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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

5
votes
0answers
129 views

Double Counting: Motivic Edition

One of the most important proof techniques in combinatorics is double counting: proving that both sides of an identity count elements of some set in two different ways. This question is an attempt at ...
0
votes
0answers
73 views

The sieve formula choosen in Zhang's breakthrough work [duplicate]

In the breakthrough work of the proof of weak twin prime conjecture, Goldstone, Pintz and Yildirim as well as Zhang use the following modified Selberg sieve: $v=\lambda^2$ where $\lambda(n)$takes ...
32
votes
4answers
947 views

Motivated account of the prime number theorem and related topics

Though my own research interests (described below) are pretty far from analytic number theory, I have always wanted to understand the prime number theorem and related topics. In particular, I often ...
5
votes
1answer
173 views

How to show that a hypersurface is a diagonal intersected with hyperplanes?

Suppose I have a hypersurface $V(F) = \{ \mathbf{x} \in k^n: F(\mathbf{x}) = 0 \}$, where $F$ is a homogeneous form of degree $d > 1$. I would like to show that there exists some diagonal form $D(...
5
votes
2answers
235 views

asymptotic for li(x)-Ri(x)

Is it true that $$\operatorname{li}(x)-\operatorname{Ri}(x) \sim \frac{1}{2}\operatorname{li}(x^{1/2}) \ (x \to \infty),$$ where $$\operatorname{Ri}(x) = \sum_{n = 1}^\infty \frac{\mu(n)}{n} \...
5
votes
0answers
139 views

Order of magnitude of extremely abundant numbers and RH

I have always been intrigued by the fact that Riemann's hypothesis is equivalent to the assertion that the inequality $$\frac{\sigma(n)}n<e^\gamma \log\log n \qquad\qquad\qquad(*)$$ holds for all $...
0
votes
0answers
68 views

How to prove a Mersenne number is not pseudoprime to the base 3?

I start think in finding a small divisor $p$ of $M_n$ then I obtain that $p\mid (3^{M_{n}-1}-1)$ hence $ord_{p}(3)\mid M_{n}-1$ Now can I prove p is the same as $M_{n}$ so that $M_{n}$ is not ...
4
votes
1answer
131 views

Local L-function $L(s,\pi_p\times \chi_p)=1$

Let $\pi_p$ be a ramified representation of $GL(n,\mathbb{Q}_p)$. Let $\chi_p$ be a ramified representation of $GL(1,\mathbb{Q}_p)$. Is it generally known that $L(s,\pi_p\times \chi_p)=1$ if $\...
11
votes
0answers
153 views

Lifting automorphic Galois representations to arithmetic fundamental groups and their quotients

Suppose $V$ is an algebraic variety over a number field $K$. Then, the absolute Galois group $G_K$ of $K$ acts by outer automorphisms on the etale fundamental group $\pi_1(V_{\bar{K}})$ where $\bar{K}$...
5
votes
0answers
259 views

Why arithmetic Langlands?

In trying to understand the import of Akshay Venkatesh' most recent work I found myself wondering anew about that old gnawing mystery: why Langlands? Why should arithmetic of polynomial equations over ...
1
vote
0answers
71 views

Counting "simultaneous squares' over the Gaussian integers

Let $n$ be a square-free integer. Then for a given integer $m$, $m$ is a square modulo $n$ if and only if the sum $$\displaystyle \sum_{d | n} \left(\frac{m}{d}\right) > 0.$$ In fact one can ...
3
votes
1answer
127 views

How do modular functions of level $N>1$ transform under the full modular group?

Let $f$ be a modular function of level $N>1$ and let $\gamma\in SL_2(\mathbf Z)$. Write $$f(\gamma\tau)=j(\gamma,\tau)f(\tau).$$ First question What can we say in general about the factor $j(\...
43
votes
7answers
4k views

Are there examples of conjectures supported by heuristic arguments that have been finally disproved?

The idea for this comes from the twin prime conjecture, where the heuristic evidence seems just so overwhelming, especially in the light of Zhang's famous result from 2014 about Bounded gaps between ...
2
votes
0answers
101 views

Categorical representations of absolute Galois groups

I am looking for interesting examples of categorical representations of absolute Galois groups of arithmetic fields. Pointers to the literature would be appreciated.
2
votes
0answers
130 views

The name for injective map $f:\mathbb{N}\rightarrow\mathbb{N}$ with $f(n)\geq n$ property

What is the name for map $f:\mathbb{N}\rightarrow\mathbb{N}$ (from natural numbers into natural numbers) with the following propeties: 1) $f$ is injective 2) $f(n)\geq n$ for every $n$?
3
votes
0answers
112 views

Modular root of $-1$

Take two distinct coprime integers $a,b$ and take $q=a^2+b^2$. Consider the equation $(xy^{-1})^2\equiv-1\bmod q$. The solutions are two roots which correspond to $(x,y)=(a,b)$ and $(x,y)=(b,a)$. Look ...
2
votes
1answer
125 views

What is the shortest length of an Egyptian fraction expansion for a given $p/q$?

