New answers tagged nt.number-theory
4
votes
Four new series for $\pi$ and related identities involving harmonic numbers
a bit long for a comment.
The "four new series for $\pi$" are examples of relationships between hypergeometric functions $_pF_{p-1}$ with rational arguments, for example, the first series is ...
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3
votes
Is there an elliptic curve analogue to the 4-term exact sequence defining the unit and class group of a number field?
As in the question $K$ is a number field and $E/K$ an elliptic curve.
Let me start by saying that I think the best analogues are the following two short exact sequences (the first two "$/n$" ...
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12
votes
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Is there any use of logarithmic derivatives of modular forms?
Application 1: in the chapter on modular forms in Serre's Course in Arithmetic, he integrates the logarithmic derivative $f'/f$ of a modular form $f$ on ${\rm SL}_2(\mathbf Z)$ around a contour (...
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5
votes
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'$\times$' or '$\otimes$' when writing $L$-functions?
The symbol $\times$ on the left-hand side is the Rankin-Selberg product. If $\pi$ and $\rho$ are automorphic representations of $\mathrm{GL}(m)$ and $\mathrm{GL}(n)$, respectively, then one can define ...
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4
votes
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Large sets of nearly orthogonal integer vectors
Let me prove the bounds
$$2^k{n\choose k+1}+\sum_{j=0}^{k}2^{j}{n\choose j}\leqslant a(n,k)\leqslant 2^{k}{n\choose k+1}+\sum_{j=0}^{k}2^{j}{k\choose j}{n\choose j}$$
which differ by $O_k(n^{k-1})$ ...
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5
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Which algebraic groups are generated by (lifts of) reflections?
Let $M$ be a division algebra of degree 3 (i.e., dimension 9) over $\mathbf{Q}$ that splits over $\mathbf{R}$, and $M_1$ its norm 1 subgroup. So $M_1$ is a $\mathbf{Q}$-anisotropic simple algebraic ...
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0
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$L^1$ norm for a product of cosines
Another idea (from the work of Maynard on primes with missing digits) to bound this integral is as follows: Since $t\in [0,1]$, we expand $t$ in base 3 as $t=\sum_{i=1}^{\infty} t_i/3^i$, where $t_i\...
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10
votes
On the number of distinct prime factors of $p^2+p+1$
There is the following theorem of Halberstam, "On the distribution of additive number-theoretic functions. III." Let $\omega(n)$ be the number of prime factors of $n$. Given any irreducible ...
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10
votes
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On the number of distinct prime factors of $p^2+p+1$
Yes. At first, there exist $c$ distinct primes $q_1,...,q_c$ which divide some $m_i^2+m_i+1$ for $i=1,\ldots,c$ respectively (induction on $c$: if you found $c-1$ such primes, take $m_c$ being equal ...
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15
votes
Computing hypergeometric function at 1
Carlo's answer is correct but doesn't show why the identity holds, so let me explain how to do this easily by hand.
Look more generally at
$$S={}_3F_2\left(\begin{matrix}-m,a,b\\a+1,b+1\end{matrix};1\...
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7
votes
Raising positive integer to $c\in\mathbb{R}-\mathbb{N}$ rarely gives an integer!
To expand on a comment of Lucia, when $c$ is irrational, we can show that there are at most $O((\log N)^2)$ values of $n\leq N$ such that $N^c$ is rational, let alone an integer.
Let $\mathcal{A}$ be ...
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6
votes
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Computing hypergeometric function at 1
For questions like this, Mathematica is your friend:
$$\, _3F_2\left(-m-\tfrac{1}{2},-m,k-m+\tfrac{1}{2};\tfrac{1}{2}-m,k-m+\tfrac{3}{2};1\right)$$
$$=\tfrac{1}{2}(k+1)^{-1}\Gamma (m+1) \left(\frac{(2 ...
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8
votes
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Does this partial sum over primes spike at all zeta zeros?
This is not a complete answer but hopefully provides some insight.
The function $$\exp\left(-\sum\limits_{p\le x},\frac{\cos(x\, \log(p))}{\sqrt{p}}\right)\tag{1}$$
seems to approximate
$$\left|\frac{...
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11
votes
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Arithmetic sequences and Artin's conjecture
It's plausible that there are infinitely many primes $p \equiv 1 \bmod 4$
of which $2$ is a primitive residue. However, this is false for
$p \equiv \pm 1 \bmod 8$, because $2$ is a quadratic residue.
...
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5
votes
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$L^1$ norm for a product of cosines
Since I was asked to in the comments, let me record here some very simple observations. They boil down to the following inequality: for all $k \ge N$ we have
$$I_{k - N} \min_{x\in \mathbb{R}} h_N(x) \...
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5
votes
The Stable Set Conjecture
Seems to have been disproven, for all $k > 1$: A counterexample to Hildebrand's conjecture on stable sets, Redmond McNamara.
