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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Request for resources or techniques for bounding the infinity norm of an infinite product convolved with a simple function
I'm attempting to bound an expression of the form.
$$
\lVert(\prod_{i=1}^{\infty} \phi_i) * s \rVert_{\infty}
$$
Where $\phi_i$ are bounded periodic step functions which can be replaced by smoothed ...
2
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0
answers
182
views
Multiplicity of zeros of partial sums of the Dirichlet Eta function
I am studying ways to approach the problem of the multiplicity of zeros of the partial sums of the Dirichlet Eta functions:
$$
\sum_{n=1}^{K}\frac{(-1)^{n-1}}{n^{s_o}} = 0
$$
more in particular, ...
2
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0
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95
views
What are the known upper bounds for $\sum_{d\leq x} a_{d}F(x/d)$?
Suppose that $a_d$ is a sequence of numbers such that $\sum_{d\leq N} |a_d| \ll \sqrt{N}$ and $F(N) \ll N(\log (N))^{-c}$ for some constant $c>1$. What are the known upper bounds for $f(x):=\sum_{d\...
2
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35
views
Are there differently normalized forms of the dual of the Turan-Kubilius innequality?
I am reading Elliot's fabulous book "Probabilistic Number Theory I", and they present that upon dualizing a variant of the Turan-Kubilius inequality one obtains that
$$\sum_{p\leq x}p \left|\...
2
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0
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205
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What problems are easier assuming zeros of a zeta function don’t behave as we expect?
What are some examples of problems which are easier to solve assuming zeros of zeta functions lie off the critical line or do not have expected vertical distribution.
There are some very well known ...
2
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0
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111
views
If $n$ is a multiperfect number, then necessarily does one of its prime factors $p$ satisfy $p \parallel n$?
My question is as in the title:
If $n$ is a multiperfect number, then necessarily does one of its prime factors $p$ satisfy $p \parallel n$?
I quote from an answer by Varun Vejalla to a closely ...
2
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159
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Galois representation absolutely irreducible after restricting to open subgroup of finite index
Let $E$ and $F$ be finite extensions of $\mathbb{Q}_p$. Let $\phi:\mathrm{Gal}(\overline{E}/E)\to GL_n(F)$ be an absolutely irreducible continuous representation. Assume that the restriction of $\phi$ ...
2
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95
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if such counter example exists for Lehmer's totient problem could we prove that there are infinity of them or just finitely?
I asked this question one month Ago in MSE but no answer for existence of argument which show if such counter example exists we would have infinity of them or just finitely many examples
Lehmer's ...
2
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167
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Sequences of very well distributed integers
The set of integers $\mathbb{Z}$ has a peculiar property: it is extremely well-distributed modulo any positive integers. For an integer $m > 1$ and integer $a$, put $\mathbb{Z}(a; m) = \{n \in \...
2
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174
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A question on $a_i(n) = a_i(\pi(n)) + a_i(n-\pi(n))$ with $a_i(n) = 1$ for $n \le i$
Let $a_i(n) = a_i(\pi(n)) + a_i(n-\pi(n))$ with $a_i(n) = 1$ for $n \le i$ where $\pi(n)$ is the prime-counting function.
By definition, it is obvious that $a_1(n) = n$ and $a_2(n)$ is https://oeis....
2
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98
views
Rankin-Selberg method and Symmetric power of elliptic curves
Let $E$ be an elliptic curve over $\mathbb{Q}$ with conductor $N$. Let $f=\sum a_n q^n$ be the weight 2 modular form corresponding to $E$.
Define $L_2(f,s)=\zeta(s-1)L(Sym^2(E),s)$.
The following ...
2
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92
views
Best known upper bound for Dedekind zeta function on line $\sigma=1$ in the $t$ aspect
What's the best known upper bound for the Dedekind zeta function $\zeta_K(s)$ of a number field $K$ for $s=1+it$ as $t\rightarrow \infty$. For example, is something like this
$$\zeta_K(1+it)=o(\log{t})...
