**10.142 nt.number-theory questions.**

Let $a_1$, $a_2$, …, $a_n$ and $b_1$, $b_2$, …, $b_n$ be $2n$ strictly positive integers not greater than $M$, with $M$ is a given positive integer, such that $$a_1+ a_2+ \dotsb+ a_n=b_1+ b_2+ \dotsb+ ...

Lagrange's four square theorem states that every non-negative integer is a sum of squares of four non-negative integers. Suppose $X$ is a subset of non-negative integers with the same property, that ...

I was reading this blog by Evan Chen about complex multiplication. He's discussing Sato-Tate Conjecture. We can have elliptic curve $E/\mathbb{Q}$ and solve it over finite fields.
\begin{eqnarray*}
...

Let $a$ be a positive integer.
Does there exist a positive integer $m$ other than 1 such that: $$a+m \ |\ a^m+1 \ $$ If not, what are the conditions of $a$ for $m$ to exist?
If there exist ...

Since the multiplication of $n$ consecutive integers is divided by $n!$, then $(n!)^n|(n^2)!$ with $n$ is a positive integer.
Are there any formula of the function $y=f(x)$ that shows the largest ...

The question is: N is an even positive integer, then $\sum_{n=0}^{N-1}\exp(-i\frac{\pi}{N}n^2)=\sqrt{N}\exp(-i\frac{\pi}{4})$. I thought the terms on the left are the solutions set of some polynomial ...

$\sum_{s=0}^n{x^sy^{n-s}}$
Does anyone know what this evaluates in the case $x, y \in N$, where can I read more about it?

Does the equation $(xy+1)(xy+x+2)=n^2$, with $n\in \mathbb{N}$, have a positive integer solution $(x, y)$? If it doesn't, how to prove it?

The equation
$$\displaystyle y^2 = f(x_1, x_2, x_3)$$
with $f$ a non-singular quartic form in three variables defines a del Pezzo surface of degree 2. I am interested in the similar construction
$$\...

Let $p$ be a prime and $\mathbb{F}$ a finite field of characteristic $p$. The theorem of Khare and Wintenberger roughly states that an irreducible, odd Galois representation $\bar{\rho}:G_{\mathbb{Q}}\...

My question has been here on MSE for a long time, but it has not received a full answer. I bring it here:
Find a prime $p$ such that $p \equiv 1 \bmod 4$ and such that the
product in the range $[...

Let $\mathbb{N}$ denote the set of positive integers. The Collatz function $f:\mathbb{N}\to\mathbb{N}$ is given by $f(n) = n/2$ for $n$ even and $f(n) = 3n+1$ for $n$ odd. Given $k\in\mathbb{N}$ we ...

Preamble: I asked this question on Math.SE with a bounty of 100, but received no replies. The bounty has now ended, and it was suggested I post this here instead.
Since Apéry we know that $\zeta(3)$, ...

I'm looking for references to (as many as possible) elementary proofs of the Weyl's equidistribution theorem, i.e., the statement that the sequence $\alpha, 2\alpha, 3\alpha, \ldots \mod 1$ is ...

Let $\{x\}$ be the fractional part of $x$, i.e. $\{x\}=x-[x]$, where $[x]$ is the biggest integer $\leq x$.
The question might be well known but I don't know where to look for: Assume $\alpha$ is an ...

This post can be seen as a continuation of this post I created on MathOverflow.
I want to understand the proof of the following Theorem from "Euler Factors determine Weil Representations" by Tim and ...

$\DeclareMathOperator{\PGL}{PGL} \DeclareMathOperator{\GL}{GL} \newcommand{\F}{\mathbb{F}} \newcommand{\p}{\mathfrak{p}}$
Let $F$ be a number field, and let $\Gamma$ be a congruence subgroup of $\...

Taking norms to be maximal norm, then the simultaneous version of the Dirichlet's approximation theorem states that given real numbers ${\displaystyle \alpha _{1},\ldots ,\alpha _{d}} $,there are ...

Let $n$ be a positive integer. The Euler $\phi$-function is defined by
$$\displaystyle \phi(n) = \# \{1 \leq a \leq n-1 : \gcd(a,n) = 1\}.$$
It is in fact a multiplicative function, and one has the ...

I'm looking for references that show almost all Dirichlet characters $\chi \mod q$ satisfy
$$|\frac{L'}{L}(1+it, \chi)|=o(\log q)$$
where $t\in \mathbb{R}$ is fixed. I have been able to adapt a method ...

Let $A = \{ a_i + b_i \mathbb{N} \}_{i=1}^{k}$, where $a_1, \ldots, a_k \in \mathbb{N} \cup \{0\}$ and $b_1, \ldots, b_k \in \mathbb{N}$ be a system of arithmetic sequences.
For a positive integer $m$...

This question is an offshoot from the following MSE post. I hope that it is appropriate for this site.
Let $\sigma(x)$ be the sum of the divisors of $x$.
An integer $a$ is said to be solitary if ...

Let $0 < k_1 < k_2 < k_3 < \cdots $ be all the zeros of the Riemann zeta function on the critical line:
$$ \zeta(\frac{1}{2} + i k_j) = 0 $$
Let $f$ be the Fourier transform of the sum ...

