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

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28 views

Number of zeros in Pascal-like triangles up to $2m+1$

Let $m \geqslant 2$ be a fixed integer. Then we have an integer sequence given by $$a(m)=\sum\limits_{n=0}^{2m+1}\sum\limits_{k=0}^{n}\operatorname{binmod}(n,k,m)$$ where $$\operatorname{binmod}(n,k,m)...
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23 views

"Residue-class generic" numbers

Let us call a set $D\subseteq\mathbb Z$ residue-class dense if for each residue class $[a]_n=\{kn+a\mid k\in\mathbb Z\}$, there is a residue class $[b]_m$ with $[b]_m\subseteq [a]_n\cap D$. Using the ...
28
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1answer
758 views

Quaternionic and octonionic analogues of the Basel problem

I asked this question in MSE around 3 months ago but I have received no answer yet, so following the suggestion in the comments I decided to post it here. It is a well-known fact that $$\sum_{0\neq n\...
28
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0answers
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A question related to the Hofstadter–Conway \$10000 sequence

The Hofstadter–Conway \$10000 sequence is defined by the nested recurrence relation $$c(n) = c(c(n-1)) + c(n-c(n-1))$$ with $c(1) = c(2) = 1$. This sequence is A004001 and it is well-known that this ...
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What is "the quotient of the universal ordinary Hecke algebra corresponding to an ordinary $\Lambda$-adic form"?

Let $\Lambda := \Bbb Z_p[[T]]$ be the usual Iwasawa algebra. In Jha and Sujatha - On the Hida deformations of fine Selmer groups on page 181, the authors refer to the quotient $\Bbb H^{\text{ord}}_{\...
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0answers
69 views

Mathematical theory of financial markets and implementations [closed]

Why are trading and financial markets not active areas of research in mathematics? Aren't there any applications of fields like probability theory, number theory, chas theory, etc to financial ...
6
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2answers
209 views

Moebius function of finite abelian groups

I am wondering if there is any literature on general formula of the Moebius function of subgroup lattices of any finite abelian group $G$? What I know is When $G$ is cyclic, the Moebius function is ...
2
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64 views

Literature on analogous arithmetic function of logarithm function

In number theoretical estimations, often we take the logarithms of a natural number to express it properly. A perfect example of this is the von-Mangoldt function. I am looking for an analogous ...
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1answer
93 views

Does Rankin-Selberg convolution preserve primitivity?

Call $L$-function any element of an L-rig (see Are there infinitely many L-rigs? for a definition). Suppose $F$ and $G$ are two primitive L-functions. Is $F\otimes G$ itself primitive? If yes, does ...
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1answer
139 views

Explicit estimates for $N(T,\chi)$ (not $N(T,\chi)+N(T,\overline{\chi})$)

Let $N(T,\chi)$ denote the number of zeros of $L(s,\chi)$ with imaginary part between $0$ and $T$, with any zero with imaginary part equal to $T$ or to $0$ (not that the latter kind really exists) ...
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129 views

Conditions under which an $\eta$-quotient becomes a **weak** modular form (reference request for theorems similar to Ligozat's theorem)

For any $z \in \mathcal{H}$, let $q = e^{2\pi iz}$; and the eta function is defined as ${\displaystyle \eta (q) =q^{\frac {1}{24}}\prod _{n=1}^{\infty }\left(1-q^{n}\right).}$ By an $\eta$-quotient ...
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125 views

Conditional results on average size of Mertens' function

Let $M(x) = \sum_{n \le x} \mu(n)$ where $\mu$ is the Möbius function. Titchmarsh, in his book on the Riemann zeta function, considers consequences of the hypothesis that $$\int_{1}^{X} \left( \frac{M(...
6
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1answer
199 views

Another generalization of parity of Catalan numbers

Recently, a question by T. Amdeberhan gathered up many enjoyable proofs that a Catalan number $C_n$ is odd if and only if $n=2^r-1$. Noam D. Elkies' answer considered $F=\sum_{n=0}^\infty C_n x^{n+1}$....
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1answer
226 views

Why do we need to represent integers as the sum of three cubes? [closed]

It is conjectured that for any integer $k\not\equiv \pm 4\pmod 9$ there are infinitely many integer solutions to $$ a^3+b^3+c^3=k. $$ Some cases for integer $k$ becomes too hard like $42$ which it ...
12
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1answer
394 views

Representing $x^3-2$ as a sum of two squares

Prove that there exist infinitely many integers $x$ such that integer $P(x)=x^3-2$ is a sum of two squares of integers. Ideally, I am looking for a proof method that also applies for other $P(x)$, ...
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53 views

Conjugacy classes in normalized unit group of a group ring

Let $V(FA_4)$ be the normalized unit group of the group ring $FA_4$, where $F$ is the field containing 4 elements and $A_4$ is the alternating group on 4 symbols. How can I find conjugacy classes of ...
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111 views

Nearby cycle is tamely ramified?