2
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101
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Why don't Dirichlet series with higher order zeros at $s=1$ have faster converging partial sums at $s=1$?
When examining the Dirichlet series generated by $\mu(n)\gcd(n,k)$ for some fixed value $k$, I found something counterintuitive. To set this up, we can expand as an Euler product to see that
\begin{...
2
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285
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On the Chowla and twin prime conjectures
I'm reading https://terrytao.wordpress.com/tag/chowlas-conjecture/ and at some point it is mentioned that, the twin prime conjecture is a variant of Chowla's conjecture that $\sum_{n\leq x} \lambda(n)\...
2
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280
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Best known primality test for the whole intervals of integers up to $10^{20}$ — like the sieve of Eratosthenes
What are the best known primality test(s) for the whole intervals of integers up to $N=10^{20}$ ? "Best" means "have minimal amortized time per tested integer".
That is, the ...
2
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0
answers
106
views
Prime gap transform
Let $n$ be a large enough composite integer, and consider an arithmetic function $f$ that maps $n$ to the sum of prime gaps making a closed interval $J_{f}(n)$ containing $n$ whose extremities are ...
2
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95
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Is there any good reference on the Bayesian view that can be helpful for reading papers on the number theory using heuristic arguments?
Nowadays there are many papers on the number theory using heuristics.
I have read some of them.
But I have no clear understanding of the Bayesian Probability(subjective probability).
The concept of ...
2
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0
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289
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Elliptic curves: about a passage in J. Silverman's "Advanced topics of elliptic curves"
Reading the proof of the main theorem of complex multiplication for elliptic curves over number fields in J. Silverman's book "Advanced topics of elliptic curves" I got stuck at a passage ...
2
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201
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Asymptotics on a double sum over primes
I am attemping to find asymptotics of
$$\sum_{p \leq n}\ln p \left( \sum_{k=1}^\infty \left(\left\{\frac{n}{p^k} \right\} - \left\{\frac{n-1}{(p-1)p^k} \right\} \right) - \left\{\frac{n-1}{p-1} \...
2
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0
answers
108
views
On variations of a claim due to Kaneko in terms of Lehmer means
This post is cross posted from Mathematics Stack Exchange, due that there was a mistake from my part (see the excellent partial answer and my thread of edits of my question on MSE) this post on ...
2
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58
views
Classicality theorems for Hilbert eigenforms of low weight
Let $p$ be a prime, $k \geq 2$ an integer and $f$ a normalized overconvergent, cuspidal eigenform of finite slope, weight $k$, and level divisible by $p$. The classicality theorem of Coleman gives ...
2
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answers
74
views
Least number of factors $\sigma(p^e)$ of representation of $\sigma(N)$ to get the least multiple of $\operatorname{rad}(N)$, for odd perfect numbers
I've cross-posted this from the post of Mathematics Stack Exchange that I've asked (Apr, 2nd 2020) with title On the least number of factors $\sigma(q^{e_q})$ to get the least multiple of $\...
2
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0
answers
146
views
Is this conjecture equivalent to Polignac's conjecture?
Under Goldbach's conjecture denote by $r_{0}(n)$ for $n$ a large enough composite integer the quantity $\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$, by $k_{0}(n)$ the quantity $\pi(n+r_{0}(n))-\pi(n-...
2
votes
0
answers
93
views
Can the cohomology group $H^1(H, E[p])$ be trivial for all subgroups $H$ of $Gal(K(E[p])/K) \simeq GL_2(\mathbb Z/p)$?
Let $p$ be a fixed odd prime, $K$ be a number field and $E$ be an elliptic curve defined over $K$. Set $L =K(E[p])$.
We know that $G=Gal(L/K)$ is a subgroup of $GL_2(\mathbb Z/p)$.
Suppose $G =GL_2(\...
2
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0
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194
views
Reciprocity theorem with $n \ge 5$
If $p \equiv 1 \pmod n$, what additional conditions are needed to ensure that
$$2^{(p-1)/n} \equiv 1 \pmod p?$$
I know:
For $n=3$ (cubic reciprocity) the form is $p=x^2+27y^2$.
