Prime numbers, Diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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2
votes
0answers
62 views

Asymptotic formula for restricted partition function

Let $p(n)$ be the partition function. Hardy and Ramanujan - and Uspensky, independently proved the asymptotic formula $$(1) \quad p(n) \sim \frac1{4\sqrt{3}} \frac{e^{c_0\sqrt{n}}}{n} \text{ as } n ...
17
votes
1answer
652 views

The conjecture of Montgomery and Soundararajan on primes in short intervals: Empirical inconsistencies?

Assume that $y/ \log x \rightarrow \infty$ and that $y/x \rightarrow 0$. Then, from a conjecture by Montgomery and Soundararajan, we expect the number of primes in the interval $[x,x+y]$ to be ...
3
votes
0answers
84 views

Arithmetic analogs of vertex algebras?

Has anyone successfully defined and studied analogs of vertex algebras where the grading of the fields is by $(\log \mathbb Q)$ rather than $\mathbb Z$? What I mean is that the usual fields $$ a(z) = ...
2
votes
0answers
260 views

Proportion of rational elliptic curves of a given rank

This morning appeared on Arxiv the following article by Manjul Bhargava et al: http://arxiv.org/pdf/1407.1826.pdf, in which the authors give a lower bound for th proportion of rational elliptic curves ...
2
votes
1answer
158 views

Extension of a formula for the quadratic Gauss sums

I am interested in the sums $$g(a,k)=\sum_{n=0}^{p-1}e^{2\pi i a n^k/p}$$ where $p\equiv1\mod k$ is a prime and $a$ is coprime with $p$. When $k=2$, it is a classical fact that $g(a,2)=\chi(a)g(1,2)$ ...
3
votes
0answers
72 views

Base change of discrete series

Let $\pi_f$ be an automorphic representation of $GL_2(A_Q)$ (attached to a modular form $f$), and suppose we want to look at its base change lift to say a quadratic imaginary field. Which are the ...
2
votes
1answer
123 views

About the restriction of a modular representation to a decomposition subgroup II

This question is a variant of this one. Let $f$ be as in the other question, but suppose that we look at the $\ell$-adic representiation attached to $f$: $$ \rho_f : G_{\mathbb Q} \to ...
3
votes
1answer
159 views

Regular singularities and the infinitesimal site

Suppose I have a smooth non-proper algebraic variety $X/\mathbb{C}$. A vector bundle with flat connection (``differential equation'') on $X$ extends, as was noted by Grothendieck, to a coherent ...
0
votes
0answers
267 views

Arithmetic progression and average of two prime numbers

Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by: $$ \ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1. $$ For all terms of $A$ greater than $\ ...
0
votes
1answer
94 views

rank of Abelian schemes under ample hypersurface section

Let $k$ be an algebraic closure of a finite field, $\ell \neq \mathrm{Char}(k)$ be prime, $S/k$ a smooth projective geometrically connected surface and $C/k$ a smooth ample connected hypersurface ...
2
votes
1answer
164 views

About the restriction of a modular representation to a decomposition subgroup

Let $f$ be an eigenform of level $\Gamma_1(N)$ and let $p$ be a prime that does not divide $N$. It is well know that there is a $2$-dimensional representation $$ \rho_f \colon G_{\mathbb Q} \to ...
2
votes
1answer
230 views

Proof of the Friedlander–Iwaniec theorem

Does anybody know where I could find the proof of the Friedlander–Iwaniec theorem. The link that I find when I search for it is http://www.pnas.org/content/94/4/1054.full.pdf+html, but this seems more ...
18
votes
0answers
499 views

Base change for $\sqrt{2}.$

This is a direct follow-up to Conjecture on irrational algebraic numbers. Take the decimal expansion for $\sqrt{2},$ but now think of it as the base $11$ expansion of some number $\theta_{11}.$ Is ...
-2
votes
1answer
135 views

Decimal digits multiplied by powers of 2: leads to mod 8? [closed]

This is more a puzzle than a research question, a puzzle to me. Perhaps it is straightforward for others. Imagine Repeatedly interpreting a number expressed with the usual base-$10$ digits as ...
0
votes
1answer
286 views

How can I calculate $ \sum_{i=1}^{n} (n \bmod i) $

I want to calculate the sum $$ \sum_{i=1}^{n} \left\lfloor\frac{n}{i}\right\rfloor, $$ and it seems to require to calculate the sum $$ \sum_{i=1}^{n} (n \bmod i). $$ How can I get an $O(1)$ ...
3
votes
0answers
92 views

Equidistribution of double coset

Let $G=PGL_n(\mathbb{R})$, $K=PO_n(\mathbb{R})$ and $X=G/K$. Also suppose $\Gamma=SL_n(\mathbb{Z})$ acts on the left of $X$. We define a typical Hecke operator on $L^2(\Gamma\backslash X)$ by the ...
11
votes
2answers
369 views

how to find cubic polynomial that an unknown subset of a set of integers satisfies

I have a set, $S$, of positive integers and I have reason to believe that some infinite subset of them may be parametrized by a cubic polynomial with integer coefficients evaluated at integer ...
0
votes
0answers
124 views

Vanishing sum of roots of unity with fixed weight

Consider the set $S_n(d) = \{(x_1, \cdots ,x_n) \in \mathbb{C}^n \mid 1 + x_1+\cdots +x_n = 0, x_i^d=1 \text{ for all } i=1,\dots,n.\}$ Many papers which I found consider the conditions on $n$ and ...
1
vote
2answers
296 views

Non-hyperelliptic curves of genus at least two

A hyperelliptic curve can be understood as the set of points satisfying an equation of the form $$\displaystyle z^2 = f(x,y),$$ where $f(x,y)$ is a binary form of degree $d = 2g+2$. In this case, $g$ ...
5
votes
0answers
240 views

Which L-functions are not “Langlands-Shahidi L-functions”?