Let $S$ be a henselization of a closed point $s$ in a smooth algebraic curve $C$ over some finite field $\mathbb{F}_q$. Then we can consider nearby cycles over $S$. Let $s$ be the closed point of $S$ ...
6
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84 views

Bounds on exponential and character sums of ratio of linear recurrences

Let $\mathbb{F}_q$ be a finite field of $q$ elements, let $\chi$ be a non-trivial additive character of $\mathbb{F}_q$, and let $\psi$ be a non-trivial multiplicative character of $\mathbb{F}_q$. Also,...
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184 views

Messing around with $e+\pi$

This question originates from the conjecture that $e+\pi$ is transcendental, and that $e$ is conjectured not to be a period. Jianming Wan in his paper Degrees of periods states that the transcendence ...
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1answer
164 views

Least number coprime to a given integer

For a positive integer $n$ let $$f(n):=\min\{m\in \mathbb N: m>1, \gcd(m,n)=1\} .$$ Equivalently, $f(n) $ is the smallest prime not dividing $n$. Is there any upper bound literature for this? It is ...
2
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1answer
129 views

On some claims on cyclic modules over Hecke algebra used in Serre's "Quelques applications du théorème de densité de Chebotarev"

I have been reading section 7 of Serre's "Quelques applications du théorème de densité de Chebotarev" (http://www.numdam.org/item/PMIHES_1981__54__123_0/), and in particular have been trying ...
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71 views

On $g^{2^x}=B$ modulo $p$

Let $p$ be prime and $g$ positive integer. Define $f(g,x)=g^{2^x} \bmod p$. Q1 Given $g,p,B=f(g,X)$, what is the complexity of finding $X$? If necessary assume $\varphi(p-1)$ is smooth. Some ...
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0answers
121 views

Exactness of a term after taking Pontryagin dual: a step in the proof of Poitou-Tate duality

I'm reading the proof of Poitou-Tate duality in the book Galois Cohomology and Class Field Theory by David Harari. After some arguments, we get a exact sequence $$ \mathbf{P}^1_S(k,M^{'})^* \...
14
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3answers
2k views

On Robin's criterion for RH [closed]

\begin{equation} \sigma(n) < e^\gamma n \log \log n \end{equation} In 1984 Guy Robin proved that the inequality is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin 1984)....
9
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1answer
468 views

Numbers $k$ with $\{\binom nk:\ n\in\mathbb N\}$ dense in $\mathbb Z_p$ for any prime $p\le k$

Let $k$ be a positive integer and let $p$ be a prime. In my 2011 PAMS paper joint with my former student W. Zhang [Proc. Amer. Math. Soc. 139(2011), 1569-1577], we studied when $$S(k)=\left\{\binom nk:...
45
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5answers
5k views

Integer-valued factorial ratios

This historical question recalls Pafnuty Chebyshev's estimates for the prime distribution function. In his derivation Chebyshev used the factorial ratio sequence $$ u_n=\frac{(30n)!n!}{(15n)!(10n)!(6n)...
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170 views

Galois group of a period

Following Jianming Wan's paper entitled "Degrees of periods", can one define the Galois group of a period as the maximal group of permutations of the variables appearing in the expression of ...
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3answers
2k views

Is there an algebraic curve over Q which is not modular?

Every elliptic curve $E/\mathbf Q$ is modular, in the sense that there exists a nonconstant morphism $X_0(N) \to E$ for some $N$. It is tempting to extend this definition in a naïve way to an ...
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0answers
357 views

Karatsuba photo [closed]

Can anyone confirm if the following link displays a photo of A. A. Karatsuba? https://commons.m.wikimedia.org/wiki/File:A.A.Karatsuba_in_Crimea.jpg
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110 views

Elliptic curves and localizations at various primes

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p$ be a prime at which $E$ has good reduction. Let $D=D_{E,p}$ be the $p$-torsion in the cokernel of the map $E(\mathbb{Q})\otimes\mathbb{Z}_p\...
2
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0answers
191 views

Two conjectures about generalised A329369

Let $m \geqslant 2$ be a fixed integer. Let $$\operatorname{wt}(n,m)=\operatorname{wt}\left(\left\lfloor\frac{n}{m}\right\rfloor,m\right)+n\bmod m, \operatorname{wt}(0,m)=0$$ Then we have an integer ...
2
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0answers
175 views

Ambiguity about the exact definition of coefficients of modular forms

You can see the parts after my questions in the boxes. I received the answer to my first question in the comments. I am confused about the definition of $a_n$ and $b_n$ in Part II below. I know the ...
20
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1answer
1k views

Primes that are sums of two squares with constraints on the squares

It is well known that there are infinitely many primes of the form $a^2+b^2$ (namely all primes congruent to $1$ modulo $4$). On the other hand, Euler raised the problem as to whether there are ...
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0answers
116 views

Can a lower bound for this weakening of Goldbach's conjecture be reached?

