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

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0
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131 views

The Diophantine equation $x^2 + bxy + cy^2 = p^z_1 \cdots p^{z_k}$

Let $b,c \in \mathbb{Z}$ and let $p_1,\ldots,p_k$ be given primes. Is there an effective algorithm to find all the solutions of the Diophantine equation $$x^2 + bxy + cy^2 = p_1^{z_1} \cdots ...
1
vote
1answer
151 views

Existence of arithmetic function satisfying a certain property

I was interested in an arithmetic function satisfying a certain property, I am not sure at the moment if such thing even exists or not. But I was wondering maybe I could get some hint or idea or input ...
5
votes
1answer
394 views

Are there infinitely many primes p such that both p-1 and p+1 have at most 3 prime factors, counted with multiplicity?

This is a subsequence of primes $p$ for which $p^2-1$ has at most 6 prime divisors counted with multiplicity. This sequence described in the question is the sequence A079153 in OEIS. I could not ...
0
votes
0answers
47 views

Arguments of Dirichlet coefficients of prime index of primitive elements of the Selberg class

Let $F$ and $G$ be two primitive elements of the Selberg class such that for $\Re(s)>1$, $\displaystyle{F(s)=\sum_{n>0}\dfrac{a_{F}(n)}{n^s}}$ and ...
12
votes
3answers
465 views

For an elliptic curve $E/\mathbb{Q}$ can the cohomology group $H^1(\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q}), E[p])$ be nontrivial?

Suppose that $E$ an elliptic curve defined over $\mathbb{Q}$ and $p$ an odd prime. Let $G=\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q})$. I am wondering whether the cohomology group $H^1(G, E[p])$ can be ...
3
votes
0answers
141 views

A Diophantine equation revisited

No integer solution of this Diophantine equation $$x^4+y^4+1=z^2$$ is known other than the trivial ones. While I was reading a paper of Don Zagier, I realized that his idea on the Euler's sum of ...
14
votes
2answers
903 views

Images of polynomials

Let $f,g \in \mathbb{Q}[x]$ be polynomials such that $\{f(a) : a \in \mathbb{Q}\} \subseteq \{g(a) : a \in \mathbb{Q} \}$. Must there be some $h \in \mathbb{Q}[x]$ such that $f(x) = g(h(x))$ for all ...
6
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0answers
159 views

Is ${\rm lcm}\{{\rm ord}_p(q)\colon q\mid p-1,\ q>2\}>\sqrt p\ \,$?

The following question is "ideologically related" to the one I recently asked here. For a prime $p$, let $M_p$ denotes the least common multiple of the orders modulo $p$ of all odd prime divisors of ...
9
votes
2answers
268 views

A back and forth Euclidean algorithm over the integers--does it have bounded length?

cLet $a,b,c,d\in \mathbb{Z}$ and suppose we have the equation $ac+bd=1$. One way of thinking about this equation is it expresses the fact $\gcd(c,d)=1$. It is well-known that all other similar ...
3
votes
3answers
410 views

Pairs of quadratic polynomials taking values pairs of consecutive squares

Let $f,g \in \mathbb{Z}[x]$ be quadratic and neither square. For $x,y,z \in \mathbb{Z}$ what is the maximal number of solutions to $f(x)=z^2,g(y)=(z+1)^2$? Solutions are integral points on the genus ...
6
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0answers
199 views

A conjecture of Erdos on consecutive differences of primes

Let $d_k = p_{k + 1} - p_k$ be the difference between consecutive primes and define \begin{equation} e_k = \left\{\begin{array}{c l} 1 &, d_{k + 1} > d_k \\ 0 &, \text{otherwise} ...
0
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0answers
123 views

Estimating the number of twin primes of given natural configuration order

This question is a follow-up from About Goldbach's conjecture. Let $$\mathrm{Co}_{k}(x):=\{n\le x:\mathrm{ord}_{c}(n):=\pi(\sqrt{2n-3})=k\},~~~\mathrm{co}_{k}(x):=\vert\mathrm{Co}_{k}(x)\vert$$ ...
7
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0answers
253 views

Lindelof Hypothesis implying Selberg Eigenvalue Conjecture?

