Questions tagged [nt.number-theory]
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
15,947
questions
6
votes
1
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425
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Three conjectural series for $\pi^2$ and related identities
Recently, I found the following three (conjectural) identities for $\pi^2$:
$$\sum_{k=1}^\infty\frac{145k^2-104k+18}{k^3(2k-1)\binom{2k}k\binom{3k}k^2}=\frac{\pi^2}3,\tag{1}$$
$$\sum_{k=1}^\infty\frac{...
- 14.5k
6
votes
1
answer
348
views
Ker of corestriction of Galois cohomology
Let $G$ be a Galois group and $H$ be its normal subgroup. Let $M$ be a $G$-module.
Consider the restriction map $res: H^1(G,M) \to H^1(H,M)$. Its kernel is given by $H^1(G/H,M^H)$.
On the other hand, ...
- 1,405
6
votes
1
answer
279
views
The error term for the second moment of Fourier coefficients of cusp forms with the level explicitly determined
There is a basis question which puzzles me for a while. The question is the following:
Let $p$ be a prime and $X\ge 2$. Let $f$ be a $GL_2$-newform of level $p$ and non-trivial
nebentypus, with the $n$...
- 553
6
votes
1
answer
502
views
Langlands-Shahidi method in classical language
The Langlands-Shahidi method says that the $L$-functions of automorphic representations appear in the constant terms of Eisenstein series. Since those Eisenstein series have analytic continuation and ...
- 1,869
6
votes
1
answer
277
views
Which $n$ have $\lvert\{2^n-2^k -1\}\cap {\mathrm{PRIMES}}\rvert=m$?
Consider numbers of the form $2^n - 2^k - 1$ with $k < n$ as considered in OEIS sequence A208083. As for A208083 I investigated how many of these numbers are prime, but turned the question around: ...
- 9,628
6
votes
2
answers
312
views
Algebraic numbers which prescribed degree which does not belong to some fields
In my research it would be great if the following result is valid. In what follows, $\overline{\mathbb{Q}}$, $\overline{\mathbb{Q}}_n$ and $\overline{\mathbb{Q}}_{<n}$ denotes the set of algebraic ...
- 515
6
votes
1
answer
271
views
How often does the omega theorem hold?
Write $\psi(x) = \sum_{n\le x} \Lambda(n)$. The classical omega theorem says that
$\psi(x) - x = \Omega_{\pm}(x^{1/2})$.
Question: How often does this hold? For example, what do we know about the ...
- 509
6
votes
1
answer
555
views
Do we have an algorithm for comparing $e^e$ with rationals?
Do we have an algorithm for comparing $e^e$ with rationals, with a known time to convergence?
In a non-constructive sense, there obviously is an algorithm.
If $e^e$ is some rational $q_0$, then we ...
user44143
6
votes
1
answer
251
views
Is there a finite extension with a non-trivial class group of any PID?
Let $R$ be a PID with infinitely many prime ideals. Does there always exist a finite extension $R\subset R'$ with $R'$ being a Dedekind domain with a non-trivial class group?
- 73
6
votes
1
answer
227
views
Is there a connection between the average 'compositeness' of a rational number and $\phi$ (golden ratio)?
Let $n\in N$, where $n = p_{1}^{k_{1}}p_{2}^{k_{2}}...p_{m}^{k_{m}}$ for $p_{i}$ prime.
Define the 'density' of $n$ as:
$d(n) = \frac{(p_{1}+1)^{k_{1}}(p_{2}+1)^{k_{2}}...(p_{m}+1)^{k_{m}}}{n}$
...
- 363
6
votes
1
answer
264
views
Limits (growth rates) of power series coefficients
Take two positive integers $m$ and $n$ and consider the rational function
$$G_{m,n}(x,t)=\frac{d}{dx}\left(\frac1{(1-x^m)(1-tx^n)}\right)$$
and the corresponding Taylor expansion as
$$G_{m,n}(x,t)=u_0(...
