number such that $q \geq 2$.
Also given is a matrix $A$ of the following form: $$A=\...

**1**

vote

**2**answers

585 views

### Statements going against the grain of Riemann Hypothesis (R.H.) [on hold]

Let $M(N) := \sum_{n=1}^N \mu(n)$
It is known that bounding $M(N)$ by $N^{1/2+\epsilon}$ implies R.H.
A bound of $M(N)$ by $K\sqrt N$ for say $ K\ge2 $ is also sufficient. $K = 1$ is excluded as ...

**7**

votes

**2**answers

332 views

### Power sums of p-th roots of unity

The following question was asked by a colleague of mine. For any prime $p$ consider
$$ M_p:=\min_{z_1,\dots,z_p}\max_{j,k}\left|z_1^k+\dots+z_j^k\right|,$$
where $z_1,\dots,z_p$ are the complex $p$-th ...

**5**

votes

**1**answer

122 views

### Stabilizers of pairs of ternary quadratic forms

Let $A,B$ be two ternary quadratic forms with real coefficients, given by symmetric matrices
$$\displaystyle 2A = \begin{pmatrix} 2a_{11} & a_{12} & a_{13} \\ a_{12} & 2a_{22} & a_{23}...

**0**

votes

**0**answers

61 views

### generic divisibility equation for two natural numbers [on hold]

Given two natural numbers N and B, so that N > B, is there a generic equation that contains only multiplications of N and B which can tell whether N is divisible by B?
Basically, something like a ...

**1**

vote

**0**answers

204 views

### Number Theory and d-Self-Contained Numbers

Given any natural number $N = a_{n}a_{n-1}\ldots a_{1}$, let us associate to it the set $S_{N} = \bigcup_{j=1}^{n}\{(a_{j},j)\}$. We're going to define a d-self-contained number as any natural number ...

**3**

votes

**3**answers

424 views

### sum of four squares with some coefficients

Is there an ordered 4-tuple of rational numbers $(a,b,c,d)$ such that $(b,d)\ne(0,0)$ and $2a^2+3b^2+30c^2+45d^2=2$?
The former (deleted) question was just about cases $(a,b,c,d)\ne(1,0,0,0)$ but it ...

**2**

votes

**0**answers

102 views

### On the conjectured nonexistence of even almost perfect numbers (other than powers of two) and odd perfect numbers [on hold]

(Note: This question has been cross-posted to MSE.)
Let $\sigma(a) = \sigma_{1}(a)$ be the sum of the divisors of the positive integer $a$.
A number $M$ is called almost perfect if $\sigma(M) = 2M -...

**13**

votes

**3**answers

565 views

### Collatz-like properties of finite fields

I was wondering what an equivalent of the Collatz conjecture might be for finite fields. In a Collatz sequence a number is moved down within a set $\{2^k n : k \in \mathbb{Z}^* \}$ for some odd $n$ or ...

**2**

votes

**1**answer

75 views

### An inequality on partitions into distinct bounded parts

Let $P(n,m)$ denote the set of all positive integer partitions of $n$ into parts that are pairwise distinct and bounded by $m$. Let $p(n,m) = |P(n,m)|$.
After some numerical experiments it appears
$...

**0**

votes

**0**answers

59 views

### Normalizing factor in Reciprocity of traces of Frobenius with solutions of equations mod p

One kind of Reciprocity tells us that we can count solutions to polynomial equations over finite fields and relate them to traces of Galois Representations with some normalization factor.
For ...

**1**

vote

**1**answer

74 views

### Parametric solutions to a system of equations

Let $s,t$ be two independent real parameters, and let $a_2(s,t), a_1(s,t), a_0(s,t)$ be linear forms in $s, t$ with real coefficients. Put $a_4 = s, a_3 = t$ and consider the quadratic form
$$\...

**-4**

votes

**0**answers

92 views

### Does the group Aut(M) preserve every Dedekind Zeta function? [on hold]

Foreword:
This excerpt of a paper of mine aims at introducing the concept of automorphism of an L-function, where by L-function we mean any element of the Selberg class that is also an automorphic L-...