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6
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Number of divisors which are at most $n$
This is a supplement to Petr Kucheryavy's very nice proof. I claim that $\tau_n$ is surjective for $n>5\cdot 10^{13}$. (In fact one can extend this range to $n>60151$ by more concrete ...
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6
votes
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Number of divisors which are at most $n$
I am going to prove that $\tau_n$ is surjective for $n$ large enough.
Denote by $P_n$ the set of primes $\le n$.
$\tau_n(x)$ depends only on powers of primes from $P_n$ in the decomposition of $x$.
...
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0
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On successive minima and basis of a lattice
I was thinking about the same question and just ran into this thread. As I was unable to find any source for this answer, I think it might benefit others to provide a construction here.
Consider the 3-...
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6
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Can any Hurwitz zeta function be written as an Euler product?
To add a little to the excellent answer above - it is known that for $0<a<1, a \ne 1/2$ the Riemann Hurwitz function has a lot of zeroes in the strip $1 < \sigma < 1+a$, so, in particular, ...
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5
votes
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Closed-form for the number of partitions of $n$ avoiding the partition $(4,3,1)$
I suggest to show this inductively via showing that the difference $a(n+1)-a(n)$ always equals $b(n+1)-b(n)$.
As mentioned in the comments, the set $S(n)$ of $(4,3,1)$-avoiding partitions of $n$ is ...
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7
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Can any Hurwitz zeta function be written as an Euler product?
As Hurwitz proved, his zeta function has meromorphic continuation to $\mathbb{C}$. It has a simple pole at $s=1$ and no other pole. Euler products define Dirichlet series with leading term $1$, so the ...
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3
votes
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Existence of tamely ramified tower of extension over $\mathbb{Q}_p$
No, such a field does not exist, since the Galois group of $k_{tr}/k_{nr}$ embeds into the multiplicative group of the residue field of $k_{nr}$, which your hypothesis implies to be finite.
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4
votes
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A cubic equation, and integers of the form $a^2+32b^2$
Update 2. Now the proof should be more readable. I deleted some content because it is replaced by more elegant version.
The equation is unsolvable. The proof requires two theorems
Theorem 1. Let $p = ...
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7
votes
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How fast can elliptic curve rank grow in towers of number fields?
I think it is a good idea to compare the growth of the rank to the degree. I would say that we have excessive growth in an extension $F/K$ if $$\DeclareMathOperator{\rank}{rank}\rank E(F) - \rank E(K) ...
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0
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A cubic equation, and integers of the form $a^2+32b^2$
The equation $(1)$ has not integer solution. An elementary proof.
Suppose $(x,y,z)=(a,b,c)$ is a solution and consider as unknowns $(X,Y)=(4,2)$ so we can form the system
$$\begin{cases}b^2X+(a+c^2)Y=-...
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2
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Cusp forms of weight 2 and level $\Gamma_0(p)$ where $p < 11$
The dimension of $S_2(\Gamma_0(N))$ is the genus of the Riemann surface $\Gamma_0(N)\backslash\mathfrak{H}^*$. See Theorem 2.24 in Shimura: Introduction to the arithmetic theory of automorphic forms. ...
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7
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Generating function over primes in an arithmetic progression
It is not a modular form. To see this, let $f(q)$ be the initial modular form, and let us assume (without loss of generality) that $a(1)=1$. Let $k$ denote the weight of $f(q)$. The Rankin-Selberg $L$-...
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3
votes
Faltings height in short exact sequences
I am sorry for answering this question nine years later, but I would like to add that a recent theorem of Rémond shows that the Faltings height is sub-additive in short exact sequences. In other ...
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2
votes
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Factorization trees and (continued) fractions?
Note that for any $m$ which isn't a leaf, due to Bertrand's postulate, there must be a prime in the range $\frac{n}{2m} < p \leq \frac{n}m$, and then $mp$ must be a leaf. Therefore, we have $p(T_{n,...
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When adding several $\sqrt[n]{p}$ to the rational numbers, what is the degree of field extension?
Let $K = \mathbf Q(\sqrt[n_1]{p_1},\ldots,\sqrt[n_k]{p_k})$ where $p_1, \ldots, p_k$ are distinct primes and $n_1,\ldots,n_k \geq 1$. Then $[K:\mathbf Q] \leq n_1\ldots n_k$. It is intuitively ...
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7
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When adding several $\sqrt[n]{p}$ to the rational numbers, what is the degree of field extension?
Yes, this statement, as well as obvious extensions for more primes, are true. For $m,n$ relatively prime, this follows as an easy application of the Tower Law. For $n=m=2$, more generally for an ...
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1
vote
Letting $S(m)$ be the digit sum of $m$, then $\lim_{n\to\infty}S(3^n)=\infty$?
A partial (relatively elementary) answer to question 2 is provided by the results proven by A. Schinzel. A good reference for this is the book W. Sierpiński, 250 problems in elementary number theory, ...