- 41.8k
6
votes
1
answer
603
views
Attempt at applying linear programming to the partial sums of the Möbius inverse of the Harmonic numbers
Let $a(n)$ be the Dirichlet inverse of the Euler totient function:
$$a(n) = \sum\limits_{d|n} d \cdot \mu(d) \tag{1}$$
and let the matrix $T(n,k)$ be:
$$T(n,k)=a(\gcd(n,k)) \tag{2}$$
It has been ...
- 1,133
6
votes
2
answers
408
views
Reduction to Lie algebra version of fundamental lemma?
Ngo famously proved the Langlands-Shelstad fundamental lemma for Lie algebras using the geometry of the Hitchin fibration.
For the purposes of the trace formula, one actually needs the fundamental ...
- 2,336
6
votes
1
answer
360
views
Is $|\{(j,k):\ 1\le j<k\le\frac{p-1}2:\ \&\ (j^{16}\ \text{mod}\ p)>(k^{16}\ \text{mod}\ p)\}|$ even for each prime $p\equiv1\pmod {16}$?
In my paper http://arxiv.org/abs/1809.07766, I determined the parity of
$$\left|\left\{(j,k):\ 1\le j<k\le\frac{p-1}2\ \&\ (j^2\ \text{mod}\ p)>(k^2\ \text{mod}\ p)\right\}\right|$$
for any ...
- 14.5k
6
votes
1
answer
548
views
Sum over characters
Take $x>0$ large, $t\in \mathbb R$, $q\in \mathbb N$ and a non-principal character $\chi $ mod $q$. If you want, take $t\leq x$. How do I bound
\[ \sum _{n\leq x}\frac {\chi (n)}{n^{it}}?\]
My ...
- 1,166
6
votes
1
answer
382
views
Existence of certain cubes in finite fields
Consider $F := GF(q)$ where $q = p^e$ and $E := GF(q^2)$ where w is a primitive element of $E$.
Fix $\theta := w^{q - 2}$.
Starting point: can I always write $1 + \theta$ as a power of $\theta$?
If $...
6
votes
1
answer
474
views
Strengthening an implication of the abc conjecture
Granville gives p.5
an implication of the abc conjecture:
Assume the abc conjecture.
Let $f(x,y)$ be squarefree homogeneous polynomial with integer
coefficients. For coprime integers $m,n$ if $q^2 \...
- 24.2k
6
votes
1
answer
507
views
$GSp(4)$ vs $PSp(4)$
After some months wandering through examples of algebraic groups in the theory of automorphic forms and number theory, I wonder why so many efforts are spent in understanding $GSp(4)$ (local newforms, ...
- 5,314
6
votes
1
answer
531
views
Analogue of j-invariant for CM fields
For any imaginary quadratic field $F$, the Hilbert class field $H$ is generated by the $j$-invariant of any elliptic curve with complex multiplication (CM) by $\mathcal O$, the ring of algebraic ...
- 528
6
votes
1
answer
448
views
Growth of Class Numbers
There is a classical formula stating:
Let $K$ be a number field with ring of integers $\mathcal{O}_K\subseteq K$ and
let $\mathcal{O}\subseteq \mathcal{O}_K$ be any non-maximal order with conductor $...
- 443
6
votes
1
answer
463
views
Twisted modular forms of half-integral weight
I am looking for references (or explainations) about the twist of modular forms of half-integral weight. I try to mimic the proof of the "integral weight case" to prove that the twist of
$$ \theta(\...
- 1,479
6
votes
1
answer
294
views
Abelian varieties with p-rank zero
Let $X$ be an abelian variety over a finite field of characteristic $p$ such that the $X[p]=0$. In other words, none of the Newton slopes are $0,1$.
QUESTIONS.
(a) Is it possible for the ...
- 337
6
votes
1
answer
514
views
Good references for K-theory of modular curves?
The title says it. I am looking for a good exposition on the K-theory of the curves $X_{i}(N)$, $Y_{i}(N)$, where $i\in\{0,1\}$.
I have some background in $K$-theory and also some background in ...
- 1,727
6
votes
1
answer
665
views
Inverse limit of $\Bbb Q/q\Bbb Z$ isomorphic to finite adeles?
Let $\Bbb Q/q\Bbb Z$, for some positive rational $q$, denote the quotient group of the discrete rationals by the subgroup of integers times $q$.