An Egyptian fraction expansion is a sum of reciprocals of integers, for example: $$\frac{4}{17} = \frac{1}{5} + \frac{1}{29} + \frac{1}{1233} + \frac{1}{3039345}$$ Every positive rational number $p/...
0
votes
1answer
130 views

Parity and number of squares taken by polynomials in a range?

I have a polynomial $f(x)=a^2x^2+bx+c\in\mathbb Z[x]$ with $f(x)$ not a constant times a square and $abc\neq0$ and I want to know how many $x$ between $-a$ and $a$ the polynomial is a perfect square. ...
28
votes
1answer
781 views

On a quantum Riemann Hypothesis

Robin's theorem (1984) states that $$ \sigma(n) < e^\gamma n \log \log n$$ for all $n > 5040$ if and only if the Riemann hypothesis is true. Recall that $γ$ is the Euler–Mascheroni ...
3
votes
0answers
120 views

Automorphy of families of motives

I have a couple of elementary questions regarding automorphy of Galois representations arising from geometric families. Suppose we have an algebraic family of varieties over a number field, and ...
-5
votes
0answers
249 views

About a pattern on prime numbers [on hold]

I have read in https://www.sciencealert.com/mathematicians-discover-a-strange-pattern-hiding-in-prime-numbers that says: "But this doesn't explain the magnitude of the bias the team found, or why ...
3
votes
0answers
173 views

Visualization of hidden structures in numbers

[Please allow me a note: The way desribed below allows to depict functions $f:X^2 \rightarrow Y$ completely in two dimensions (without hiding or omitting any information). This allows for depicting ...
-2
votes
0answers
61 views

On the generalized taxicab number Taxicab(4,2,3) [closed]

Is there a number that can be represented as a sum of two fourth powers in three different ways?
-6
votes
0answers
86 views

About inverse function of π(x) (the prime counting function) [closed]

If for a given π(prime(n)) or f(π(prime(n))), we can find prime(n), is this enough to help on solve RH?
8
votes
1answer
825 views

Are there infinitely many primes of this form?

The semiprime $87 = 3*29$ has a curious property: it's the fact that both $87^2 + 29^2 + 3^2 = 8419$ and $87^2 - 29^2 - 3^2 = 6719$ are prime numbers. This intrigued me and led me to wonder if ...
3
votes
1answer
183 views

A specific Diophantine equation restricted to prime values of variables.

Consider the following Diophantine equation $$x^2+x+1=(a^2+a+1)(b^2+b+1)(c^2+c+1).$$ Assume also that $x,a,b,c,a^2+a+1,b^2+b+1, c^2+c+1$ are all primes. We'll call such a quadruplet $(x,a,b,c)$ a ...
10
votes
0answers
156 views

Recognizing the Galois group from the field discriminant

Along the lines of the general question "How much does the discriminant of a number field reveal about the field?", I was wondering how often it happens that the discriminant of some number field ...
7
votes
0answers
58 views

Is anything known about this class of series involving the divisor function?

I hope it is OK to ask the following reference request. If my question is not suitable, please let me know and I will do my best to modify it! Let $N\in\mathbb{N}$, let $q$ be a point in the open ...
24
votes
1answer
459 views

Has the $E_8$-based generating function for squares numbers been proven?

In his 2004 paper Conformal Field Theory and Torsion Elements of the Bloch Group, Nahm explains a physical argument due to Kadem, Klassen, McCoy, and Melzer for the following remarkable identity. Let $...
0
votes
0answers
65 views

Infinite difference length of integer subsets

Let $A$ be a set of integers. In our recent researches, we've faced to the following property and definition: We say $A$ has infinite difference length, if (a) For every integer $n$ there exist a ...
3
votes
1answer
87 views

Higher roots modulo prime complexity best algorithm

Given integers $a,\ell$ and prime $p$ we need to find the roots of the algebraic equation $x^\ell\equiv a\bmod p$. We know there are at most $\ell$ such $x$. What is the best method to find all such ...
-3
votes
0answers
86 views

Random numbers between 0 and 1 [closed]

John Derbyshire in his book PRIME OBSESSION says on page 366: CHAPTER 3 10. "Here is an example of e turning up unexpectedly. Select a random number between 0 and 1. Now select another and add ...
2
votes
0answers
188 views

The Hilbert Symbol and real algebraic geometry

Let $(a,b)_K$ be the quadratic Hilbert symbol in a local field $K$. Let $a$ be a rational number. By a consequence of the quadratic reciprocity law we have: $$\prod_{p} (a,-1)_{\mathbb{Q}_p}=\mathrm{...
0
votes
1answer
93 views

On the asymptotics of a certain sum involving the prime counting function

Let $\pi(x)$ denote the number of primes $\leq x$. What is the asymptotic form for $$\sum_{r=1}^{\pi(x)-1} \Bigg(\frac{(\pi(x))!}{r!(\pi(x)-r)!}-\frac{(\pi(x))!}{(r-1)!(\pi(x)-r+1)!} \Bigg) $$ ? The ...
5
votes
1answer
291 views

