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
2,275
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Analytic equivalents for primes in arithmetic progressions
By way of context: it is known that the prime number theorem $\pi(x) \sim x/\log x$ is (nontrivially) equivalent to the statement that $\zeta(s)$ does not vanish on the line $\Re s=1$.
I would like ...
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Is there a mistake in Mochizuki's proof of Theorem 1.10 in IUTT IV? [closed]
In Global character of ABC/Szpiro inequalities, Peter Scholze says that he thinks Joshi's proof of the abc conjecture in his paper has a mistake in Proposition 6.10.7. However, for the proof of ...
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Density and sums of reciprocals
Is there a notion of density for a (strictly increasing) sequence of natural numbers that decides whether the sum of the reciprocals of that sequence converges?
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Prescribed values for the uniform density
Strauch & Tóth [1] Georges Grekos [3][4] showed that for any choice of upper and lower density, there is some subset of $\mathbb{N}$ with the chosen densities, provided the lower is no more than ...
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How comes vanishing of the real part of function involving zeta is very well approximated by algebraic curve?
In this question
Agno asked about the zeros of $\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$.
I fixed $a=2$ and the minus sign and defined:
$$
f(s)=\Re \left( \zeta\left(\frac{...
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What would be the consequences of $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log k}$?
The question is in the title: what would be the number theoretic consequences if we managed to establish the conjectured asymptotic equality $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log ...
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Parity of class number of pure cubic fields
A pure cubic field is an algebraic number field of the form $K = \mathbb{Q}(\theta)$ with $\theta^3 = m$, $m \neq \pm 1$.
What can be said about the parity (odd or even) of the class number of a pure ...
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Invariant means on the integers
Let $A\subseteq\mathbb Z$, as usual we define the lower Beurling density $d^{-}(A)=\lim\inf_{n\rightarrow\infty}\frac{|A\cap[-n,n]|}{2n+1}$ and the upper Beurling density $d^+(A)=\lim\sup_{n\...
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Double sum of reciprocal powers of integer LCMs
I am looking for approximations, or a closed form, if available, for the sum
$S(n,k)=\sum_{1\leq i,j,\leq n} lcm(i,j)^{-k}$
where lcm(i,j) is the least common multiple of integers $i,j$ and $k$...
- 10.1k
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Is $n=6$ the only integer satisfies ${\sigma}_x(n) \equiv 0\bmod{n}$ for every odd integer $x > 0$ and $2 (\bmod n)$ if $x$ is even integer? [closed]
After a few computations in wolfram alpha about the divisor function for some values of $n$ to look the behavior of $\sigma_x(n)\bmod n$ for $\,n=6,\,$ i got this result : $\sigma_x(6)=0 \bmod 6$ for $...
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Is there a similar formula like Ramanunjan's Eisenstein series identity for $\sum_{k=1}^{n-1}k^2 \sigma(k)\sigma(n-k)$?
This question is related to the last question about van der Pol's identity for the sum of divisors.
In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$ satisfies the following ...
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Are all complex zeros of $\zeta(s) \pm \zeta(-s)$ on the line with $\Re(s)=0$?
My conjecture is that all zeros in the strip $-1 \le \Re(s) \le 1$ of $\zeta(s) \pm \zeta(-s)$ are on the line $\Re(s)=0$.
I did find three complex zeros for $\pm =+$ (i.e. 12 in total) and two ...
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Unprovable sentence about integers
Is there any natural* statement S about the natural integers such that if PA contains no contradictions then neither PA+S nor PA+not S contains a contradiction?
If unknown, where can I read about the ...
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What is wrong with this deterministic algorithm efficiently generating large primes?
According to PolyMath
(Strong) conjecture. There exists deterministic algorithm which, when given an integer k, is guaranteed to find a prime of at least k digits in length of time polynomial in k. ...
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A question on BSD conjecture
If $E$ is an elliptic curve over $\mathbb{Q}$, and $K$ is an imaginary quadratic field. If $rank E(K)\leq 1$ and both $E$ and the quadratic twist of $E$ by $K$ satisfy the full BSD conjecture, does ...
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References for the result that $\sqrt{n}$ is equidistributed mod 1
It is not difficult to show (even without Weyl criterion) that the sequence $\sqrt{n}$, $n=1,2,\ldots$ is equidistributed mod 1. However, I need a reference to this result. Can you help me? Thanks.
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Arbitrarily long composite anti-diagonals?
Plot points $(a,b) \in \mathbb{N}^2$ if gcd$(a,b) \neq 1$.
Call such points composite points.
Call a sequence of points $(a+i,b-i), i=0,\ldots,k$
a composite anti-diagonal if all are composite points.
...
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Counterexample to Proposition of Granville related to abc conjecture
Looks like there is counterexample to Proposition related to
abc conjecture. Confusion is likely.
From RATIONAL AND INTEGRAL POINTS ON QUADRATIC TWISTS OF A GIVEN HYPERELLIPTIC CURVE, Andrew ...
