All Questions
Tagged with reference-request nt.number-theory
1,408 questions
0
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Existence of smooth integers in every residue class with large modulus
Let us say that a positive integer $x$ is $y$-power smooth, if the largest prime power divisor of $x$ is at most $y$. In what follows, let $C$ be any real number larger than $1$ and, for an integer $x$...
- 1,663
5
votes
1
answer
228
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Lifting mod $p$ representations of arithmetic fundamental groups of a non-affine scheme over a finite field of characteristic $p$
Let $X$ be a geometrically irreducible scheme (not necessarily affine) over $\mathbb{F}_{p}$ and let $ \pi_{1}(X) $ be the arithmetic etale fundamental group of $ X $. Let $ \overline{\mathbb{F}}_{p} $...
- 863
5
votes
1
answer
187
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Reference Request: Test vectors for local Rankin-Selberg L-factors in ramified cases
Let $F$ be a global number field, i.e. a finite extension of the field of rational numbers. Let $\sigma$, $\pi$ be automorphic representations of $\mathrm{GL}_n(F)$ and $\mathrm{GL}_{n+1}(F)$ ...
- 639
6
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0
answers
176
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Fundamental lemma of sieve theory in function fields
Is there any literature concerning the fundamental lemma of sieve theory in $\mathbb{F}_q[T]$?
In integers there are various versions of the lemma (bases on different sieves); I would be happy with ...
- 14.6k
1
vote
0
answers
75
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automorphisms and mellin transforms
If a real analytic function $f$ is involutive i.e. $f(f(x))=x$ and its Mellin transform can be taken on a section of the real axis, and is analytic for $x>0$, in certain cases can this imply that $\...
- 101
3
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0
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164
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Cardinality of the set $\#\{ 1 \leq n \leq N: \| \alpha n^2/N \| < 1/N \}$
Let $\alpha \in I$ where $I$ is some closed interval that does not contain $0$.
I am interested in upper bound for
$$
M(\alpha) = \#\{ 1 \leq n \leq N: \| \alpha n^2/N \| < 1/N \}
$$
where $N$ ...
- 3,625
5
votes
1
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310
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Is there a statement in Presburger arithmetic about primes this simple heuristic fails for?
I came up with the following conjecture while thinking about ways to formulate some heuristics about primes:
Conjecture: Given a statement $s$ in Presburger arithmetic, using an additional unary ...
- 3,319
2
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0
answers
278
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On $(k,\ell)$-sumfree sets
Call a set $\mathcal S \subset \mathbb N$ to be $(k,\ell)$-sumfree if there are no non-trivial solutions to the equation
$$x_1+\dots +x_k = y_1+\dots +y_\ell$$
in the set (for distinct $x_i$'s and $...
- 791
3
votes
0
answers
186
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Bourgain-Gamburd-like theorems in the non-algebraic case
For $\mu$ a Borel probability measure on the compact group $G=\operatorname{SU}(d)$, Bourgain-Gamburd prove that the spectral radius of the associated operator on $L^2(G)$ is strictly less than one, ...
- 31
1
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127
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Multiplicities of Galois representations in the semisimplification of the reduction of a Tate module
Let $C$ be a smooth proper curve, of genus $g$ over a number field $K$. Let $v$ be a prime of good reduction for $C$ above $p>2$, and let $T_pJ$ denote the $p$-adic Tate module of $J$, the Jacobian ...
- 2,907
1
vote
0
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158
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A question and reference about Bombieri's article continued fraction of algebraic numbers
Above the Comments in the article continued fraction of algebraic numbers, there are some words on the unboundedness/cycle of coefficients of continued fraction of algebraic numbers "Thus, ...
- 3,094
1
vote
0
answers
59
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A question on generalized bases
I just came to know that it is possible to define a generalized base as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\...
3
votes
1
answer
320
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Counting points on elliptic curves
Consider the Legendre family of elliptic curves
$$E_a: y^2=x(x-1)(x-a).$$
Let $p$ be an odd prime.