Say a non negative integer $r$ is a primality radius of $n$ if both $n-r$ and $n+r$ are prime, and that a non negative integer $w$ is a weak primality radius of $m$ if $\omega(m-w)=\omega(m+w)=1$, ...
2
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1answer
82 views

Modulo $2$ binomial transform of A243499 applied $k$ times

Let $m \geqslant 1$ be a fixed integer. Let $f(n)$ be A007814, exponent of highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$. ...
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1answer
148 views

Are Li's numbers $\lambda_n$ absolutely convergent for $n>1$?

Li's numbers $\{\lambda_n\}$ are defined as $$\lambda_n=\frac{1}{(n-1)!}\frac{d^n}{ds^n} [s^{n-1}\log\xi(s)]_{s=1} $$ for all positive integers $n$. Also $\lambda_n$ is given as a sum over the non ...
5
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0answers
192 views

The Tate-Nakayama theorem and inflation

Let $K$ be a nonarchimedean local field, and $L/K$ be a finite Galois extension with Galois group $G={\rm Gal}(L/K)$ of order $n=[L:K]$. By local class field theory, there is a canonical isomorphism $$...
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0answers
58 views

Correspondence between class group of binary quadratic forms and the narrow class group via Dirichlet composition: an elementary approach?

I have been trying to explore and learn about connections between the form class group and the ideal class group. To be on the same page, we define the form class group of a negative discriminant $D \...
1
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1answer
174 views

Euler quotients modulo $n$

For odd integer $n$, define the Euler quotient modulo $n$ to be $a(n)$: $$ a(n)=\frac{(2^{\phi(n)}-1) \bmod n^2}{n}=\frac{2^{\phi(n)}-1}{n} \bmod n$$ $a(n)=0$ for OEIS sequence Wieferich numbers ...
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0answers
91 views

Can PARI compute class numbers without factoring the discriminant?

When calculating properties of algebraic number fields, one of the hardest steps is factorizing the discriminant of a defining polynomial. This is necessary in the Pohst-Zassenhaus algorithm for ...
3
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0answers
104 views

Global Vogan A-packet is infinite set?

For an cuspidal automorphic representation of general linear group, we can attach its global Vogan A-packet. Though I thought that it is finite set, in some paper, it is written that there are ...
0
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0answers
160 views

A paper 'the rank of elliptic curves' by Brumer [closed]

I want to download a paper: Brumer, Armand; Kramer, Kenneth The rank of elliptic curves. Duke Math. J., but I have no access to Duke journal. I want to read the paper for two years. Please help me!
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2answers
2k views

Status of the $x^2 + 1$ problem

It is a long-standing conjecture (probably just as old as the twin prime conjecture, which has gotten a lot of attention as of late since Zhang and Maynard's breakthrough results this year) that ...
2
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1answer
274 views

Tannakian fundamental group of automorphic representations

Let $\mathcal{C}_{\mathrm{aut}}(G, F)$ be the category of automorphic representations of a connected reductive group $G$ over a number field $F$. If this is a Tannakian category, it has an associated ...
15
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2answers
1k views

Existence of polynomials of degree $\geq 2$ which represent infinitely many prime numbers

To my knowledge it is open so far whether the polynomial $x^2+1 \in \mathbb{Z}[x]$ takes infinitely many prime numbers as values. Is it known so far whether there is at all any polynomial $P \in \...
2
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0answers
99 views

Finding elements in a real extension of $\mathbb{Q}$ that are close to some number in $\mathbb{R}$

Let's consider a set of numbers that one knows to high precision, and one knows or has a strong suspicion that `exact versions of these numbers' (see below) belong to a real extension of $\mathbb{Q}$. ...
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1answer
293 views

Does $n \mid \sigma(n^2)$, if $q^k n^2$ is an odd perfect number?

Let $\sigma(x)=\sigma_1(x)$ be the classical sum of divisors of the positive integer $x$. It is known that $$\gcd(\sigma(q^k),\sigma(n^2))=\frac{\bigg(\gcd(n,\sigma(n^2))\bigg)^2}{\gcd(n^2,\sigma(n^2))...
4
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0answers
130 views

Magic hourglass of squares hyperelliptic equation

I have been looking into the problem of the magic square of squares, or more specifically, the magic hourglass of squares, like so: $a^2$ $b^2$ $c^2$ $ $ $ $ $ $ $ $ $ $ $d^2$ $e^2$ $f^2$ $g^2$ ...
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0answers
50 views

Modulo $2$ binomial transform of A124758

Let $f(n)$ be A153733, remove all trailing ones in binary representation of $n$. Here \begin{align} f(2n)& = 2n\\ f(2n+1)& = f(n)\\ \end{align} Then we have an integer sequence given by \begin{...
0
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1answer
251 views

Factorial primes: expected finite or infinite?

A factorial prime is of the form $n! \pm 1$. The first $14$ factorial primes are listed in the Online Integer Sequences (OEIS) A088054: $$ 2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, ...

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