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Reduction mod $p$ of units in a ring of integers
Let $\mathcal{O}_k$ be the ring of integers in an algebraic number field $k$ and let $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_k$. I'm looking for conditions on $k$ and $\mathfrak{p}$ which ...
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What is the shortest proof of the existence of a prime between $p$ and $p^2$ ? other examples? [closed]
1) It is well known that between a prime $p$ and $p^2$ always exist a prime, but what is the shortest proof of that (by elementary methods or not)?
(One can say that we can have it as a collorary of ...
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Re: Mordell's Equation $y^2 = x^3 + k$ and Perfect Numbers
I have already tried a somewhat exhaustive search of the literature, but couldn't find anything close to the problem that I am working on.
My question is: When does Mordell's Equation
$$y^2 = x^3 +...
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The distribution of fractional parts $\Big\{ \frac{N}{n} \Big\}$
Let $N$ be a large integer. What is known about distribution of fractional parts $\Big\{ \frac{N}{n} \Big\} \in [0,1)$ after division of $N$ by all odd numbers $n$ in the range $3 \leq n < \sqrt{N}$...
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On a certain $(-1)$-Eulerian polynomials of type $B$
Let $(q)_n=(1-q)(1-q^2)\cdots(1-q^n)$ with $(q)_0:=1$. Define a $q$-exponential by
$$e_q(z)=\sum_{n\geq0}\frac{z^n}{(q)_n}.$$
There is a notion of $q$-Eulerian polynomials of type $A$, see the ...
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Difficult elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?
Similar to the case $x^4+y^4+z^4 = 1$ discussed in this MO post, define the system,
$$x^4+y^4+z^4+1 = (x+y+z+1)^4\tag1$$
$$\frac{x^2+x+1}{(x+y+1)(x+z+1)}=u\tag2$$
$$\frac{y^2+y+1}{(y+z+1)(y+x+1)}=v\...
- 12.1k
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CM fields and Hilberts 12th problem
According to the main theorem of CM, for every abelian variety $A$ associated to
a CM field $K$, one obtains a certain unramified abelian extension of the reflex field $K^\times$ given by the field of ...
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Trisecting $3$-fold sumsets: is the middle part always thick?
Here is a truly minimalistic and seemingly basic question which should have a simple solution (I hope it does).
Let $A$ be a finite set of integers with the smallest element $0$ and the largest ...
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Dirichlet series of the reciprocal radical function
Define $ rad(n):=\prod_{p|n}p $ and $a_n:=\frac{n}{rad(n)}.$ For example $a_n=1$ whenever n is a squarefree integer. The associated Dirichlet series $$F(s):=\sum_{n} \frac{a_n}{n^s}=\prod_{p} (1+\frac{...
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Are these continued fractions for the "tails" of $\zeta(3)$ and of the Catalan constant known?
For polynomials $a=a(x)$ and $b=b(x)$, define the continued fraction $$f(a,b):=a(1)+ \lower 2pt\overset{\infty }{\underset{n=1}{\mathbb{\LARGE K}}}~\dfrac{b(n)}{a(n+1)}=a(1)+\cfrac{b(1)}{a(2) + \cfrac{...
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What is the best known upper bound for $\frac{1}{\zeta'(\rho)}$ assuming the SZC but not the RH?
Let $\zeta$ denote the Riemann zeta function and let $\rho$ denote one of its complex zeros. What is the best known upper bound for $\frac{1}{\zeta'(\rho)}$ assuming that all zeros are simple (SZC), ...
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Is there a connection between the closed forms of these two infinite products?
Take the following two infinite products that have closed forms.
Assume: $\gamma_n > 0 \in \mathbb{R};s,a,x \in \mathbb{C}; x \ne 0,a \pm ix\gamma_n \ne 0$
The first product:
$$\displaystyle H_{...
- 4,169
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Minimum number of unit fractions to sum up a given positive rational
For any positive $p,q\in\mathbb{N}$ there is a finite subset $S$ of $\{\frac{1}{n}:n\in\mathbb{N}, n\geq 1\}$ such that $\sum_{s\in S} s=\frac{p}{q}$, see this article by Paul Erdös and Sherman Stein (...
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Positive integer solutions to the diophantine equation $(xz+1)(yz+1)=z^4+z^3 +z^2 +z+1$
Let $P(z) = z^4 +z^3 +z^2 +z+1$ where $z$ is a positive integer.
While working with the diophantine equation $(xz+1)(yz+1)=P(z)$, I was able to construct a seemingly infinite and complete solution set ...
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Bound on a scaled sum of the Liouville function
Terence Tao has shown see his blog post that
$$\left| \sum_{n\leq x} \frac{\mu(n)}{n} \right|\leq 1,$$
for $x$ a positive real number, where $\mu(n)$ is the Möbius function. Let $\lambda(n)$ denote ...