QUESTION. Is the following true? If $p\equiv 3\pmod4$ then number of solutions to $E_2$
over the ...
- 43.2k
3
votes
1
answer
98
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Reference Request: Possible generalizations of the stability of $\gamma$-factors
$\DeclareMathOperator\GL{GL}$
Let $F$ be a nonarchimedean local field. Suppose $\pi, \sigma$ are irreducible admissible representations of $\GL_{n}(F)$ and $\GL_{m}(F)$ respectively, with $n \geq m$. ...
- 639
1
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64
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$F$-structure implies regular singularities + unipotent local monodromy?
Let $(\mathcal{E},\nabla)$ be a vector bundle with an integrable connection on a smooth quasi-projective $K$ scheme $X$, with $K$ a $p$-adic number field of characteristic $0$. Let $F$ denote a semi-...
- 2,907
3
votes
1
answer
212
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Isocrystal with no $F$-structure
$\DeclareMathOperator\Isoc{Isoc}$Let $X_k$ be a quasiprojective $k$ scheme, with $k$ finite, and let $X_K$ be the rigid analytic space lifting it to the fraction field of its Witt ring, which I denote ...
- 2,907
2
votes
0
answers
129
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Imaginary quadratic fields with prime class number
Let $K$ be an imaginary quadratic field, with class number equal to an odd prime, say $h_K = p$.
In the proof Proposition 2.4 of this paper, Fukuda and Komatsu write,
"Since $h_K = p$, there ...
- 707
6
votes
2
answers
755
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Prove positivity of a binomial sum
Some problems appear easy on the face of it, but perhaps they are not. Here is an instance of a certain calculation which is slightly reformulated from its original encounter in a current work. I have ...
- 43.2k
2
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0
answers
221
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Squares whose differences are squares
EDIT. I've just noticed a thread from 2011 in the "Related" column on the right (click me), where a closely related question is being discussed (the main difference seems to be that, in ...
- 10.5k
6
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0
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200
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Empirical bounds on $\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right|$
It is reasonable to expect that $$\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right| < 2 \log \log t$$
for all $t\geq 4$ (say): a somewhat stronger bound is known for $t\geq 10^{165}$ or so (Theorem 5 ...
- 20.2k
1
vote
1
answer
155
views
Iwahori action on the $p$-ordinary line of a principal series representation
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\diag{diag}\DeclareMathOperator\Ind{Ind}\newcommand\Iw{\mathrm{Iw}}\DeclareMathOperator\ord{ord}$Let $F$ be a $p$-adic local field, i.e. a finite ...
- 639
4
votes
2
answers
486
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Reference request - Pillai-Selberg Theorem
I want to find a proof of the following claim. Let $\Omega(n)$ denote the number of prime factors of an integer $n$ counted with multiplicity. Then $\Omega(n)$ equidistributes over residue classes. ...
user344548
3
votes
1
answer
459
views
Limit of an infinite series with quadratic arguments
I have encountered a limiting process on some infinite series. So, I would like to ask:
QUESTION. Assume $n$ is an even positive integer. Is this true?
$$\lim_{r\rightarrow1^{-}}\sum_{j=1}^{\infty}\...
- 43.2k
42
votes
4
answers
4k
views
Are these fast convergent series for $\log(2)$, $\log(3)$ and $\log(5)$ already known and proven?
Now that some of the previously MSE formulae that I left here have been applied Dec.2023 to compute high precision record values ($10^{12}$ decimal digits) of trascendental constants $\Gamma(1/3)$ (Eq....
- 2,836
2
votes
2
answers
293
views
Cohomology of $S$-arithmetic groups with trivial coefficients such as $H^n(\rm{PGL}_2(\mathbb{Z}[1/N]);\mathbb{Z})$
As $\rm{PSL}(2,\mathbb{Z})=(\mathbb{Z}/2\mathbb{Z})*(\mathbb{Z}/3\mathbb{Z})$, its cohomology groups $H^n(\rm{PSL}(2,\mathbb{Z});\mathbb{Z})$ are easy to get.