**1**

vote

**0**answers

88 views

### Relative Leopoldt defect

Let F be a totally real number field such that the Leopoldt conjecture holds at a prime number $p$ and $M$ be a quadratic totally real extension of $F$.
Is there a bound of the Leopoldt defect of $M$ ...

**1**

vote

**0**answers

75 views

### The Linnik problem for dimension $2$

For $N$ an integer, let
$$\Omega_N:=\left\{\frac{\alpha}{\|\alpha\|}|\alpha \in\textbf{Z}^n~\text{and}~\|\alpha\|^2=N\right\}.$$
For $n=3$, Linnik asked if the set $\Omega_N$ was uniformly distributed ...

**2**

votes

**0**answers

67 views

### On the size of residue class

Let $n \in \mathbb{N}$ be a odd number. Let $S \subseteq \{1,3,5,7,...,n-2,n\}$ and $|S|$ is even number. Let $R_i^k=\{a \mid a \in S \text{ } \&\text{ } a\equiv i \text{ }(mod \text{ } k)\}$ ...

**1**

vote

**0**answers

204 views

### Computational number theory

Suppose that $p$ is prime and $q$ is an even number divides $p-1$, such that $q<\frac{p-1}{q}$ and $u$ has order $q$ modulo $p$. Let $S$ be the subgroup of $Z^*_p$ consisting of the powers of $u$. ...

**-3**

votes

**0**answers

77 views

### Considering a matrix with integrer entries over $\mathbb{Z}/p \mathbb{Z}$, does it remain full rank? [closed]

Suppose I have an $m \times n$ matrix $M$ with integer coefficients, and suppose it has full rank. Let $p$ be a prime and now consider the matrix $\bar{M}$ over $\mathbb{Z}/p \mathbb{Z}$. Is it true ...

**13**

votes

**1**answer

361 views

### Does Littlewood's bound on $\zeta(1+it)$ extend to all the partial sums?

Littlewood established that $2e^{\gamma} \geq \limsup_{t \to \infty} |\zeta(1+it)| / \log{\log{t}} \geq e^{\gamma}$, the lower bound unconditionally and the upper bound on RH. It now seems to be ...

**10**

votes

**1**answer

284 views

### Polylogarithm sheaves

In many different places, I could find the notion on ''(poly)logarithm sheaves''. As is indicated in the name of it, I guess that it should have something to do with (poly)logarithm function: $\mathrm{...

**4**

votes

**2**answers

205 views

### Exceptional isomorphisms between finite simple Chevalley groups

Steinberg's "Lectures on Chevalley Groups"
https://math.depaul.edu/cdrupies/research/papers/chevalleygroups.pdf
contain ``a complete list of isomorphisms" among the various finite simple Chevalley ...

**0**

votes

**0**answers

67 views

### On semi-complete ring K[X_1,X_2,…,X_∞]] and Popescu theorem

Let $P_n \colon= K[X_1,...,X_n]$ be a $n$-variables polynomial ring. We define 'semi-complete' polynomial ring $P_{\infty}$ by the following$\colon$
$P_{\infty} = K[X_1,...,X_\infty]] \colon = \...

**2**

votes

**1**answer

136 views

### Criteria for the surjectivity of the reduction map of the $SL_n$-group scheme

Let $R$ be a commutative ring and let $I\subseteq R$ be an ideal. We have a natural projection map
$$
\pi:SL_n(R)\rightarrow SL_n(R/I)
$$
(In the original question I had put $GL_n$ instead of $SL_n$ ...

**4**

votes

**1**answer

213 views

### Hilbert modular forms twist-equivalent to their conjugates

Let $L / K$ be a solvable (or cyclic) Galois extension of totally real fields, and let $f$ be a Hilbert modular newform over $L$.
Suppose that, for every $\sigma \in Gal(L / K)$, the conjugate ...

**1**

vote

**0**answers

100 views

### Combinatorial splitting in number rings

The goal of this problem is to see if there is a structured way to factor numbers constructed from a set of distinct odd primes $p_1$ through $p_n$ in a number ring.
Take an arbitrary non empty ...

**3**

votes

**3**answers

320 views

### Jacobi's theorem on sums of two squares (reference request)

One of Jacobi's theorems states that the number of representations of a positive integer $n$ as a sum of two squares of integers equals
$$4(d_1(n)-d_3(n)),$$
where the function $d_i$ counts the number ...