(General) Lindelof Hypothesis which says for any $L$-function we have $$L(1/2+it)\ll Q(t)^{\epsilon}$$ for any $\epsilon>0$ where $Q(t)$ is the conductor of $L(s)$ at $t$. For a Maass form $\phi$ ...
11
votes
0answers
187 views

What's the dimension of the space of CM cusp forms?

I would guess that the following is very well known, but I don't know the answer and I couldn't find anything with some googling. Let $\Gamma \subset \mathrm{SL}(2,\mathbf Z)$ be a congruence ...
-1
votes
1answer
110 views

CD - continuous development

Let K/Q be a galois extension, p an odd prime and L/K a Z_p extension, different from the cyclotomic one. Let H/K be finite abelian and linearly disjoint from L. a) Are there infinitely many primes ...
4
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1answer
151 views

Is $p$ is square modulo $F_p$ when $p=4k+1 > 5$?

$F_n$ are the Fibonacci numbers. In On computing factors of cyclotomic polynomials p.1 for odd square-free $n>1$ the cyclotomic polynomial $\Phi_n(x)$ satisfies: $$ 4 \Phi_n(x)=A_n(x)^2 - ...
0
votes
0answers
69 views

solving system of equations like Goormaghtigh equation

I will be so thankful for any comment about solving the following system of equations. Could some one find a result for the following or prove that it has no solution? (MATLAB software could not find ...
5
votes
2answers
305 views

A central limit theorem for a trigonometric series involving primes

In some recent work I found I needed to prove a central limit theorem for the interesting series: $\sum_{n=1}^\infty \cos (u \log p_n) $ where u is a random variable uniform on the interval ...
4
votes
3answers
530 views

Is $\lceil \frac{n}{\sqrt{3}} \rceil > \frac{n^2}{\sqrt{3n^2-5}}$ for all $n > 1$?

An equivalent inequality for integers follows: $$(3n^2-5)\left\lceil n/\sqrt{3} \right\rceil^2 > n^4.$$ This has been checked for n = 2 to 60000. Perhaps there is some connection to the ...
2
votes
1answer
140 views

Tower-of-squares sequence divides linear recurrent A001921 sequence?

Let $(a_n)$ be the A001921 sequence $$ a_0 = 0,\ a_1 = 7, \quad a_{n+2} = 14a_{n+1} - a_n + 6. $$ Let $(b_k)$ be the (almost)"tower-of-squares" sequence defined by $$ b_0=2, \quad ...
5
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0answers
175 views

On a claim of Deligne about representations of Weil-Deligne groups

In Deligne's article 'Les constantes des equations fonctionelles des fonctions L' http://publications.ias.edu/sites/default/files/Number20.pdf, we find the following claim: Proposition 8.9 (ibid.): ...
6
votes
3answers
376 views

Asymptotic formulas for Monster-related modular functions?

Define the following, $$j(\tau) = \Big(\tfrac{E_4(\tau)}{\eta^8(\tau)}\Big)^3 = {1 \over q} + 744 + \color{blue}{196884} q + 21493760 q^2 + 864299970 q^3 + \cdots \tag{1}$$ $$j_{2A}(\tau) ...
3
votes
0answers
120 views

Computing local volumes : the case of Hecke p-adic subgroups

I am quite interested in knowing how to compute some volumes of groups defined on local fields $K$, mainly in order to evaluate the identity term in trace formulas. It is something well done in the ...
11
votes
1answer
877 views

Is a number field uniquely determined by the primes which split in it?

Let $K/\mathbb{Q}$ be a number field. We say that a rational prime $p$ splits in $K$ if there exists a prime $\mathfrak{p}$ of $K$ above $p$ of interia degree $1$. Is a number field $K$ ...
5
votes
0answers
118 views

When is the ratio of Jacobi theta functions algebraic?