- 10.1k
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Is every square root of an integer a linear combination of cosines of $\pi$-rational angles?
For example, $\sqrt 2 = 2 \cos (\pi/4)$, $\sqrt 3 = 2 \cos(\pi/6)$, and $\sqrt 5 = 4 \cos(\pi/5) + 1$. Is it true that any integer's square root can be expressed as a (rational) linear combinations of ...
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Is every polynomial a factor of a trinomial?
We say that $T(X) \in \mathbb{Q}[X]$ is a trinomial if there exist $A,B,C \in \mathbb{Q}$ such that $T(X) = AX^n + BX^m + C$ for some $n \geq m \in \mathbb{N}$.
Is it true that for each irreducible ...
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Langlands conjectures in higher dimensions
Geometric class field theory (curves over a finite field) has been generalized
to higher dimensional varieties over a finite field (and other arithmetical fields). Some of the key names here are Lang, ...
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The $\zeta$-word [closed]
I was wondering about classical notations in number theory. I will not ask here about special functions in general but about the more ubiquitous number theory functions. That which made me wonder ...
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Eulerian ordering of the integers modulo n
Let $n>1$ be an integer. Consider the set $C_n := \{0,1, \dots , n-1\}$.
An Eulerian ordering of $C_n$ is an ordering $r_1, \dots, r_n$ of its elements such that:
$$\forall i \le n \ \forall j&...
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Optimizing a smoothing function with the Prime Number Theorem in mind
Let $f:[0,\infty)\to \mathbb{R}$ be a function with $f(x)=1$ for $0\leq x\leq 1$. Write $Mf$ for the Mellin transform of $f$. Let $c>0$, $T>10^6$ be constants. We are interested in minimizing ...
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Transforming the Kondo quintic $5T2$ into the Lehmer quintic $5T1$?
I. Kondo-Brumer quintic
The deceptively simple solvable quintic,
$$x^5 + (a - 3)x^4 + (-a + b + 3)x^3 + (a^2 - a - 1 - 2b)x^2 + b x + a=0$$
is quite important for imaginary quadratic fields. For ...
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348
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(Variation of an old question) Are these functions polynomials?
This is a followup to the question here: How to show that the following function isn't a polynomial over Q?.
As before, let $b_1,b_2,\dots$ be an enumeration of $\mathbb Q$. The question might ...
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Differential Galois number theory
Following https://math.stackexchange.com/questions/635659/can-the-error-term-involved-in-the-pnt-be-expressed-in-a-galois-theoretic-framew?noredirect=1#comment1341143_635659, I vainly tried to find ...
- 6,932
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A truncated divisor function sum
Let $d(n)$ be the number of divisors function, i.e., $d(n)=\sum_{k\mid n} 1$ of the positive integer $n$. The following estimate is well known
$$
\sum_{n\leq x} d(n)=x \log x + (2 \gamma -1) x +{\cal ...
- 10.1k
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Ternary quadratic form theta series as Hecke eigenforms and class number one
At
Simple comparison of positive ternary quadratic form representation counts
Jeremy answered:
"The reason is that the theta series for the sums of three squares form is an eigenfunction for all the ...
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3
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The rank of a class elliptic curves
For elliptic curve $y^{2}=x(x+a^{2})(x+(1-a)^{2})$,($a$ is a rational number and does not equal 0,1,1/2),is its rank always 0?
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Zeta function double product
Is it possible to write the following double product in terms of the zeta function?
\begin{align}
&\prod_{i=1}^{\infty}\prod_{j=1}^{\infty} \frac{1}{1-(p_i\ p_j)^{-s}}
\end{align}
Extending the ...
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263
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Are continuous self-maps of the Golomb space $\mathbb G$ dense in the space of all self-maps of $\mathbb G$?
The Golomb space $\mathbb G$ is the set $\mathbb N$ of positive integers endowed with the topology generated by the base consisting of arithmetic sequences $a+b\mathbb N_0:=\{a+bn:n\ge 0\}$ with $a,b$ ...
- 40.9k
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425
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consecutive prime gaps and explicit bound
I am aware of the theorem that $p_{n+1}- p_n \leq n^{0.525}$ which is true for all sufficiently large numbers due to Baker, but if i want to make the implicit "for all sufficiently large numbers" ...
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Hamming distance between $a+b$ and $a \oplus b \oplus ((a \land b) \ll 1)$
Motivation. In their paper about the cryptographic scheme NORX, the authors use a fast approximation of + by bitwise operations (taking fewer CPU cycles than proper addition) using the formula $$a+b "=...
- 45.5k
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There at least 4 divisors of $n-1$ which do not divide $\phi(n)$ if $n$ is a composite of the form $6k+1$
If $n$ is composite then $\phi(n) < n-1$ (Euler's totient function) hence there must be one or more divisors of $n-1$ which do not divide $\phi(n)$. For lack of a better terminology, let us call ...
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