Let $N$ be a product of distinct primes.
...
- 391
1
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0
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131
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Analytic properties of $L$-functions attached to a compatible system of $\ell$-adic Galois representations
Let $F$ and $E$ be number fields, $G_F$ be the absolute Galois group of $F$, and $S$ be a finite set of primes of $F$. For $\lambda$ a prime of $E$ we denote by $\ell$ its residual characteristic. We ...
- 663
5
votes
5
answers
751
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The diophantine equation $ \sum_{n=1}^{N} \frac{1}{x_{n}} = \prod_{k=1}^{N} \left(1-\frac{1}{x_{k}} \right) $
Background
I wonder if there are any rational numbers such that their Egyptian fraction (sum) representations are equal to their Egyptian product analogue. In other words, I am curious1 about ...
- 4,829
3
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0
answers
117
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Reference Request: Local decomposition of GGP period integrals of cuspidal forms on unitary groups
Setup: Let $E/F$ be a CM-extension of global number fields. Let $(V,\phi)$ be an Hermitian space of dimension $n$ over $E$. Let $(V^{\flat}, \phi^{\flat})$ be a subspace of $V$ of dimension $n-1$ on ...
- 639
4
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1
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308
views
3 divides coefficents of this $q$-series
Denote $\phi(q):=\prod_{j\geq1}(1-q^j)$ and let $\xi=e^{\frac{2\pi i}3}$ be a cube root of unity.
Define the sequence $u(n)$ by
$$\prod_{n\geq1}\prod_{s=1}^2(1-q^n\xi^{ns})(1-q^{2n}\xi^{ns})
=\sum_{n\...
- 43.2k
11
votes
0
answers
290
views
Color your partitions by parity
Let $a_c(n)$ be the number of ways to partition a positive integer $n$ where each even part comes in $c$ colors. Then, we can supply the generating function
$$\sum_{n\geq0}a_c(n)q^n=\prod_{k\geq1}\...
- 43.2k
13
votes
0
answers
328
views
Upper bound on prime powers in interval
I just spent a full day on the brutish and thankless task of proving that the Brun-Titchmarsh bound holds for prime powers (including primes), and not just for primes, in the following senses:
(a) the ...
- 20.2k
1
vote
0
answers
89
views
Finiteness of elliptic curves with trivial conductor over function fields
Let $K=\mathbb{F}_q(C)$ be the function field of a smooth projective curve $C$ over a finite field $\mathbb{F}_q$ with $\text{cha}(K)>3$ and let $E$ be an elliptic curve over $E$. To $E$ we may ...
- 121
1
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0
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174
views
Upper bound on sum of Lambda(n) over short interval
I am looking for a bound of type
$$\sum_{x<n\leq x+y} \Lambda(n) \leq \frac{\log(x+y)}{\log y} \cdot 2y$$
(or better). Of course such a bound has to exist: the idea of the proof of Brun-Titchmarsh (...
- 20.2k
6
votes
1
answer
249
views
Syndetic sets and Banach limits: reference request
First of all, let us give a few definitions. Suppose that $A$ is a subset of natural numbers. We say that $A$ is syndetic if there is a constant $M$ such that every set of $M$ consecutive natural ...
- 7,193
1
vote
1
answer
232
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Transfer for the group of coinvariants: a reference request
Let $G$ be a group and $M$ be a $G$-module,
that is, an abelian group written additively on which $G$ acts:
$$ (g,m)\mapsto g m.$$
We consider the group of coinvariants
$$ M_G:=G/\langle g m -m\ |\ g\...
- 14.2k
5
votes
0
answers
100
views
Buchi's conditional proof of the non-existence of finite algorithm to decide solubility of system of diagonal quadratic form equations in integers
I am doing some literature review regarding Buchi's problem. In particular, I am reading the relevant section in this survey paper by Mazur (Questions of Decidability and Undecidability in Number ...