For $n=4$ (...
2
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0
answers
69
views
GCD of polynomial values over a box
Let $f,g$ be two polynomials with integer coefficients and equal degree $r \geq 3$, and co-prime as polynomials. I am interested in the following question: what is the distribution of $\gcd(f(m), g(n))...
2
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91
views
Elementary Iwasawa module
Let $k$ be a given number field. What is the importance and applications of knowing that the Iwasawa module $X_\infty$ of $k$ is an elementary $\Lambda$-module?
2
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124
views
Analytic properties of motivic L-functions twisted by Dirichlet characters
Let $M$ be a pure motive over $\mathbb{Q}$ and consider the (completed) $L$-function $\Lambda(M, s)$ attached to its $\ell$-adic realization. Let us assume that this $L$-function admits analytic ...
2
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0
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78
views
Number of solutions to linear diophantine equations, with natural coefficients in a box
Let c, k, d $ \in \mathbb{N} $, let a, x $ \in \mathbb{N}^k $ suppose for all i $ \leq $ k, $ x_i \leq d $, $ a_i \in \mathcal{O}(d2^i) $ and $ \sum{a_ix_i} = c $ my question is for the value of c ...
2
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123
views
Geometrization/sheafification of elementary formulas
Looking in admiration at Deligne's re-definition of Kloosterman sums as traces of Frobenii acting on stalks of certain complexes of sheaves defined via pull-push from $\mathbb{G}_a$ to $\mathbb{G}_m$ ...
2
votes
0
answers
113
views
Reducible polynomial among sequence of polynomials
Let $a_1$ and $a_2$ be two elements of a finte field $\mathbb{F}_{2^m}$ of even characteristic and $a_1^2\neq a_2$. Is it true that there always exists an element $a\in\{a_1,a_2,a_3,\ldots,a_{2^m}|a_{...
2
votes
0
answers
90
views
Improved upper bound for second moment of reduced residues modulo $q$?
The question asked here is a follow up to this question, which was answered by user GH from MO. [EDIT: The question is edited for clarification after receiving a comment on the original posting.]
As ...
2
votes
0
answers
101
views
Sumset of $k$-smooth numbers
Take the set $T(k,n)=M_1(k,n)$ of all $k$ smooth numbers less than $n$.
What is the cardinality of $$\{1,\dots,n\}\cap M_2(k,n)$$ where every integer in $M_2(k,n)$ is the sum of two integers in $M_1(...
2
votes
0
answers
139
views
Double Diophantine approximation
Let $0 < \alpha < 1$. For any $n$ there is a closest lower Diophantine approximation $\max p / q \leq \alpha$ with integer $0 \leq p < q \leq n$. It can be found efficiently, e.g., with Stern-...
2
votes
0
answers
71
views
Write large $n$ as $n_1+\ldots+n_k\ (n_1<\ldots<n_k)$ with $\varphi(n_1),\ldots,\varphi(n_k)\in\{x^k:\ x\in\mathbb Z\}$
Let $\varphi$ denote Euler's totient function.
QUESTION. Is it true that for each positive integer $k$ large integers $n$ can be written as $n_1+\ldots+n_k$ with $n_1,\ldots,n_k$ distinct positive ...
2
votes
0
answers
133
views
Slopes of classical newforms
Let $f$ be a newform in $S_k(\Gamma_1(Np^r))$ with $r\geq 1$, and let $a_p$ be the $U_p$ eigenvalue of $f$.
If $f$ is in $S_k(\Gamma_0(Np^r))$, it seems to be a well known consequence of Atkin-...
2
votes
0
answers
151
views
On the smallest solution of a linear congruence
I have the following question. First, consider the following congruence for primes $p\geq 5$:
$24x\equiv -1\;(\mbox{mod}\;p)$.
The smallest $x$, that is, $1\leq x\leq p-1$ for which the above ...
2
votes
0
answers
83
views
The number of admissible tuples with last element equal to $h_{k-1}$?
Let $k \geq 2$ and $(h_1, h_2,\cdots,h_{k-1}) \in \mathbb{N}^{k-1}$.