For any $q_1, q_2 \in \Bbb Q^+$ and $n \in \Bbb N^+$ ...
- 4,525
6
votes
2
answers
1k
views
Motivation for Hirzebruch-Jung Modified Euclidean Algorithm
Let $a,b \in \mathbb{N} \ \ s.t. \ \ a > b$ have $\gcd(a,b) =1$. We can define the Hirzebruch-Jung modified euclidean algorithm as follows:
Let $e_i \in \mathbb{N} >2$, and $ r_k \in \mathbb{N}$...
6
votes
1
answer
462
views
Logarithmic weights on number theoretic sums
Suppose we are interested in the sum
$\sum _{n\leq x}a_n.$
The study of the sum
$\sum _{n\leq x}a_n\log (x/n)$
may be easier.
What can one say about the first sum from knowing the behaviour of ...
- 1,166
6
votes
1
answer
593
views
Converse to Modularity II: Maass cusp forms
(This comes from this other question. You can find more details there)
The following bijection is now a theorem:
Odd irreducible 2-dim Galois repn $\longleftrightarrow$ weight 1
newforms
note: ...
- 17.4k
6
votes
1
answer
411
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. ...
- 621
6
votes
4
answers
808
views
Uniform upper bound for the sum over primes $\sum_{p \leq x} p^{-1+\varepsilon}$
I am reading the article D. M. Gordon and C. Pomerance, The distribution of Lucas and elliptic pseudoprime, Math. Comp. (1991) (click).
In equation (27) the authors, apparently, used the following ...
user40023
6
votes
1
answer
739
views
Parity of primes [duplicate]
While working on a completely different (combinatorial) problem, I ran a simple program to calculate the parity of the first ~50000 primes (number of 1s in their binary representation modulo 2). The ...
- 1,602
6
votes
1
answer
751
views
Can the generalized divisor summatory function $D_z$ be expressed explicitly in terms of Zeta Zeros?
Mertens function has, by residues, an explicit formula of
$M(n)=\displaystyle\sum_{\rho}\frac{x^\rho}{\rho\zeta'(\rho)}-2+\sum_{n=1}^\infty\frac{(-1)^{2 n}(2\pi)^{2n}}{(2n)! n \zeta(2n+1)x^{2n}}$
...
- 607
6
votes
1
answer
314
views
Periods of Twists of Modular Forms
Let $f \in S_2(\Gamma_1(N))$ be an eigenform. By a theorem of Shimura, there are associated "periods" $\Omega_f^\pm$ such that, after normalizing by these periods, the L-function associated to $f$ ...
- 1,412
6
votes
2
answers
776
views
Argument of a Gauss sum
It is well known that the Gauss sum of a Dirichlet character modulo $N$
$$G(\chi)=\sum_{a=1}^N\chi(a)e^{2\pi ia/N}$$
Moreover,$$\vert G(\chi)\vert=\sqrt{N}$$
when $\chi$ is primitive.
Question :...
- 3,317
6
votes
3
answers
2k
views
References on techniques for solving equations with discontinuous functions such as floor and ceiling?
Here I describe the sort of reference I'm after with a motivating example. I am not seeking solutions to my equations on this forum; I'm quite happy to do that myself. Rather, I'm asking for some good ...
- 524
6
votes
1
answer
1k
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A Universal Elliptic Curve
I'm working through Deligne's "Formes modulaires et representations l-adiques" paper and I find one of his constructions particularily ambigious. I'm hoping someone can give me a bit of clarification ...
- 61
6
votes
1
answer
1k
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Must the $j$-invariant of an elliptic curve with an isogeny be integral?
Let $K$ be a quadratic field, and $E/K$ a non-CM elliptic curve with a $K$-rational $p$-isogeny, for $p$ a prime. I would like to say the following:
For large enough $p$, the $j$-invariant $j(E)$ ...
- 2,797
6
votes
2
answers
378
views
Lattice-cube minimal blocking sets
Let $C_d(n)$ be the lattice cube consisting of the $n^d$ points with
each of its $d$ coorindates in $\lbrace 1,2,\ldots,n \rbrace$.
Define a blocking set for a lattice cube to be a set of points
in ...