**14**

votes

**2**answers

358 views

### Can you use Chevalley‒Warning to prove existence of a solution?

Recall the Chevalley‒Warning theorem:
Theorem. Let $f_1, \ldots, f_r \in \mathbb F_q[x_1,\ldots,x_n]$ be polynomials of degrees $d_1, \ldots, d_r$. If
$$d_1 + \ldots + d_r < n,$$
then the ...

**2**

votes

**3**answers

269 views

### Geometry of numbers argument: counting integers with some linear condition

I am interested in the proof of the following result:
Suppose that $A > 1$, $\lambda \in \mathbb{R}$, and for $0 < Z \leq 1$, let $U(Z)$ be the number of integer solutions $v$ of
\begin{...

**11**

votes

**4**answers

662 views

### Smallest solution to $x^2 \equiv x\pmod{n}$

Given $n$, is it possible to upper bound the smallest $x > 1$ that satisfies the congruence $x^2 \equiv x\pmod{n}$? Obviously when $n$ is a prime power $x = n$, and we are in the worst situation. ...

**1**

vote

**0**answers

77 views

### Reference request: Bounding exponential sum $\sum_{x \in [0,X]} \Lambda(x) e(\beta_d x^d + \ldots + \beta_1 x )$

Let $1 \leq i \leq d$, $q \in \mathbb{N}$, and $0 \leq a_{i} < q$. Let
$$
\mathfrak{M}^{(i)}_{a_{ i}, q} (C) =\{ \beta_{i} \in [0,1) : | \beta_{i} - a_{i}/q | \leq (\log X)^{C} X^{-i} \} .
$$
We ...

**0**

votes

**0**answers

110 views

### Troost-Bourget identity $ N \sum_{d|N} 1 = \sum_{d| N} \sum_{l=1}^d \mathrm{gcd}(d,l) $ [closed]

In the process of evaluating a "supersymmetric index", Bourget and Troost establish a rather elementary identity:
$$ \frac{N}{m} \sum_{d| N} \sum_{l=1}^{\mathrm{gcd}(d,m)} \mathrm{gcd}\left[ \mathrm{...

**-1**

votes

**0**answers

72 views

### Sum-free sets of powerful numbers

For $n=p_1^{\alpha_1}\cdots p_r^{\alpha_r}$ with distinct primes $p_i$, call $\alpha= (\alpha_1,\dots,\alpha_r)$ the type of $n$ and denote by $N_\alpha$ the set of all naturals of this type.
We ...

**-1**

votes

**0**answers

93 views

### A question about arithmetic progressions and prime numbers

"I took number $3$ and observed:
$3$ is an arithmetic progression of length one.
$3,5$ is an arithmetic progression of length two.
$3,5,7$ is an arithmetic progression of length three.
Then I took ...

**0**

votes

**1**answer

137 views

### How to count fixed-sized subsets of pairwise co-prime numbers less than a prime, satisfying an additional constraint?

In part of my research, I need to count (or find a sharp bound for) the number of possible ways to select $n$ distinct integers less than the prime $p$, say $r_1, r_2, …, r_n$, which are pairwise ...

**12**

votes

**2**answers

615 views

### Congruence equation and quadratic residue

The following observation makes me quite confused when I am trying to count the number of solutions of the equation:
$$\sum_{k=0}^{M}{M \choose k}^2x^k=0$$
on finite fields $\mathbb{F}_p$ with the ...

**-3**

votes

**1**answer

103 views

### Is a positive integer determined by its sequence of typical primality radii?

This question is a follow-up to About Goldbach's conjecture . Assuming the truth of Goldbach's conjecture, suppose $n$ and $m$ are two positive integers such that $N_{2}(n)=N_{2}(m)=:N$ and that ...

**5**

votes

**1**answer

160 views

### A $p$-adic sum of reciprocals of powers

Let $p$ be a prime number and $k\geq 2$ an even integer. Consider the following $p$-adic integer:
$$
S_{p,k} := \lim_{r\to+\infty} \sum_{a=1}^{p^r} \big(\frac{p^r}{a}\big)^k
$$
Convergence is easy to ...