**-1**

votes

**0**answers

15 views

### How to find the continuous version of this $f$?

I was doing some maths and required a function which mimics the following function:
$$ f(k,c) = \mid \sin(k/2) \sin(k/3) ... \sin(k/c) \mid $$
I know that $ f(k,c) $ is a discrete function. So how ...

**28**

votes

**0**answers

433 views

### Mod sequences that seem to become constant; and the number 316

Define a "mod sequence" of nonnegative integers
based on one start parameter $s$, its first term,
as follows.
$A(s)=(a_1,a_2,\ldots,a_n,\ldots)$
with $a_1 = s$
and
$$ a_n = \left(\sum_{k=1}^{n-1} a_k ...

**0**

votes

**1**answer

88 views

### More on Vojta's exceptional set for a more general abc conjecture

Likely a mistake, but got very large exceptional set in Vojta's
more general abc conjecture.
In A more general abc conjecture, p. 7 Paul Vojta conjectures:
If $x_0,\ldots x_{n-1}$ are nonzero ...

**2**

votes

**5**answers

314 views

### Mean of a vector

Be a set of numbers $v=(a_1, a_2, \ldots, a_n)$
I want to form the following average vector $\mu = (\frac{\sum a_i}{n}, \frac{\sum a_i}{n}, \ldots, \frac{\sum a_i}{n})$.
If I do it iteratively step ...

**6**

votes

**0**answers

131 views

### Is the set of numbers $\{ [n^{3/2}] \mid n\text{ an integer}\}$ a basis of order 3?

A set of integers is called a basis of order h if every nonnegative integer can be written as the sum of h not necessarily distinct elements of the set.(Nathanson, Additive Number Theory, page 7)
...

**0**

votes

**0**answers

112 views

### E- and A-algorithms for finite arithmetic prime progressions and other sets

(EDIT from scratch).
Let $\ \mathbf a := (a_1\ \ldots\ a_n)\ $ be an increasing non-constant arithmetic progression of odd positive numbers. The goal here is to resolve efficiently one of the two ...

**0**

votes

**0**answers

65 views

### Abscissa of absolute convergence of the product of two Dirichlet series

I first asked the following question on Mathematics StackExchange (a few weeks ago), since the content of MathOverflow is mostly above my academic level. I didn't want to bother people on this forum ...

**5**

votes

**1**answer

450 views

### Any way to prove Prime Number Theorem using Hyperbolic Geometry? [on hold]

The prime number theorem says that the density of prime numbers is inverse as the number of digits of $n$:
$$\displaystyle \frac{\{1 \leq k \leq n : \text{ prime } \}}{n} \approx \frac{1}{\log n}$$
...

**19**

votes

**3**answers

665 views

### Number of elements of “$\mathrm{SL}_n(\mathbb{F}_p^\times)$” mod $p$

How many elements of $\mathrm{SL}_n(\mathbb{F}_p)$ have all nonzero entries? Just the answer mod $p$ would be fine as well. This seems like it should be easy/in the literature but I couldn't find it.

**-1**

votes

**1**answer

96 views

### Algorithm Involving Quadratic Diophantine Equation

Let $a,b,c,d\in\Bbb Z$.
$abcd\neq 0$ and $a>0$.
What is the best standard low complexity procedure to find the set of solutions for $x,y,t\in\Bbb Z$ such that:
$$-axy+bx+cy+d=t^2$$
where ...

**-4**

votes

**0**answers

71 views

### Can the divisibility-by-7 test be extended to preserve the remainder? [on hold]

The divisibility-by-7 test (in base 10) performs iterations whereby the last digit is doubled and subtracted from the number formed by all other digits. If the original number is divisible by 7, then ...

**2**

votes

**0**answers

101 views

### Over which fields is a $G$-module reducible?

I have asked this question at math.stackexchange, but have not received an answer so far. Also, I'm not entirely sure that this is a suitable question for mathoverflow... See this link for the ...

**21**

votes

**7**answers

7k views

### Collecting proofs that finite multiplicative subgroups of fields are cyclic.

I teach elementary number theory and discrete mathematics to students who come with no abstract algebra. I have found proving the key theorem that finite multiplicative subgroups of fields are cyclic ...

**1**

vote

**1**answer

135 views

### Prime divisors of $p^n+1$

Let $p$ be a rational prime and $n$ be a positive integer.
It can be easily deduced from Zsigmondy's theorem that $p^n+1$ has a prime divisor greater than $2n$ except when $(p,n)=(2,3)$ or ...

**3**

votes

**1**answer

228 views

### Improving the bound $q < n\sqrt{3}$ for an odd perfect number $N = {q^k}{n^2}$ given in Eulerian form

(Note: This was cross-posted from MSE.)
Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q, n) = 1$).
Therefore, ...

**4**

votes

**1**answer

174 views

### Constructing a family of convergents from continued fractions formed by a set of prime partial quotients

For a given real number $x$, the continued fraction representation $x = [a_0; a_1, a_2, \cdots]$ where $(a_n)_{n \geq 0}$is defined by setting $x = \alpha_0$, then $a_i = \lfloor \alpha_i \rfloor$, ...