The Langlands-Shahidi method, among other things, obtains certain L-functions from the constant term of Eisenstein series attached to so-called $(G,M)$ pairs, where $G$ is a reductive group, $M$ a ...
1
vote
1answer
299 views

What is the value of $\sum _{n=1}^{\infty \:}\frac{n!}{n^n}$? [closed]

I have a question: What is the value of $\sum _{n=1}^{\infty \:}\frac{n!}{n^n}$? Only I've calculated the following identity: $$\sum _{n=1}^{\infty \:}\frac{n!}{n^n}=\int _0^{1}\left(1+x\cdot \ln ...
3
votes
1answer
370 views

Orders of Finite Simple Groups

Which finite simple groups have order N so that N+1 is a proper power? As an example: the simple group of order $168=13^2-1$.
2
votes
4answers
387 views

Equivalent binary forms

Two binary forms $f, g \in k[x, y]$ are equivalent when there exists an $M \in GL_2 (k)$ such that $f^M = g$. For simplicity we take $k$ such that $char (k) =0$ and $k=\bar k$. The equivalence ...
1
vote
1answer
168 views

What can be said about the asymptotics of this sequence?

This is a more general version of a question I have asked before. Let $a_1,\dots,a_r\in\mathbb C$ be algebraic numbers and suppose that for every natural number $n$ the sum $$ F(n)=\sum_{j=1}^r a_j^n ...
0
votes
0answers
268 views

Are these roots of unity?

Suppose you are given complex numbers $a_1,\dots,a_N$ which are algebraic over $\mathbb Q$ and satisfy $1=|a_1|=\dots=|a_k|>|a_{k+1}|\ge\dots\ge|a_N|$. Assume that $$ \sum_{j=1}^Na_j^m $$ is a ...
9
votes
2answers
469 views

Bound on gcd of two integers

Well this is a problem I was fiddling with. I came up with it but it probably is not original. Suppose $a\in \mathbb{N}$ is not a perfect square. Then show that : ...
1
vote
2answers
363 views

Chinese Remainder Theorem backwards

I have the following situation, that is much alike the Chinese Remainder Theorem. Let $\phi_d(\alpha)$ be the $d^{th}$ cyclotomic polynomial in the variable $\alpha$ (I'm not specifying the ...
2
votes
0answers
224 views

schanuel's conjecture and real root of $x+e^x=0$

Schanuel's conjecture states: -If $\alpha_1,\alpha_2,...,\alpha_n$ are complex numbers linearly independent over $\mathbb{Q}$ then the trascendence degree of the field ...
2
votes
0answers
81 views

Primality Criterion for Specific Class of Numbers of the Form kb^n-1

Let $N=k\cdot b^n-1$ where $b$ is an even integer , $3\nmid b$ , $3\nmid N$ , $k \equiv 1,5 \pmod{6}$ , $k< b^n $ and $n>2$ . Let $S_i=P_b(S_{i-1})$ with $S_0=P_{k\cdot b/2}(P_{b/2}(4))$ , ...
4
votes
0answers
96 views

Complexes of arithmetic $\mathcal{D}$-modules with Frobenius structure

This is a question about the category $F\text{-}D^b_\mathrm{coh}(\mathscr{D}^\dagger_{\mathscr{X},\mathbb{Q}})$ of complexes of arithmetic $\mathscr{D}$-modules with Frobenius structure on a smooth ...
16
votes
1answer
380 views

Covering a set with geometric progressions

Consider the set $S_n=\{1,2,\cdots ,n\}$. What is the minimum number of distinct geometric progressions that cover $S_n$? Let us call this number $a_n$. I was wondering about this number after doing a ...
2
votes
0answers
76 views

Berkovich Analytification of the transseries

I am looking for references to articles about the following subjects: Connections from the field of (real) transseries to the field of surreal numbers (mentioned very briefly in the introduction of ...
2
votes
0answers
86 views

Primality Criterion for Specific Classes of Generalized Fermat Numbers

Let $F_n(b)= b^{2^n}+1 $ where $b$ is an even integer , $ 3\nmid b , 5\nmid b $ and $n\ge2$ Let $S_i=P_b(S_{i-1})$ with $S_0=P_{b/2}(P_{b/2}(8))$ where ...
8
votes
0answers
154 views

