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

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Existence of Euler product on critical line for $L(\chi,s) L(\overline{\chi},1-s)$?

Generally there is no Euler product for Dirichlet L-functions $L(\chi,s)$ in the critical strip.(cf Is the Euler product formula always divergent for 0<Re(s)<1?) But I would like to know if ...
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48 views

Are all complex zeros of $\Psi^{(s)}(1) \pm \Psi^{(1-s)}(1)$ on the critical line?

The balanced polygamma function $\Psi^{(s)}(x)$ for $x=1$ can be expressed as: $$\Psi^{(s)}(1)=\dfrac{\big(\Psi(-s)+\gamma\big)\,\zeta(s+1)+\zeta'(s+1)}{\Gamma(-s)}$$ Note $\Psi(s)$ is the digamma ...
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63 views

Collecting sufficient conditions for Sorli's conjecture on odd perfect numbers

Sorli's conjecture predicts that, for an odd perfect number $N$ given in the Eulerian form $N = {q^k}{n^2}$ (where $q$ is prime with $\gcd(q, n) = 1$ and $q \equiv k \equiv 1 \pmod 4$), the condition ...
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1answer
397 views

On a result attributed to W. Ljunggren and T. Nagell

I've read in a number of places that, building on previous work of T. Nagell, W. Ljunggren proved in 1 that the Diophantine equation $$\frac{x^{n}-1}{x-1} = y^{2}$$ doesn't admit solutions in ...
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68 views

Expliciting the distance between consecutive Goldbach numbers assuming it's finite

In this paper, the author shows unconditionally that at least one of the following statements holds: i) the distance between two consecutive Goldbach numbers is finite, i.e. there exists an ...
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1answer
94 views

Uniform convergence of infinite sum with Dirichlet characters

I would like to prove uniform convergence of function series like : $$\sum\limits_{n=1}^{\infty} \chi(n) f(nx)$$ where $\chi$ is a primitive character and $f(x)$ a function decreasing to zero in ...
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196 views

Upper bound on Betti numbers of an intersection of hypersurfaces (or quadrics)

I have a problem I have been stuck with since several weeks now, and yet I believe it should be easy to specialists. Let $k$ be an algebraically closed field, $m$ and $n$ two integers. Let ...
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137 views

Second differences of primes determined by increasing first differences: every positive even integer?

Suppose that $p(1)$ and $p(2)$ are primes. For $n > 2$, let $p(n)$ be the least prime $p$ such that $p-2p(n-1)+p(n-2)>0$. Does $p(n)-2p(n-1)+p(n-2)$ range through all the even positive ...
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151 views

On the natural density of almost perfect numbers

This question is pretty basic, so I apologize in advance if it is unsuitable for MO. If so, please do let me know and I will migrate it over to MSE. Essentially, by work of Kanold, we know that the ...
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151 views

Some continued fractions for transcendental numbers

In a chapter of Computational Algebra and Number Theory called Continued Fractions of Algebraic Numbers (Available at ...
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268 views

Langlands reciprocity for C*-algebras

I just came across this paper which, judging by what I understood, establishes the Langlands reciprocity conjecture for a certain Shimura variety. My question, regardless of the validity of the proof, ...
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140 views

Looking for a copy of Algebraic Number Theory in honor of Iwasawa

I am looking for an electronic copy of this volume: Advanced studies in Pure Mathematics, Volume 17 Algebraic Number Theory - in honor of K. Iwasawa Edited by J. Coates, R. Greenberg, B. Mazur and I. ...
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70 views

Low height integer points on a rank variety

Let $M_i$ be fixed rectangular matrices with integer coefficients less than $n$. Consider the variety defined by the condition $$ \mathrm{rank}(\lambda_1M_1 + \lambda_2M_2 + ... + \lambda_kM_k) = 1. ...
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1answer
305 views

How to find representatives of $SL(2,\mathbb{R})/SL(2, \mathbb{Z})$

While reading about the Teichmuller flow, I am reading about the space of lattices $SL(2,\mathbb{R})/SL(2, \mathbb{Z})$. I could not a find a good way of computing the Teichmuller flow on this ...
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1answer
141 views

On the number of 3-Selmer elements of rational elliptic curves

I am trying to understand a step in the proof of Theorem 39 in the recent work of Bhargava and Shankar, "Ternary Cubic Forms having bounded invariants, and the existence of a positive proportion of ...
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47 views

Must a multivariate polynomial be zero subject to constraint on the values of all pairs of distinct variables at solutions?