**0**

votes

**0**answers

84 views

### $\zeta$ function for ambiguous class group

Let $K$ be a Galoisian number field with ring of integers $O_K$. I consider $\zeta_{A,K}(s)$ the function defined by
$$\zeta_{A,K}(s)=\prod_{\substack{p\in\mathbb N\\p\text{ ...

**5**

votes

**1**answer

291 views

### Ruth-Aaron triples, etc

A Ruth-Aaron pair is two numbers $(n,n+1)$ such that
their sum of prime factors is equal, counting repeated prime factors.
(The name refers to Hank Aaron's 715 homeruns surpassing Babe Ruth's 714!)
So
...

**2**

votes

**1**answer

141 views

### Chebyshev polynomials factoring uniformly modulo all primes

Consider the Chebyshev polynomial of the first kind $T_n(x)$ and its factorization in $\mathbb F_p$ for a given prime $p$. Most often, this factorization is not uniform (meaning that the irreducible ...

**10**

votes

**1**answer

501 views

### More elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?

(Note: See also the $a^4+b^4+c^4 = 1$ version in this old MSE post.)
The equation discussed in a paper by Jacobi and Madden,
$$a^4+b^4+c^4+d^4 = (a+b+c+d)^4 = z^4\tag1$$
or equivalently,
$$(p-2q + ...

**7**

votes

**1**answer

344 views

### Tate's thesis for Artin L-functions

As far as I know, Tate's thesis has been successfully applied in two fronts:
Hecke L-functions, by Tate and Iwasawa (and Teichmüller, Witt, Schmid)
Automorphic L-functions, by Jacquet, Shalika, ...

**2**

votes

**3**answers

246 views

### Growth of $r_{2}(n)$

Let $n$ be a positive integer. From Jacobi's two-square theorem we know that the number $r_{2}(n)$ of representations of $n$ as a sum of two squares is given by
$$
r_{2}(n)=4(d_{1}(n)-d_{3}(n)),
$$
...

**0**

votes

**0**answers

78 views

### Relation between cyclotomic character and fundamental character of level

The question I have is the following: is it true that the $p+1$ exponentiation of the fundamental character of level $2$ gives us the reduction (mod $p$) of the cyclotomic character?
For a review of ...

**6**

votes

**1**answer

217 views

### what is the intersection of all congruence subgroups of the profinite completion of SL(2,Z)?

Let $\widehat{SL(2,\mathbb{Z})}$ be the profinite completion of $SL(2,\mathbb{Z})$. Let $\Gamma(N)$ denote the typical principal congruence subgroup of $SL(2,\mathbb{Z})$ (ie, all matrices congruent ...

**8**

votes

**1**answer

545 views

### What is a totient?

In addition to the Euler totient function, there are a great many generalizations and related functions which go by the "totient", usually with some name: Jordan, Lehmer*, Schemmel, Nagell, Alder, ...

**1**

vote

**1**answer

143 views

### Bombieri-Vinogradov in short intervals

In 1985 Perelli, Pintz & Salerno proved a short-interval form of the Bombieri-Vinogradov theorem with $\theta \in (7/12, 1]$. Have there been any improvements on this, in particular with the ...

**2**

votes

**3**answers

727 views

### Where do Set Theory and Number Theory meet together?

As all know, by absoluteness theorems in Set Theory, most of theorems in number theory are $ZFC$-provable if and only if they are consistent with $ZFC$, it's because of absoluteness of essence of ...

**7**

votes

**0**answers

358 views

### Application of the Riemann hypothesis and the ABC conjecture to independence results

In Old Home Page of Andreas Weiermann Andreas Weiermann has stated the following:
Quite recently I submitted a preprint about an application of the Riemann hypothesis and the ABC conjecture to ...

**0**

votes

**1**answer

702 views

### A letter from J. P. Serre

Which is the letter where J. P. Serre present "Analogues Kählériens de certaines conjectures de Weil" to Weil?

**10**

votes

**0**answers

270 views

### Applications of $p$-adic Hodge theory

I am trying to learn $p$-adic Hodge theory. I found some materials explaining main theorems (or aspects) of the theory. However, I could not find references which explaining how to use the theory. ...

**2**

votes

**1**answer

588 views

### Efficient algorithms to determine the roots of: $p(x) = r^x $ in the finite field $GF(q)$, where $r$ a primitive root of the field

I need to make sure that no efficient (i.e., polynomial time) algorithm exists for the following problem:
Exponentiating Polynomial Root Problem (EPRP)
Let $p(x)$ be a polynomial with $\deg(p) \geq ...

**3**

votes

**2**answers

200 views

### Sets of squares representing all squares up to $n^2$

Let $S_n=\{1,2,\ldots,n\}$ be natural numbers up to $n$.
Say that a subset $S \subseteq S_n$
square-represents $S_n^2$ if every
square $1^2,2^2,\ldots,n^2$ can be represented by adding or subtracting
...