Probably this is well known. $\theta_2$ and $\theta_3$ are Jacobi theta functions as defined in mathworld (31) and (32). For natural $n$ define $$ f(n) = ...
4
votes
1answer
155 views

Asymptotic behaviour of $K$-Bessel function in transition range

It is known that the famous mistake of Iwaniec-Sarnak in their paper of $L^\infty$ norm of eigenfucntion of non-cocompact arithmetic surfaces in lemma (A1) is because of they did not consider the bump ...
0
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0answers
98 views

Lower bound on number of smooth values of polynomial at primes

Given a polynomial $f$, it is known believed that the number of smooth values of $f$ has a positive proportion (for fixed $u$, $\lim_{X\rightarrow\infty} \frac{|\{ n < X\ :\ f(n)\ is\ X^u\ smooth ...
2
votes
1answer
127 views

DL-problem on abelian variety

Let $A$ be an abelian variety over $\mathbb{F_q}$ with dimension $n$. Let $q$ be a constant. Is there polynomial algorithm of finding discrete logarithm in $A$? UPD: really I don't undestend: can we ...
1
vote
0answers
126 views

Simultaneous vanishing of convolutions of Mertens function with itself

By Landau's theorem on Dirichlet series, we know that all the step functions ($k\geq 1$) $$M_k(x)=\frac{1}{2\pi i}\int^{2+i\infty}_{2-i\infty}\frac{x^sds}{\zeta^k(s)s}=\sum_{n\leq ...
10
votes
2answers
470 views

Faltings height in short exact sequences

Let $K$ be a number field and $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ a short exact sequence of abelian varieties over $K$. Let $h(A)$ denote the logarithmic Faltings height ...
10
votes
1answer
401 views

Parity of the Prime Counting Function

I am interested in the distribution of the parity of $\pi(x)$, the prime counting function, over the natural numbers. Let: $\ \ E_n := \left\{ k \in \left\{1,\dots,n\right\} : \pi(k) \equiv 0 \mod 2 ...
14
votes
1answer
371 views

Special fiber of $X(p)$ in characteristic $p$

Let $p \geq 5$ be a prime. Let $Y(p)$ be the fine moduli space representing elliptic curves + basis of the $p$-torsion over $\mathbb{Q}_p$ and let $Y_1(p)$ be the fine moduli space representing ...
3
votes
0answers
146 views

Argument againts the $abcd$ conjecture with extra gcd conditions

Got an argument and numerical support againts the $abcd$ conjecture with extra gcd conditions (observe that this is different from the $abc$ and the $abcd$ conjectures). This thesis p. 20 defines the ...
5
votes
0answers
162 views

The sum of all the elements of every non empty subset of $A$ is not a multiple of $n$

Let $N=\{1,2,\ldots ,n\},n>1$. We wish to construct a set $A\subseteq N$ with the property: The sum of all the elements of every non empty subset of $A$ is not a multiple of $n$. ...
2
votes
0answers
102 views

Tensor product of two elements of the Selberg class

Maybe too easy a question for most members of this site, but suppose whenever $F$ and $G$ belong to the Selberg class, then so does $F\otimes G$ where the considered tensor product of $F$ and $G$ is ...
0
votes
1answer
115 views

A conjectural convergence condition for a weakened Elliott-Halberstam conjecture

For $a$ and $q$ positive integers such that $a\lt q$ and $(a,q)=1$, let $\pi(x;q,a)$ be the number of primes $p\equiv a\pmod q$ below $x$. One can show that $\pi(x;q,a)\sim \dfrac{\pi(x)}{\varphi(q)}$ ...
11
votes
0answers
214 views

Why $\gcd \{ {\rm ord}_p(q)\colon q\mid p-1 \}$ likes to be large?