Does every prime $p$ appear in a $p$-term arithmetic progression of primes? [duplicate]

This is a follow-up to an earlier question. The answer to that question was found on this page. The discussion on OEIS seems to suggest that, for any prime $p$, there should exist a $p$-length ...
29
votes
4answers
3k views

Is there an 11-term arithmetic progression of primes beginning with 11?

i.e. does there exist an integer $C > 0$ such that $11, 11 + C, ..., 11 + 10C$ are all prime?
2
votes
1answer
96 views

Twisted modular equation

Let $\gamma_2(\tau)=j(\tau)^{1/3}$. The modular equation shows that the functions $$j\left(\frac{a\tau+b}{c\tau+d}\right),\qquad ad-bc=n$$ are integral over $\mathbf Z[j]$. Under what conditions is ...
1
vote
1answer
204 views

Sum of log over friables

Let $x$ and $y$ be two positive real numbers. What is the mean value of the function $\log$ on $y$ friables integers less than $x$ i.e the value of the following sum $$\sum_{\substack{n \leq x \\ P(n)...
3
votes
0answers
114 views

Extension of Galois representations from geometry

Let $K$ be a number field. Take two representations $A$, $B$ of $G_K=\mathrm{Gal}(\bar{K}/K)$ over some ring $\Lambda$ (in fact, I only consider the case $\Lambda$ is a field, usually of ...
4
votes
4answers
299 views

Fixed Points of the Weyl Group action on a Maximal Torus and the Center of a Reductive Group

Suppose $G$ is a connected reductive group over an algebraically closed field. Then given a maximal torus $T$, we can define a Weyl group $W$ and consider $T^W$, the Weyl-invariants of $T$. This ...
11
votes
1answer
713 views

A curious valuation of this sequence

The sequence $a_n$ given by $$a_n=\sum_{k=0}^n\frac{n!}{k!}$$ is found at A000522 on OEIS with a description: total number of arrangements of a set with $n$ elements. Let $\nu_2(x)$ denote the $2$-...
12
votes
1answer
255 views

Determine $2^{\frac{p-1}{4}}\equiv 1\pmod p$ or $2^{\frac{p-1}{4}}\equiv -1\pmod p$ when $p\equiv 1 \pmod 8$

Let $p=8k+1\equiv 1\pmod 8$ be a prime, thus $2$ is a quadratic residue module $p$. Euler's criterion show that $$2^{\frac{p-1}{2}}\equiv 1 \pmod p.$$ So we must have $$2^{\frac{p-1}{4}}\equiv \...
2
votes
0answers
90 views

Expressing modular functions of level 9 and 32 as rational functions

Let $$\gamma_2(\tau)=j(\tau)^{1/3},\qquad \mathfrak f_1(\tau)=\frac{\eta(\tau/2)}{\eta(\tau)},$$ where $j$ is the modular invariant and $\eta$ is the Dedekind eta function. Cox in his Primes of the ...
1
vote
1answer
95 views

Cryptography with general RSA type integers?

Denote $\mathcal N_r=\{n\in\mathbb Z:\exists\mbox{ distinct equal bit primes }p_1,\dots,p_r:n=p_1p_2\dots p_{r-1}p_r\}$. $\mathcal N_1$ refers to primes and $\mathcal N_2$ referes to balanced ...
10
votes
1answer
608 views

Books with exercises to learn Langlands program, Galois representations, modular forms

I want to develop a basic background in number theory with a goal towards contributing to some part of the Langlands program (which I know is vast, and has many different aspects; I want to do ...
-2
votes
1answer
522 views

Can this criterion to indicate the randomness some numbers? [closed]

John Derbyshire in his book PRIME OBSESSION says on page 366: CHAPTER 3 10. "Here is an example of e turning up unexpectedly. Select a random number between 0 and 1. Now select another and add ...
5
votes
1answer
472 views

An asymptotic formula for this sum

Let $X$ be a positive real number. Can someone help me by providing an asymptotic formula for this sum. $$\sum_{n \leq X, \; n\, \equiv\, a \mod{b}} \log{n},$$ where $a$ and $b$ are two coprime ...
1
vote
0answers
53 views

Weak Leopoldt Conjecture for the Split Prime $\mathbb{Z}_p$-extension

In his 1973 Annals paper, Iwasawa proved that the weak Leopoldt Conjecture holds for the cyclotomic $\mathbb{Z}_p$-extension of any number field. If $K$ is an imaginary quadratic field and $F/K$ is ...
6
votes
2answers
358 views

Primality test for specific class of Proth numbers

Can you provide a proof or a counterexample for the following claim : Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$ Let $N=k\cdot 2^n+1$...
0
votes
0answers
91 views

About the multiplicative group of p-adic complex

I was studying the multiplicative group of the $\mathbb{C}_p$. I'm interesed in the ring $\mathcal{O}_p$ of elements in $x\in\mathbb{C}_p$ such that $|x|_p\geq 1$. I have three questions. The first ...