**1**

vote

**1**answer

117 views

### Cardinality of the prime divisor set of a k-power sum

Let $a_{1},\dots,a_{n}$ be positive natural numbers ($n>2$) such that $a_{i}\neq a_{j}$ if $i\neq j$. I want to prove that
$$ \left\lvert \left\{ p \text{ prime} \; : \; p \mid \sum_{i=1}^n ...

**0**

votes

**0**answers

53 views

### The hypertriangular function of $n$

I'm looking for papers or recent results on the hypertriangular function of $n$:
$$H_t(n)= \displaystyle\sum\limits_{k=1}^{n} k^k$$
This is A001923 in the OEIS.
I don't have much experience with ...

**3**

votes

**1**answer

136 views

### Pathological behavior of Lie algebra under a map of abelian schemes

I am trying to understand Example 7.5/9 from the book "Neron models". There one has a discrete valuation ring $R'$ that is the localization of $\mathbb{Z}[\zeta_p]$ at $p$, so that the absolute ...

**2**

votes

**1**answer

213 views

### Primes $p=x^2+27y^2$ and Ramanujan's $x_1^{1/3} + x_2^{1/3} + x_3^{1/3}$

I was trying to generalize,
...

**0**

votes

**0**answers

65 views

### Arithmetic functions associated with Hurwitz Zeta function raised to arbitrary complex powers, $\zeta(s,q)^z$ for $q \in \mathbb{N}$?

If $\zeta(s)$ is the Riemann Zeta function, then $\zeta(n)^z$, with $z \in \mathbb{C}$, $\Re(s)>1$, can be represented as
$$\zeta(s)^z=\sum_{n=1}^\infty \frac{d_z(n)}{n^{-s}}$$
where $d_z(n)$ ...

**-2**

votes

**1**answer

137 views

### Prove that $\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2})$ [closed]

Let $a,b\in\mathbb{Z}$, and $f\in C^2([a,b])$ such that $|f''(t)|\asymp \lambda$ for $a\le t\le b$. Prove that
$$\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2}).$$
...

**6**

votes

**1**answer

250 views

### abelian $\ell$-adic representations in $\widehat{SL(2,Z)}$

In Grothendieck's Esquisse he claims that the action of
$$\text{Gal}(\mathbb Q)\to\text{Out}(\pi_1(M_{1,1})=\text{Out}(\widehat{SL(2,Z)})$$
obtained from the homotopy exact sequence of the étale ...

**-4**

votes

**0**answers

45 views

### Height of a tripod [closed]

I am trying to determine the height of a tripod when the length on the tripod's legs (81") and the distance between the ends that touch the ground are 57" apart.
My thought process so far: When the ...

**1**

vote

**1**answer

232 views

### Number of solutions in a sum of squares Diophantine equation

Let $n$ be an integer. I'm interested in upper bounding the number of integer solutions of
\begin{equation*}
x_1^2+x_2^2+\ldots+x_p^2=y_1^2+y_2^2+\ldots+y_p^2,
\end{equation*}
where for all ...

**5**

votes

**1**answer

139 views

### When is $\vartheta(x)>x$? [Skewes number analog]

Let $\vartheta(x)=\sum_{p\le x}\log p$. What is known about the first time $\vartheta(x)>x?$
Bays & Hudson give good upper bounds (slightly improved by Chao & Plymen) on the first crossing ...

**5**

votes

**0**answers

510 views

### 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 ...

**5**

votes

**1**answer

161 views

### Which criteria for “uniformly splitting” polynomials?

Let $P(x)$ be an irreducible monic polynomial of degree $\ge4$ with integer coefficients. We all know that over a finite field $\mathbb F_p$, $P$ will often split, and I am interested in polynomials ...

**4**

votes

**1**answer

705 views

### Possibly an easy application of Tate-Nakayama duality and local CFT

Let $F$ be a non-archimedian local field and $F^{ur}$ the maximal unramified extension. Let $T$ be a torus defined over $F$ such that $T$ splits over a tamely ramified extension and is anisotropic ...

**4**

votes

**1**answer

426 views

### Modular forms and “too many symmetries”

How do we interpret Barry Mazur's quote of
Modular forms are functions on the complex plane that are inordinately symmetric. They satisfy so many internal symmetries that their mere existence ...

**0**

votes

**0**answers

110 views

### Shimura reciprocity law

I have spent quite some time understanding class fields generated by Kummer extensions and class fields generated by modular forms. Now, I am turning the notch of sophistication a bit to study class ...

**1**

vote

**1**answer

350 views

### Numerical evidence and argument against Littlewood conjecture

This is joint work with someone. We got numerical evidence and argument
against Littlewood conjecture, though mistakes are certainly possible.
Littlewood conjecture states that for any two real ...

**11**

votes

**4**answers

579 views

### Zeros of the derivative of Riemann's $\xi$-function

The Riemann xi function $\xi(s)$ is defined as
$$
\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s).
$$
It is an entire function whose zeros are precisely those of $\zeta(s)$.
Since $\xi$ is real ...

**17**

votes

**10**answers

3k views

### New proofs to major theorems leading to new insights and results?

I am wondering, historically, when has a new proof of an old theorem been particularly fruitful. A few examples I have in mind (all number theoretic) are:
First example is classical... which is ...