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2
votes
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Unramified composition for every extension
The Abhyankar construction can be generalized naturally as follows: For each $p\in S$, take the (finite) set of Galois extensions $F_p/K_p$ occurring as completions of the Galois closure of a degree-$...
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0
votes
What are the properties of umbra with moments $\{1,1/2,1/3,1/4,1/5,...\}$?
This is not a full answer, but I want to point out an interesting observation. The umbra in question can be represented as the set of integrable functions on interval $(0,1)$ with usual operations ...
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12
votes
Smallest prime factor of numbers
The number of integers $1\leq n\leq x$ with smallest prime factor exceeding $y$ is usually denoted by $\Phi(x,y)$. It has been studied thoroughly. See, for example, Chapter III. 6 (Integers free of ...
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1
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Density of monogenic number fields?
It might we worthwhile to talk about this density instead
$$\frac{\ln(|\{K: K \text{ is monogenic} \Delta(K)< x\}|)}{\ln(|\{K: \Delta(K) < x\}|)}$$ which is expected to be $$\frac{1}{2}+\frac{1}{...
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16
votes
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binomial coefficients are integers because numerator and denominator form pairs?
The kind of pairing sought does not always exist. Take, for example,
$$\binom{8}{4}=\frac{8\cdot7\cdot6\cdot5}{4\cdot3\cdot2\cdot1}.$$
The pair of $4$ must be $8$, the pair of $3$ must be $6$, and ...
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5
votes
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Proof of remark in algebraic number theory
$\def\aroof{\overline{\hspace{-1pt}\smash[t]{|}\hspace{1pt}\alpha\hspace{1pt}\smash[t]{|}\hspace{-1pt}}}$One has the $h\times h$ linear system
$$\alpha^{(i)} = a_1\beta_1^{(i)} + \ldots + a_h\beta_h^{(...
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5
votes
Proof of remark in algebraic number theory
Consider the distinct embeddings $\sigma_\ell\colon K \to \mathbb{C}$ of the number field $K$ in the complex numbers (up to equality on $K$, not just up to their image): by standard facts in Galois ...
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6
votes
Reference book on Riemann zeta function and random matrices
There is a substantial literature; for an introduction, you could start with the chapter by Keating and Snaith in the Handbook of Random Matrix Theory. For a more specialized overview, take a look at ...
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5
votes
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A non-$p$-adic proof of a congruence of Bernoulli numbers
$\newcommand{floor}[1]{\left\lfloor #1 \right\rfloor}$
I doubt that this answer is still useful to you, since this question is one year old. Anyway, I'll leave the answer here, and maybe it will help ...
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5
votes
Closed formula for number of ones in a proper factor tree
A simple generating function, though not a closed formula, for $\gamma(N)$ is given by $\sum_{N\geq 1}\frac{\gamma(N)}{N^s} = \frac{1}{2-\zeta(s)}$, where $\zeta(s)$ is the Riemann zeta function. See ...
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1
vote
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Quadratic unramified extension of a p-adic field
You only need (1). A Hensel's-lemma type approximation, using surjectivity of $\operatorname{tr}_{k_F/k_E}$ (where $k$ denotes residue fields), shows that $N_{E/F} : 1 + \mathfrak p_E \to 1 + \...
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2
votes
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A Mordell equation $y^3=x^2+20$
This equation corresponds to the (Mordell) elliptic curve with LMFDB label 10800.i1. The LMFDB confirms that $(14, 6)$ is the unique integral point with positive entries.
(The way that the LMFDB ...
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6
votes
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Closed formula for number of ones in a proper factor tree
This seems to be Sloane's A002033, namely the number of perfect partitions of $n-1$. Since the OEIS doesn't give any closed formula, there probably isn't one, but it's probably worth checking the ...
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5
votes
Closed formula for number of ones in a proper factor tree
Take the formal product $g(x_1,x_2,\ldots)=\prod_{i\ge 1} (1-x_i)$ and
define
$$f(x_1,x_2,\ldots) = \frac{g(x_1,x_2,\ldots)}{2g(x_1,x_2,\ldots)-1}.$$
Then
$\gamma(\prod_i p_i^{\alpha_i})$ is the ...
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3
votes
Integer solutions to $x^2 + x + 1 = y^z$?
You should definitely take a look at this previous answer of mine:
https://mathoverflow.net/a/251637/1593
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3
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A cubic equation, and integers of the form $a^2+32b^2$
details, details. From $x$ odd and
$$
x^4-32x-16 = (x^2 + 4)^2 - 2(2x+4)^2.
$$
we see that $x^4-32x-16$ is not divisible by any prime $q \equiv 3,5 \pmod 8.$ That is, $x^2 + 4$ is also odd. ...
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3
votes
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Integer solutions to $x^2 + x + 1 = y^z$?
The only solution is $x=18$, $y=7$, $z=3$. See my response at What is prime power of this equation of p?
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