- 149k
6
votes
1
answer
499
views
Equidistribution on the unit circle of particular sequences of finite subsets
Given a strictly convex function $g : [0, 1] \to \mathbb{R}$, I'm curious about the asymptotic distribution of the points $\exp{(2 \pi i N g(n / N))}$ for $n = 1, 2, \dots, N$, counted with ...
6
votes
1
answer
570
views
Are there examples of sets containing no primes but for which both Type I and Type II information can be proven?
In Harman's book "Prime Detecting Sieves," he describes a method to prove that a set contains primes if we have enough Type I and Type II information for it. As shown by Selberg's example of the set ...
- 721
6
votes
2
answers
853
views
Number of integers coprime to l
A long time ago I've seen a paper considering, given $\ell$ fixed, estimates for
$$
\sum_{n \leq x, (n, \ell) = 1} 1
$$
Of course, this is easy to estimate with a trivial error term of $O(\varphi(l))...
- 293
6
votes
1
answer
1k
views
The resultant of an arbitrary polynomial and a cyclotomic polynomial
This is a natural generalization of this question.
Let $f$ be a monic irreducible polynomial over $\mathbb Z$. Let $S_f$ be the set of natural numbers $n$ such that one of the three equivalent ...
- 137k
6
votes
2
answers
2k
views
Image of a Galois representation
Notation:
$E$ is a non-CM Elliptic curve over $\mathbb{Q}$.
$p$ is an ordinary prime.
$f$ - cuspidal eigenform of weight $k$ = 2 attached with $E$.
$\rho_f$ - the global 2-dimensional $p$-adic ...
- 461
6
votes
2
answers
2k
views
Computing the fixed field of an automorphism of a function field
Let say we have a function field $k(x,y)$ defined by $f(x,y)$ over $k$, with $\sigma \in Aut(k(x,y)/k)$ and. Suppose, I'm not that out of luck, so that either of $\prod \sigma^i(x)$ or $\sum \sigma^i(...
- 601
6
votes
1
answer
602
views
Lorentzian characterization of genus
Suppose we take the "even" indefinite lattice from page 50 in Serre A Course in Arithmetic (1973)
$$ U \; = \;
\left( \begin{array}{cc}
0 & 1 \\\
1 & 0
\end{array}
\right),$$
...
- 25.4k
6
votes
2
answers
1k
views
Question related to Diophantine approximations and Roth's theorem
The following question came up in my arithmetic geometry course yesterday. Suppose $\alpha$ is an irrational real algebraic integer, and suppose $\epsilon >0$ is given. Then by Roth's theorem there ...
- 1,362
6
votes
3
answers
556
views
Any rigorous way to claim that sums with repeat summands are few?
Let $B \subset \mathbb{Z}^+$. Define $r_{B,h}(n)$ to be the number of ways of writing $n$ as the sum of $h$ elements of $B$ and $R_{B,h}(n)$ the number of ways to write $n$ as the sum of $h$ DISTINCT ...
- 25.5k
6
votes
1
answer
463
views
Theory of addition and a predicate that recognizes powers of 2
What is the complexity of the theory of addition (Presburger arithmetic) augmented by a unary predicate that recognizes powers of 2?
- 17.5k
6
votes
2
answers
2k
views
Algebraic integers on the unit circle
Consider a set of algebraic integers which lie on the unit circle, they will generate a multiplicative subgroup of $\mathbb S^1$. Do these objects have a name?
I would guess they contain useful ...
- 1,775
6
votes
1
answer
661
views
Residues of $1/\zeta$
Are there any bounds on residues of $1/\zeta$ in roots of $\zeta$ in critical strip, which may use RH, but do not use the conjecture on simplicity of roots or something similar? I did not find such ...
6
votes
1
answer
476
views
How to see that Eisenstein series are eigenfunctions of the laplacian?
Let $\Gamma$ be a discrete subgroup of $PSL_2(\mathbb{R})$ of finite type. Let $c_1,\ldots,c_h\in\mathbb{R}\cup\{\infty\}$ be a set of representatives of the $\Gamma$-equivalence classes of cusps. For ...
- 7,161