Symmetric Fifth Power Lift of GL(2) Automorphic Form

Let $\pi$ be an automorphic representation of $GL(2)/\mathbb{Q}$. For simplicity, you can take it to be a Maass form for $SL(2,\mathbb Z)$. Kim, Shahidi, Gelbart-Jacquet prove that $$L(s, \pi, ...
0
votes
0answers
118 views

Relations among the height of algebraic numbers

I would like to find effective upper bound for the height of a-b in terms of the heights of a and b. Thanks in advance. Here the height of an algebraic number a is the maximum of the absolute values ...
8
votes
1answer
185 views

Self-dual automorphic forms on GL(4)

as is known among experts, all self-dual automorphic form on GL(3) comes from symmetric square lift from GL(2). You can find this from Ramakrishnan ...
0
votes
1answer
257 views

A number theoretic inequality

Is this inequality true? : $$\displaystyle \prod_{i\le \left\lfloor{\frac{n-1}{2}}\right\rfloor}\left\lfloor{\frac{\left\lfloor{\frac{n-1}{2}}\right\rfloor}{i}}\right\rfloor\le ...
4
votes
1answer
183 views

Iwasawa logarithm and analytic continuation

I am reading Number Theory vol. 1 by Henri Cohen (among other things) and I am curious about the Iwasawa logarithm. Let $\mathbb{C}_p$ be the completion of the algebraic closure of $\mathbb{Q}_p$. ...
3
votes
2answers
390 views

Primes from a Dirichlet sequence and an irrational number

From Dirichlet's theorem on arithmetic progressions, if $\text{gcd}(a,b)=1$ we know $\{ak+b\}_{k\ge 0}$ contains infinitely many primes. Let those primes be $p_1,p_2,\cdots$. Then the real ...
2
votes
1answer
169 views

Newform and Galois representation (Shimura-Deligne Reciprocity Law)

Shimura-Deligne Reciprocity Law implies that to a newform $f \colon =f(z) \in S^{\mathrm{new}}_k(\Gamma_0(N))$ of weight $k$, one can associate Galois representation $\rho_{f,\lambda} \colon ...
3
votes
1answer
144 views

Normality property of powers of integers?

Inspired by this question, is there some conjecture stating that $$ \limsup_{n \to \infty} \frac{d_j(2^n)}{dc(2^n)} = \frac{1}{10} $$ where $d_j(m)$ counts the number of $j$s in the digits of $m$, and ...
11
votes
1answer
244 views

Cantor set intersecting a geometric sequence

I was working on a problem involving finding all points in the intersection of the Cantor set $C$ and the geometric sequence $\{ (2/3)^i \}_{i=1}^\infty$. The only points I have in this intersection ...
30
votes
3answers
2k views

Fixed points of $x\mapsto 2^{2^{2^{2^x}}} \mod p$

Let $p$ be a prime. For how many elements $x$ of $\{0,1,\dotsc,p-1\}$ can it be the case that $$2^{2^{2^{2^x}}} = x \mod p?$$ In particular, can you find a simple proof (or, even better, several ...
5
votes
2answers
227 views

Absolutely irreducible p-adic representation of the absolute Galois group of Q_p

Let $p$ be a prime number, $\mathbb{Q}_p$ the field of $p$-adic numbers, $G_p$ the absolute Galois group of $\mathbb{Q}_p$ and $V$ a finite dimensional vector space over $\mathbb{Q}_p$. Assume we are ...
10
votes
1answer
352 views

Primes dividing $2^a+2^b-1$

From Fermat's little theorem we know that every odd prime $p$ divides $2^a-1$ with $a=p-1$. Is it possible to prove that there are infinitely many primes not dividing $2^a+2^b-1$? (With ...
0
votes
0answers
77 views

Mellin transform of time-shifted function

The Mellin transform of a function $f(x)$ can be written as $$ \mathcal M[f(x);z]=\int_0^\infty f(x)x^{z-1} dx $$ Is there a simple expression for the Mellin transform of the function $f(x-x_0)$? ...
5
votes
1answer
307 views

Easiest way to distinguish $E_8 \oplus E_8$ from $E_{16}$

For $n$ an integer divisible by $8$, let me denote by $E_n$ the "usual" even non-degenerate positive definite integral symmetric bilinear form over $\mathbf Z^n$. It is well known that in dimension ...
5
votes
0answers
215 views

On simple normality to co-prime bases

Question : Do there exist infinitely many rational numbers $0\lt p\lt 1$ which are simply normal to two co-prime bases? How about three or more co-prime bases? Remark : This question has been ...
-4
votes
1answer
159 views

A general question on nonnegative integer sequence [closed]

Let $A=\{x\ |\ x\in\mathbb Z_{\ge 0},\ x\ $ with some conditions$\ \}$. Let $B=\mathbb Z_{\ge 0}-A$. Define $\ 2A= \{a+b : a \in A,\ b \in A\}$. Define $\ 2B=\{a+b : a \in B,\ b \in B\}$. Then the set ...
7
votes
1answer
263 views

Fast quadratic norm algorithms

Given rational numbers $a$ and $b$, what is the fastest way to determine whether there are any rational solutions to $a=x^2+by^2$? I am interested in the case where the numerator and denominator of ...