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
4,673
questions with no upvoted or accepted answers
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Is there a result that connects the discriminant of an order to the discriminants of the generators of an integral basis for the order?
Let $\left\{ 1 , \omega_1, \dots , \omega_{n-1} \right\}$ be an integral basis for an order $\mathcal{O}$ of a number field of degree $n$ over $\mathbb{Q}$, let $\Delta $ be the discriminant of $\...
2
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115
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Conjectural bound on gaps between values assumed by quadratic forms
Let $D$ be a discriminant, i.e., $D \equiv 1 \pmod{4}$ or $D \equiv 0 \pmod{4}$. Let $\mathcal{S}(D)$ be the set of positive integers for which there exists a binary quadratic form $f$ with integer ...
2
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161
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Prime ideals being finitely-generated implies coherence?
Let $R$ be a non-noetherian local domain. Suppose that the following two conditions hold for $R$$\colon$
$(*)$$~\quad$An arbitrary prime ideal ${\frak P}$ of $R$ such that ${\mathrm{ht}}({\frak P}) &...
2
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67
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Discrepancy related independent vector from tensor product?
Here discrepancy is from $(2.4)$ in https://www.ricam.oeaw.ac.at/files/people/siambook_nied.pdf given by 'The discrepancy $D_N(P) = D_N(x_l,\dots,X_N)$ of
the point set $P$ of $N$ points in $\mathbb Z^...
2
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166
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Solving solutions to systems of polynomial equations over $\mathbb Z$
Macaulay Resultants help identify common root in $\mathbb P^{n-1}(\mathbb K)$ of $n$ homogeneous polynomials in $n$ variables when $\mathbb K$ is algebraically closed. Is it possible that some type of ...
2
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57
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Classes of curves with "determinant-like operation"
Consider a motivating example:
Let $E\in \mathbb{Q}[y][x]$ be of degree $n=2$ (in $x$) and separable when viewed as a member of $\mathbb{Q}[x,y]$. Therefore we can calculate it's roots in $\mathbb{Q}[...
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95
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Polynomials passing through points with tangential conditions
In corollary here http://math.mit.edu/~lguth/Exposition/erdossurvey.pdf on polynomial methods it is said "(Parameter counting) If $S\subset\mathbb F^n$ is a finite set, then there is a non-zero ...
2
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182
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Evaluate a curious determinant with Legendre symbol entries
Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre symbol. R. Chapman's conjecture on the exact value of the determinant of
$$C_p:=\left[\left(\frac{i-j}p\right)\right]_{0\le i,j\le (p-...
2
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139
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Combination of irrationals
Fix a very small $\epsilon>0$; and irrationals $a_1,a_2>0$. Now suppose we look at all integer combinations of these irrationals which has a small norm; that is,
$$
S=\{(m_1,m_2)\in\mathbb{Z}\...
2
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135
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What is the best known upper bound for $( \gamma_{n+1}-\gamma_{n})\max_{\{T\in(\gamma_{n},\gamma_{n+1})\}}(\vert\zeta(1/2+iT)\vert) $?
For $ n $ a positive integer, denote by $ L(n) : =\gamma_{n+1}-\gamma_{n} $ with $ \gamma_{n} $ the imaginary part of the $ n $-th critical zero of the Riemann zeta function and by $ M(n) : =\...
2
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154
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Categorical representations of absolute Galois groups
I am looking for interesting examples of categorical representations of absolute Galois groups of arithmetic fields. Pointers to the literature would be appreciated.
2
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254
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The Hilbert Symbol and real algebraic geometry
Let $(a,b)_K$ be the quadratic Hilbert symbol in a local field $K$. Let $a$ be a rational number. By a consequence of the quadratic reciprocity law we have:
$$\prod_{p} (a,-1)_{\mathbb{Q}_p}=\mathrm{...
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126
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Expressing modular functions of level 9 and 32 as rational functions
Let
$$\gamma_2(\tau)=j(\tau)^{1/3},\qquad \mathfrak f_1(\tau)=\frac{\eta(\tau/2)}{\eta(\tau)},$$
where $j$ is the modular invariant and $\eta$ is the Dedekind eta function. Cox in his Primes of the ...
2
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239
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Weak Leopoldt Conjecture for the Split Prime $\mathbb{Z}_p$-extension
In his 1973 Annals paper, Iwasawa proved that the weak Leopoldt Conjecture holds for the cyclotomic $\mathbb{Z}_p$-extension of any number field.
If $K$ is an imaginary quadratic field and $F/K$ is ...
2
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86
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local gamma factors and their compatibility with the local langlands correspondence
Let $F$ be a p-adic field and $(\rho,V)$ be an n-dimensional indecomposable representation of the local Weil-Deligne group $W_F'$. Then we know that $\rho\simeq \rho'\otimes Sp(m)$, where $\rho'$ is ...
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124
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Number of partitions of $\{1,2,\ldots,n\}$ whose blocks are arithmetic progressions of length $t$ or more
This question was inspired by the earlier question here,where no lower bound on arithmetic progression size was given. In particular, $t\geq 3,$ is assumed here.
The set $\{1,\ldots,n\}$ has $2^n$ ...