I suppose this is both easy and false. Let $f \in \mathbb{F}_2[x_1, \ldots x_n]$. Suppose in all solutions of $f(x_1, \ldots x_n)=0$ all pairs of variables $(x_i,x_j),i \ne j$ can take all possible ...
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1answer
125 views

Subconvexity bound for Hecke $L$-functions in the $s$-aspect

Let $L(s,\chi)$ be the $L$-function of a non-trivial Hecke character of a general number field $K$, so that $L(s,\chi)$ which has no pole or zero at $s=1$. I am looking for a reference for upper ...
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297 views

An algorithm for Poincare recurrence time

Define the function $[0,+\infty) \rightarrow R$: $$ f = \cos (t) + \cos (\sqrt{2} t) + \cos (\sqrt{3} t) + \cos (\sqrt{5} t ) . $$ I want a number $t $ bigger than $10^7$ such that $$ f(t) > 4 ...
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1answer
361 views

Multiplicity one theorem

I am reading Dorian Goldfeld's book Automorphic forms and L functions for the groups GL(n,R) ...
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1answer
339 views

A property of e?

Define $f(n) = \lfloor {ne}\rfloor$ if $n$ is odd and $f(n) = \lfloor {n/e}\rfloor$ if $n$ is even. Is the set $\{n, f(n), f(f(n)),\dots\}$ bounded for every $n$? Computer sampling suggests that ...
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45 views

Asymptotics on number of bounded prime gaps [duplicate]

It's been over 2 years since the groundbreaking paper by Yitang Zhang in which he has shown that infinitely many prime pairs are by some constant $H$, with $H\leq 70000000$. Over the course of the ...
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1answer
79 views

Weil group of a local field, small notational problem

In Bushnell and Henniart, The Local Langlands conjecture for GL(2), there is a proposition on p. 184 in which they prove the following: Let $F$ be a non-archimedean local field, $\mathcal W_F$ its ...
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85 views

A question on the Siegel's theorem in specific condition

I want to know the number of expressions such that \begin{align} x=p+aq \end{align} for sufficiently large even number $x$, where $p$ and $q$ are prime numbers and $a$ is a positive odd integer which ...
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1answer
332 views

A question on the bounds of the $n$-th composite $c_n$

While trying to prove the inequality $$c_{p_n-m}+c_{m-n}>p_n+2$$ I tried the bounds of $c_n$ (denotes the $n$-th composite number) given in this paper to prove that the sum $c_{p_n-m}+c_{m-n}$ ...
5
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1answer
211 views

Are all complex zeros of $\dfrac{\zeta'}{\zeta}(s) \pm \dfrac{\zeta'}{\zeta}(1-s)$ on the critical line $\Re(s)=\frac12$?

Numerical evidence suggests that all complex zeros (real ones exist as well) of: $$\frac{\zeta'}{\zeta}(s) \pm \frac{\zeta'}{\zeta}(1-s)$$ reside on the critical line with $\Re(s)=\frac12$. I made ...
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1answer
389 views

Completely multiplicative functions with values in $\{-1,1\}$

This question is from Eric Saias and myself: Let $A$ be the set of abscissas of convergence of Dirichlet Series $\sum_{n\ge 1} \frac{f(n)}{n^s}$ where $f(n)$ is completely multiplicative and $f(n) \in ...
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246 views

Are there always at least *five* divisions?

@JosephO'Rourke asked a question about a Collatz like function related to primes: $f(n) = \begin{cases} n^2 & \text{if} \;n \;\text{is prime} \\ \lfloor n/2 \rfloor & \text{if} \;n ...
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1answer
1k views

Is there an arbitrarily long arithmetic progression whose members are palindromes?

I suspect there isn't for some simple reason, but I could not find anything on it and if the opposite would hold in a more general sense, then that would solve this question.
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152 views

Roots of not-necessarily reciprocal polynomials

Consider (irreducible) monic polynomials with integer coefficients which satisfy some fixed linear condition $L$ (for example, the coefficient of $x$ is $1,$ or something more complicated). The ...
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2answers
242 views

roots of reciprocal polynomials

Consider the set $S = \{x \in \mathbb{R} \left| f(x) = 0\right. \},$ where $f$ is a reciprocal monic irreducible polynomial with integer coefficients (reciprocal means that the sequence of ...
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333 views

Elliptic curves and supercuspidal representations of conductor $p^2$

Let $E$ be an elliptic curve defined over $\mathbf{Q}$. Let $p \geq 5$ be a prime of additive reduction for $E$. Let $f$ be the newform associated to $E$, and let $\pi$ be the irreducible admissible ...
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45 views

Explicit twisted Padé approximants

This is a follow-up of Twisted Padé approximants Let $z\in\mathbb Z_p$ with $v_p(z)>0$. One puts $f_z(x)=(1+z)^x$ for all $x\in\mathbb Z_p$. I try to determine the twisted Pade approximants ...
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85 views

What are the differences between p-adic Whittaker functions and archimedean Whittaker functions? [closed]

What are the differences between p-adic Whittaker functions and archimedean Whittaker functions? Are there some references about the differences? Thank you very much.
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1answer
173 views

Explicit description of rings of Witt vectors

I have some basic questions on the rings of Witt vectors. The first example one looks at is $W(\mathbb{F}_{p})= \mathbb{Z}_{p}$. Is it known if $W(\mathbb{F}_{p}[x]/(x^{n})) = ...
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254 views

What is a “generalized zeta function”?