- 26.9k
5
votes
0
answers
131
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Taking integer values of a sequence of Beurling primes
Let $P=(p_j)_{j=1}^\infty$ be an increasing sequence of real numbers with $1<p_1$ and $\lim_{j\to\infty}p_j=\infty$. As mentioned in [1], Beurling proved that if the multiplicative group $N_P$ ...
- 51
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0
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246
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Video abstracts for mathematical papers
I recently published a video abstract of a manuscript of mine (number theory), finding that more people are interested in its content than when I uploaded the preprint on arXiv.
Now, my main question ...
- 1,451
0
votes
0
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73
views
Decrease of $(1/\zeta)^{(r)}(\sigma + i T)$ as $\sigma\to -\infty$?
What is a standard reference for the simple fact that, for $T$ fixed and $\sigma\to -\infty$,
every derivative $|(1/\zeta)^{(r)}(\sigma+i T)|$ of the Riemann zeta function decreases faster than any ...
- 20.2k
2
votes
0
answers
489
views
Are these finite semirings known?
I am trying to prove the properties below, and by doing this, I hope to find a way to speed up the computation of the below defined addition and multiplication. I am also interested if these finite ...
- 3,064
6
votes
2
answers
685
views
Number of divisors which are at most $n$
I’m interested in the function $\tau_n:\mathbb{N}\to\{1,2,3,\cdots, n\}$ defined by
$$\tau_n(x)=\sum_{k=1}^n \mathbf{1}_{k\mid x},$$
the number of divisors of $x$ which are at most $n$. Question 6 of ...
- 541
0
votes
1
answer
292
views
Factorization trees and (continued) fractions?
This question is inspired by trying to understand the lexicographic sorting of the natural numbers in the fractal at this question:
Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , ...
- 3,064
5
votes
1
answer
737
views
Smallest prime factor of numbers
The literature refers to smooth integers as \begin{equation}\Psi(x,y):=\#\{n\le x:P_1(n)\le y\},\end{equation} where $P_1(n)$ is the largest prime factor of $n$. There are lots of results studying $\...
user344548
3
votes
3
answers
382
views
Closed formula for number of ones in a proper factor tree
Edit [2023 Dec 7]: One of my specific wonders, along with that of students, is around when a recursive formula might have – or be expected to have – an explicit or closed formula. What is the ...
- 7,831
1
vote
0
answers
126
views
Reference request: unfolding of Integral representation of an L-function
Are there any text or papers that thoroughly address unfolding of integral representation of an L-function such as D. Ginzburg's On Spin L-function for Orthogonal Groups page 762-763 and page 774 (or ...
- 43
10
votes
0
answers
598
views
Does the interior of Pascal's triangle contain three consecutive integers?
This question defeated Math SE, so I am posting it here.
Consider the interior of Pascal's triangle: the triangle without numbers of the form $\binom{n}{0},\binom{n}{1},\binom{n}{n-1},\binom{n}{n}$.
...
- 3,527
2
votes
1
answer
198
views
Series with the smallest number whose square is divisible by $n$
I was looking into this sequence. And I'm particularly interested in the asymptotic behavior of the following series (which is stated on the site) $$\sum_{k=1}^n \frac{1}{a(k)} \sim \frac{3(\log n)^2}...
2
votes
2
answers
432
views
Expositions of the classical approach to local class field theory (Brauer group and Hasse invariant)
I've posted this question already on MSE and didn't get much out of it, so I hope it's OK to repost here.
I'm an undergraduate trying to learn local class field theory from the corresponding chapter ...
- 121
5
votes
2
answers
541
views
When are two elliptic curves with zero j invariant isogenous?
Consider elliptic curves of the form $E_B\colon y^2=x^3+B$ for $B\in\mathbb Q$. These are exactly the elliptic curves with zero $j$-invariant. I would like to know when are two elliptic curves $E_B$ ...
- 89
0
votes
0
answers
200
views
Kato's explicit reciprocity law paper
Does anyone have a copy of Kato's article Generalized explicit reciprocity laws in Advanced Studies in Contemp. Math which is used heavily in his paper constructing his eponymous Euler system? I used ...
- 2,044