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161
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Dirichlet series as rational zeta expressions
Let $D(f,s):=\sum_{n=1}^\infty \frac{f(n)}{n^s}$, otherwise known as a Dirichlet series. When $f$ is a multiplicative, number theoretic function, $D(f,s)$ tends to be expressed as a rational product ...
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46
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Counting intersections of rectilinear lattices
The following proposition
Let $g>0$ be an integer and let $\Lambda \subset \mathbb{R}^g$ be a rectilinear lattice (possibly shifted) with mesh $d$ at most $D$. Then we have
$$
\left| \#(\Lambda \...
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57
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Making a certain coefficient of a generating function for partitions as small as posssible
Begin by writing the generating function for unrestricted partitions as follows:
(1+x+x^2+x^3+…)(1+x^2+x^4+x^6+...)(1+x^3+x^6+x^9)⋯= 1+p(1)x+p(2)x^2+...
Now change some of the coefficients from plus ...
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114
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Does each odd prime $p$ have a primitive root $g < p$ which is the sum of two central binomial coefficients?
The central binomial coefficients are those integers
$$\binom{2n}n=\frac{(2n)!}{(n!)^2}\ \ \ (n=0,1,2,\ldots).$$
QUESTION: Does each odd prime $p$ have a primitive root $g<p$ which is the sum of ...
2
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292
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Zagier's algebraicity of singular moduli
Let $\mathcal M_m\subset M(2,\mathbb Z)$ be the set matrices with determinant $n$. The modular group $\Gamma$ acts on $\mathcal M_m$ from the left and we have the following finite set as a set of ...
2
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300
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Conjectured initial values of Inkeri's primality test for Fermat numbers
This is a repost of this question .
Can you provide a proof or a counterexample to the claim given below ?
First , we shall give a definition of the Inkeri's primality test for Fermat numbers :
...
2
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82
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Can a ring always be found for a list of primes and how they split in the desired ring?
Given a set $\{(p_1, e_1),(p_2, e_2)\dots(p_n, e_n)\}$, does there exist a Galois extension $R$ of the natural numbers such that $p_i$ splits into $\frac{k}{e_i}$ ideals for all $i$ and some $k$?
In ...
2
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166
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Möbius function in every short interval
I am looking for references on conjectures or heuristics concerning cancellations of the Möbius function in very short intervals, namely how small is it believed one can take $H$ so that
$$\sum_{X \...
2
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114
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Sieving the values of an arithmetic sequence which is infinitely many times $1$
I have a sequence of positive integers $a_n$ which assumes infinitely many times the value $1$. I want to estimate the cardinality of the following set:
$$\#\{n\leq x : a_n>1 \text{ and } (a_n, \...
2
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75
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Bounding algebraic numbers away from the Gaussian integers
Let $\alpha$ be an algebraic number with degree $\leq d$ and (absolute multiplicative) height $\leq H$. Then we can say a couple of things about such $\alpha$:
(1) We know the set of all $\alpha$ ...
2
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128
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Can this construction generate bounded aperiodic functions?
This question is based on this old MathOverflow question: How this set of functions is ordered?
In that question, Vladimir Reshetnikov asked about a class $S$ of functions $f:\mathbb{N}\to\mathbb{N}$ ...
2
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132
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Quick computation of a certain exponential sum
Is there a quick way to compute (somewhat accurately) for large $X$, the following exponential sum, where $\Lambda$ is the von Mangoldt function?
$$\int_0^1 \bigg|\sum_{n\le X} \Lambda(n)e(n\alpha)\...
2
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170
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Does this follow from a version of the Strong Approximation Theorem?
I am hoping someone can help me with this.
Define $\Lambda(x)$ to be the set of all 4-tuples $(A,B,C,D)$; $A,B,C,D \in \mathbb{F}_2[x]$ (where $\mathbb{F}_2$ is the field with 2 elements)
such that $...
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163
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Galois action on posets of number fields and $p$-adic fields
In the theory of group actions on posets one studies the action of a group $G$ on a poset $\mathcal{P}$ (via order-preserving permutations) partly through its derived action on the order complex of ...
2
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208
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Cyclotomic ring of integers proof via matrix theory
Not sure of a better title (and I'm open to suggestions). But when following Laffey's notes on Integer Matrices, found here (and in particular, starting on page 13), I came across an alternative proof ...
2
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142
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Genus Zero Diophantine Equations and Infinite Valuations
I'm interested in an explicit upper bound for the integral solutions of a certain genus zero curve $F(X,Y)=0$. I found some papers that address this problem:
[1] Solving genus zero diophantine ...
2
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0
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104
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Dual sheaf of universal pointed unipotent connection and the canonical de Rham torsor
I'm trying to understand a particular point in the paper The Unipotent Albanese Map and Selmer Varieties on Curves by Kim.
We fix a basepoint $b \in X(L)$ on a curve $X$ over $L$ a characteristic 0 ...
2
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81
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Continuity of the conductor of automorphic representations
I am interested in properties of (semi-)continuity of the conductor of automorphic representation. Let $F$ be a number field.
The meta question is, given a function on the unitary dual of $PGL(2, F)$ ...