Out of procrastination I computed $$\sum_{k=1}^\infty k^{-k^2}\sim 1.06255080549625593786944593879.$$ The inverse symbolic calculator identified this number as "From generalized Zeta function". I do ...
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372 views

Are There Infinitly Many $n$ Which $a\times n!+1$ Be Composite?

$ax+1$ is a linear polynomial with integral coefficients. Are there infinitly many $n$ which $a\times n!+1$ be composite? As I know this problem is true for polynomials with degree greater that 1, ...
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45 views

Distribution of smooth values of polynomials

Let $\xi$ be a positive parameter. We say a positive integer $n$ is $\xi$-smooth, or friable, if for all primes $p$ dividing $n$, we have $p \leq \xi$. Let $T(X, \xi)$ denote the set of $\xi$-smooth ...
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1answer
179 views

Siegel-Walfisz for the Möbius function

I am working through the proof of the Bombieri-Vinogradov theorem in Analytic Number Theory (Iwaniec, Kowalski). My problem is that on page 424, it is said that $\mu(m)$ satisfies $D_f(x;q,a)\ll ...
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1answer
443 views

A Collatz-like function that bifurcates on primes

This is likely piling one mystery on another, but ... I was exploring a function $f(n): \mathbb{N} \mapsto \mathbb{N}$ defined as follows: $$ f(n) = \begin{cases} n^2 & \text{if} \;n \;\text{is ...
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1answer
96 views

Twisted Padé approximants

Let $f$ be a continuous function defined on $\mathbb Z_p$. By Mahler theorem, there exists a sequence $(\gamma_k)_{k\in\mathbb N}$ of $\mathbb Z_p$ such that for every $z\in\mathbb Z_p$ ...
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713 views

A Question about Palindromic Numbers and System of Arithmetic Progression

Based from Harminc and Sotak's result, www.fq.math.ca/Scanned/36-3/harminc.pdf We know that under certain condition, an arithmetic progression can contain an infinitely many palindromes. My question ...
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1answer
163 views

Sums of two squares: positive lower density? [duplicate]

This question was (indirectly) raised in this post. A set $A\subseteq \mathbb{N}$ has positive lower density if $$\text{lim inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} > 0.$$ Does the set ...
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347 views

genus 2 Siegel theta series of 3-dimensional lattices

Let $(V,f)$ be a $3$-dimensional positive definite quadratic space over $\mathbf Q$. Let $G(V)$ be a set of representatives of the isometry classes of maximal integral lattices on $V$. To an ...
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60 views

Representations of Hamilton's real/complex quaternions algebra

A lot of works and questions deal with classifying representations of a simple central algebra of given dimension over a non-archimedean field, for instance here. But do we know precisely such a ...
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331 views

Sums of sets of lower density 0

We say that a set $A\subseteq \mathbb{N}$ has lower density 0 if $$\text{lim inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} = 0.$$ Given $A,B\subseteq \mathbb{N}$ we set $A+B = \{a+b: a\in A, b\in ...
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1answer
121 views

Simultaneous approximation by rationals with relatively prime numerators

The following seems hard to me (or perhaps just not true), but perhaps I am mistaken. It is known that given irrational numbers $x_1$ and $x_2$, there are infinitely many simultaneous rational ...
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230 views

n torsion groups of quadratic twists of elliptic curves

If $E$ is an elliptic curve over a number field $K$ and $E^{F}$ is a quadratic twist of $E$. Then it is stated in ``Ranks of twists of elliptic curves and Hilbert’s tenth problem" due to Mazur and ...
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471 views

Which irrationals yield bounded sets of iterates?

For $r > 0$, define $f(n) = \lfloor {nr}\rfloor$ if $n$ is odd and $f(n) = \lfloor {n/r}\rfloor$ if $n$ is even. For which real numbers $r$ is the set $\{n,f(n), f(f(n)),\dots\}$ bounded for every ...
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585 views

Distribution of composite numbers

I have moved this question to math.stackexchange.com. People who are interested in this question can discuss at :http://math.stackexchange.com/questions/1272431/distribution-of-composite-numbers ...
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Mod 2 modular forms in levels 5 and 25--how to account for this Hecke isomorphism?

The space $P1$ of my earlier question 203755 "Two spaces attached to mod 2 level 9 modular forms...", is essentially the space of mod 2 level 3 modular forms. That such a space should appear inside ...