Consider the $k$-tuple : $\mathcal{H}_k=(0,h_1,\cdots,h_{k-1})$ with $0<h_1<\cdots<h_{k-1}$.
The $k$-tuple $\mathcal{H}...
2
votes
0
answers
110
views
Inequality about exponential integrals
I am reading about Dirichlet polynomials in the book Analytic Number Theory by Iwaniec-Kowalski.
During the proof of Theorem 9.1 for any positive real numbers $T, N$ they define a piecewise linear and ...
2
votes
0
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529
views
Explicit formula for $n$th prime in terms of Riemann zeros:
We all know there exists an explicit Formula for prime counting function in terms of Riemann zeros.
I'm wondering if similar formula exists for $n$th prime in terms of Riemann zeros?
Or any other ...
2
votes
0
answers
232
views
If some numbers satisfy this divisibility condition with $\sigma$ and $\varphi$, are they necessarily multiples of $6$?
After doing some computations of the divisibility of $\sigma(n)$ by $n+ \varphi(n)$, mostly with Peter´s help, we found these solutions:
$n=2, 456, 828, 7584 ,33462 , 1357440, 1596048 ,1964544 ,...
2
votes
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153
views
Does Langlands use the geometric Frobenius or the classical Frobenius in his papers?
In several of Langlands' papers: Representations of Abelian Algebraic Groups, On Artin's L-functions, On the Functional Equation of Artin's L-functions, Langlands takes a finite Galois extension $K/F$ ...
2
votes
0
answers
135
views
Hasse invariant of abelian varieties with complex multiplication
Is there a good way to compute Hasse invariants of elliptic curves or higher dimensional Abelian varieties with complex multiplication?
For example, if $E$ is an elliptic curve with CM by an ...
2
votes
0
answers
123
views
Another generalization of the Gauss circle problem
In this question I asked for a generalization of the Gauss circle problem, the type of generalization I am asking is to view the Gauss circle problem as one about counting algebraic integers of ...
2
votes
0
answers
122
views
Coefficents of these partition-based polyomials are $0, \pm1$
This is a follow up on my earlier MO question.
Given an integer partition $\lambda=(\lambda_1,\dots,\lambda_{\ell(\lambda)})$ of $n$ where $\ell(\lambda)$ is the length of $\lambda$, associate $\...
2
votes
0
answers
88
views
Cubic extensions of number fields and their local nature
Let $F$ be an irreducible squarefree cubic polynomial over a number field $K$. Let $L:=K[x]/{(F(x))}$ be a cubic extension of $K$.
Suppose that $\alpha \in L^\times$ such that $N_{L/K}(\alpha)=\beta^...
2
votes
0
answers
191
views
The multiplicative constant in the estimate for $S_a(x)=\sum_{n\leq x} d(n)^a$
Let $a$ be a positive real constant and let $d(n)$ denote the number of divisors of $n.$ Define
$$
S_a(x)=\sum_{n\leq x} d(n)^a.
$$
For $a=1,$ the following is well known
$$
S_1(x)=\sum_{n\leq x} d(n)...
2
votes
0
answers
266
views
Cartier duality and Frobenius on Witt vector schemes
Suppose for simplicity we are working over $\mathbb{F}_p$. Cartier duality is an antiequivalence between formal groups and affine group schemes over $Spec(\mathbb{F}_p)$. Let $\mathbb{W}_p(-)$ denote ...
2
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0
answers
170
views
Funny questions about Moebius Function
I need to firstly claim that my research is not about number theory, however, I am pretty interested in it, especially funny questions in number theory, e.g. Kollatz Conjecture. Three years ago, I ...
2
votes
0
answers
118
views
Analogues over finite fields of certain integers defined multiplicatively in $\mathbb Z$
For any irreducible polynomial $f$ of degree $d$ in finite field $\mathbb F$ we have that $$x^{|\mathbb F|^d}-x\equiv0\bmod f$$ and thus the polynomial $x^{|\mathbb F|^d}-x$ is the analog of factorial ...