For a prime $p$, let $F_p$ denote the greatest common divisor of the orders modulo $p$ of all prime divisors of $p-1$: $$ F_p = \gcd \{ {\rm ord}_p(q)\colon q\mid p-1 \}; $$ thus, for instance, ...
6
votes
1answer
284 views

Cusps forms for $\Gamma (N)$

I know how to build a basis of the vector space of cusp forms for the congruence subgroups $\Gamma_1 (N)$ and $\Gamma_0 (N)$, but I couldn't find in the literature how to build a basis for ...
2
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0answers
88 views

Orders of Clifford algebra

Let $C_n$ be the Clifford algebra over $\mathbb{Q}$ associated to negative definite quadratic form $-I_n$ (i.e. $-x_1^2-\dots-x_n^2$). Let $\mathcal{O}$ be a $\mathbb{Z}$-order of $C_n$. Q1) Is it ...
13
votes
1answer
650 views

Proving the Irrationality of this Number

I found this problem on Math.SE: Prove that $\log_35+\log_25$ is irrational. http://math.stackexchange.com/q/986227/173397. I labored on it for a few days, and couldn't find an algebraic ...
0
votes
1answer
256 views

Is there a formula that can predict the primes in the sequence of ratios of consecutive superior highly composite numbers? : $2, 3, 2, 5, 2, 3, 7,…$

This is the sequence of prime numbers which are the elementary building blocks for the superior highly composite numbers: $2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 2, 23, ...$ The $n^{th}$ ...
1
vote
1answer
166 views

About Section 4.2 in 'introduction to the spectral theory of automorphic forms' by Iwaniec

In Section 4.2 in 'introduction to the spectral theory of automorphic forms' by Iwaniec, the author said that the kernel of the invariant integral operator $$ (Lf)(z)=\int_{\mathbb{H}}k(z,w)f(w)d\mu w ...
14
votes
0answers
495 views

Positive binary quadratic form plus univariate monic cubic (giving Hilbert class field)

We have the Lucas numbers, $$ L_1 = 1, \; L_2 = 3, \; L_3 =4, \; L_4 = 7, L_5 = 11, \; L_{n+2} = L_{n+1}+ L_n \; . $$ Question: is it the case that $$ f(x,y,z) = 4 x^2 + 3 x y + 9 y^2 + z^3 + 3 z ...
7
votes
2answers
381 views

Visibility interpretation of Riemann zeta zeros on the critical line?

This is a long shot, but ... The fraction of $\mathbb{Z}^2$ lattice points visible from the origin $1/\zeta(2)=6/\pi^2 \approx 61$%. The fraction of $\mathbb{Z}^3$ lattice points visible from the ...
1
vote
0answers
98 views

Skew symmetry for the Hilbert symbol

Let $K$ be a local field containing the group $\mu_n$ of $n$th roots of 1 and the $\theta_K:K^*\to G_K^{ab}$ be the reciprocity map. The we know that the Hilbert symbol $$K^*\times K^*\to \mu_n$$ ...
4
votes
3answers
401 views

Textbook request for class field theory [duplicate]

I am studying class field theory. I need good reference books, notes, or other materials which explain the following topics: ideles and ideals, Haar measure and integration on local fields, Fourier ...
8
votes
0answers
964 views

The zeta function and classical mechanics

In this paper, Guilherme França and André LeClair show that $$\gamma_{y}\sim 2 \pi \left(y-11/8\right)/W\left((y-11/8)e^{-1}\right)$$ where $W$ is the Lambert W function, and $\gamma_{y}$ is the ...
3
votes
0answers
145 views

Effective version of the Bombieri-Vinogradov theorem

Is there an effective version of the Bombieri-Vinogradov Theorem, in that have bounds on the implied constant been found?
5
votes
1answer
276 views

Is there a simple proof that Milnor $K_2$ of a number field is torsion?

This is a theorem of Garland. I had a look at the original paper which looks pretty complicated. I was wondering if the proof has been simplified over the years or if a different approach is nowadays ...
2
votes
1answer
229 views

Can the Gaussian integers be covered by restricted recurrences?

Relaxation of the second question here. Let $a(n)$ be recurrence of the form $a(n)=f(n,a(n-1)\ldots(a(n-k))$ with fixed initial terms. (Observe that it might depend on $n$). $f$ may contain ...