2
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103
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Bound for the number of solutions to a system of congruence relations
Suppose I have $n$ polynomials in $n$ variables $G_j(x_1, \ldots, x_n)$ with integer coefficients. Let $u_j$ be some fixed $p$-adic integers.
Consider the system of congruences
$$
G_j(\mathbf{x}) \...
2
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348
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The Breuil-Mezard Conjecture and Generalizations (Survey)
What's the current state of the Breuil-Mezard conjecture? Has the original version (from the 2002 paper) been solved in its entirety? What are some of the new directions being explored?
2
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462
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Maximal gaps between squarefree numbers
In this question here, I asked for the moment statistics of the gaps in squarefree numbers which got an immediate and excellent answer by @Lucia.
The answers to two questions in mathstackexchange ...
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142
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Reference request: Regarding the image of inertia group being a subgroup of Aut($\widetilde{E}$)
Let $E$ be an elliptic curve over $\mathbb{Q}_p$ with potential good reduction. I was told that if $F$ is the smallest Galois extension over $\mathbb{Q}_p$ such that $E$ has good reduction then the ...
2
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146
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Skewes' number and the ratio $\frac{\operatorname{li}(x^{1/2})}{\operatorname{li}(x)-\pi(x)}$
(A complementary post is here.)
Given the prime counting function $\pi(x)$ and the logarithmic integral $\operatorname{li}(x)$, we have Table 1,
$$\begin{array}{|c|l|}
\hline
x&\operatorname{li}...
2
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65
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Can this function satisfy Song conditions?
Let $r_{s,k}(n)$ be the number of representation of a naturel number $l$ as a sum of $s$ positive $k$-th powers.
Joung Min Song introduced some conditions to study asymtotic behavior of some positive ...
2
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241
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An exponential sum like the Kloosterman sums
I encounter a tricky sum like the Kloosterman sum
$${\sum_{x \mod P}}^\ast e\left(\frac{ax+\overline{x}}{P}+\frac{lx^2}{P^2}\right),$$
where $l$ is a positive integer co-prime with $P$ and here $P$ ...
2
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111
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Is there an algorithm for numerical approximation of (naive) period integrals
Let $d$ be an integer, and let
$f_1, \ldots, f_m$ and $g_1, \ldots, g_n$ in $\mathbb{Q}[X_1,\ldots,X_d]$
be rational polynomials.
Then
$$D = \{ (x_i)_{i=1}^d \in \mathbb{R}^d \mid f_j(x_1,\ldots,x_d) =...
2
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107
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Is there an explicit version of Morse Lemma used in stationary phase method?
In the proof of the stationary phase method (at least the one I have seen) Morse lemma shows up, which states: Let $g:\mathbb R^n\to \mathbb R$ be a function of class $C^\infty$ for which $0$ is a ...
2
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193
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A high dimensional generation of Dirichlet approximation theorem, linear case and nonlinear case
I am working with something on Diophantine approximation, and I found a high dimensional generation of Dirichlet approximation theorem which may be true; I will be very happy if this is true. The ...
2
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94
views
Identifying an ideal class, identify its norm form, and vice versa
Let $n > 1$ be a positive integer. By a norm form of degree $n$ we mean a homogeneous polynomial $F$ in $\mathbb{Z}[x_1, \cdots, x_n]$ which is irreducible over $\mathbb{Q}$ and splits completely ...
2
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228
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Question about the term $\sum_{ \rho} \frac{X^{\rho}}{\rho}$ in the explicit formula of $\sum_{n \leq X} \Lambda(n) \chi(n)$
Let $\Lambda$ be the von Mangoldt function and $\chi$ a primitive character mod $q$, then we have the explicit formula
$$
\sum_{n \leq X} \Lambda(n) \chi(n) = \delta_{\chi} X - \sum_{ |Im \ \rho| \leq ...
2
votes
0
answers
108
views
Can Davenport's estimate be extended to cubic polynomials with non-zero discriminant?
In 1961 Davenport showed that $H$ large enough there is a constant $c > 0$ such that
$$
\sum \lvert D(P) \rvert^{-1/2} < c H^2
$$
where the sum is taken over the irreducible polynomials of ...
2
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0
answers
167
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About transfer from Hilbert modular forms to Siegel modular forms
Suppose $F$ is a totally real field of degree $d$. Is there an explicit way (like theta series or so) to construct automorphic forms on $\mathrm{Gsp}(2d)$ from Hilbert modular forms of ${\rm GL}_2(F)$?...
2
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0
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217
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Number of solutions to $n^ 2 = x^2 + y^2 + 2z^2$
I was intrigued by this result in Journal of Number Theory from 2013. Let $n = 2^{\lambda_2}\prod p^{\lambda_p}$.
Theorem The number of $(x,y,z) \in \mathbb{Z}^3$ such that $n^ 2 = x^2 + y^2 + 2z^...
2
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0
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479
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Good place to learn about arithmetic schemes?
Where is a good place to learn about arithmetic schemes? There is discussion in Eisenbud-Harris's book The Geometry of Schemes (and also Mumford's red book) and I hear that there is discussion in Liu'...