The notes here https://web.eecs.umich.edu/~cpeikert/lic13/lec04.pdf have the note 'Small decryption exponent $d$: so far the best known attack recovers $d$ if it is less than $N^{.292}$. This uses a ...

Suppose that $f$ and $g$ are two coprime polynomials in $\mathbb Z[x]$. I'm interested in any sort of upper bound on $gcd(f(a),g(a))$, in terms of the integer $a$. Are there any results of this type?...

(I asked this in MSE but did not find resonance, there is also a relation to an older discussion here on summability see here and a followup formulating an $\text{ais}()$ already here)
...

There is a lot of fascination with the Vandermonde determinant and for many good reasons and purposes. My current quest is more of number-theoretic.
QUESTION. Let $p\equiv 3$ (mod $4$) be a prime. ...

Let $\xi\in\mathbb{R}$ and consider the average value (with respect to hyperbolic length) of the $j$-invariant ($j(z)=q^{-1}+744+196884q+\ldots$, $q=e^{2\pi iz}$) along a geodesic aimed at $\xi$:
$$
\...

Can one prove by elementary means (such as Paul Erdös' proof of Bertrand's Postulate) that every prime greater than 5 is less than the sum of the two primes immediately preceding it?

In their 2001 paper defining periods, Kontsevich and Zagier (pdf) without further comment state that $e$ is conjecturally not a period while many other numbers showing up naturally (conjecturally) are....

It is well-known that the Euler $\phi$-function is multiplicative: that is, for co-prime positive integers $m,n$ we have $\phi(mn) = \phi(m)\phi(n)$. Thus it is defined by its values on prime powers. ...

Let $K$ be a cyclic cubic number field with conductor $f$ and ring of integers $\mathcal{O}_K$.
Define $K$ to be blue if and only if $$\operatorname{Norm}_{K/\mathbb{Q}}(w) = \operatorname{Norm}_{K/...

Results are known in many different cases to bound powers of L-functions on average over a wide enough family. I am interested in results for general number fields, not only for the rationals, for ...

Looking at @Lucia's answer to this question it appears $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$ converges for $\sigma > \frac{1}{2}$. Can someone point me to a proof or provide proof for this? If I ...

Is there an algorithm which on input "$(a,p)$" (where $0\leq a<p$ are integers) takes time polynomial in $\log p$ and outputs "NOT PRIME" if $p$ is not prime and otherwise outputs the Legendre ...

This question is part of series of the questions, as follows:
Coefficients in the sum $\sum_{k=0}^{n-1}\sum_{j=0}^{m}A_{j,m}(n-k)^jk^j=n^{2m+1}$,
Coefficients $U_m(n,k)$ in the identity $n^{2m+1}=\...

Given two positive integers $a<b$, can we always find an integer $c\in [a, b]$ that is coprime to $\sum_{a\le i\le b} i$?

Let $L$ be a dihedral quartic field. That is, we have $[L : \mathbb{Q}] = 4$ and the Galois closure $M$ of $L$ is a degree 8 extension of $\mathbb{Q}$ with Galois group isomorphic to the dihedral ...

In the Corvallis article Number Theoretic Background, here is what John Tate has to say on the local Langlands correspondence for a $p$-adic field $F$:
So, granting a correspondence between ...

I am looking for an upper-bound of the Euler's totient function $\varphi$ which would be equivalent to the Riemann hypothesis (RH). There is the following Nicolas' criterion about primorial numbers $...

Yesterday I came across the following one-paragraph summary of the history of the Law of Quadratic Reciprocity in Roger Godement's Analyse mathématique, IV, p.313 (perhaps the only treatise on ...

Let $K$ be a number field and $\pi$ be a regular algebraic cuspidal automorphic representation on $\mathrm{GL}_2(\mathbb{A}_K)$. Let $\lambda$ be a prime of the field of Fourier coefficients of $\pi$ ...

Let $L(s)$ be an automorphic L-function (attached to a self contragredient automorphic representation on $GL(3)$), according to the following notations for $s$ of sufficiently large real part:
$$L(s) =...

Take natural numbers $A_1,B_1,A_2,B_2$ random pairwise coprime in $[n,2n]$ for $n$ large enough and consider the space of solutions to $A_1a+B_1b=0$ and $A_1A_2a+A_1B_2b+B_1A_2c+B_1B_2d=0$ spanned by $...

Let us call a positive integer $n\in\mathbb{N}$ consecutively summable if there are positive integers $m, k < n$ such that $$n=\sum_{i=0}^k (m+i).$$For $A\subseteq \mathbb{N}$ we set the lower ...

The title is a quote from a Jim Holt article entitled, "The Riemann zeta conjecture and the laughter of the primes" (p. 47).1
His example of a "long-standing conjecture" is the Riemann hypothesis,...

How to prove the following two congruences?
Question1: Let $p\equiv 1 \pmod 3$ be a prime, then
$$\sum_{k=0\atop k\neq(p-1)/3}^{(p-1)/2}\frac{\binom{2k}k}{3k+1}\equiv 0 \pmod p.$$
...

Let $d(n)$ denote the number of positive divisors of the positive integer $n$.
Pick some positive $X,h \in \mathbb{R}$ and consider the sum
$$ S(X,h) := \sum_{X \leq n \leq X+ h} d(n).$$
In view of ...

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