**1**

vote

**0**answers

37 views

### Do the zeros of the sum/difference of these series all reside on the line $\Re(s)=\frac12$?

The following series seems convergent for all $s\in \mathbb{C}$:
$$\displaystyle f(s):=\sum_{n=1}^\infty \frac{(-1)^n}{(n+s)^{n+s}}$$
The function itself does not appear to have any real or complex ...

**6**

votes

**0**answers

76 views

### Are there irreducible polynomials with all zeros on two concentric circles?

This is somewhat similar to this recent question, but extending in a different direction.
Let $f(x)$ be an irreducible polynomial of degree $n$ with integer coefficients. Call such $f$ a bicycle ...

**4**

votes

**0**answers

57 views

### Young-Fibonacci lattice and purely periodic continued fractions

The Fibonacci lattice $\mathcal{F}$ is the poset of all finite words consisting of 1's and 2's where a word $v$ covers a word $u$ if $v$ is obtained from $v$ by either (a) inserting a 1 in $u$ prior ...

**12**

votes

**1**answer

342 views

### Erdös-Turán via Hardy-Littlewood circle method?

For any set $B\subseteq \mathbb{N}$ one can associate the formal series
$$f_B(z) = \sum_{b\in B}z^b$$
and obtain
$$f_B(z)^k = \sum_{n\geqslant 0} r_{B,k}(n)z^n,$$
where $r_{B,k}(n) = ...

**4**

votes

**2**answers

249 views

### Inverse Galois problem for simple Lie type groups

Progress towards the Inverse Galois problem over $\mathbb{Q}$ is very well documented for sporadic groups ($M_{23}$ is the only case open) and for $PSL_n(q)$ (a lot of cases known, but wide open in ...

**4**

votes

**1**answer

298 views

### Intuition behind salient numbers in number of h-cobordism classes of smooth homotopy n-spheres

The Wikipedia article on Exotic Sphere displays the sequence of numbers (see also OEIS A001676 and the Milnor link therein) for the order of the classse as
$$1, \;1, \;1,\; 1,\; 1, \;1, \;28,\; 2,\; ...

**4**

votes

**1**answer

235 views

### Listing all solutions to $n = x^2 + y^2 + z^2 $ with integers

I would like to list all ways of writing $n$ as the sum of 3 squares. This is slightly different from finding just one:
Is there an algorithm for writing a number as a sum of three squares?
...

**2**

votes

**1**answer

288 views

### Does Borel's proof for existence of normal numbers make an essential use of axiom of choice?

A normal number is a real number whose infinite sequence of digits in every base $b$ is distributed uniformly in the sense that each of the $b$ digit values has the same natural density $\frac{1}{b}$, ...

**8**

votes

**1**answer

161 views

### On Bailey–Borwein–Plouffe formula for irrational numbers

A BBP-type formula for an irrational number $\alpha$ in the integer base $b\geq 2$ is a formula in the form $\alpha=\Sigma_{k=0}^{\infty}\frac{1}{b^k}\frac{p(k)}{q(k)}$ ($p, q$ are polynomials in ...

**9**

votes

**1**answer

172 views

### The density of integers represented by a binary form

Suppose that $F(x,y)$ is a binary form of degree $d \geq 3$ with integral coefficients, and non-zero discriminant. It is known (from a paper due to Erdős and Mahler from 1938) that the density of ...

**1**

vote

**0**answers

99 views

### A Question on Chinese Remainder Theorem [on hold]

Let $p_1,p_2,\ldots,p_n$ be odd primes.
Let $y$ be the unique solution of the Chinese Remainder Problem ( $0 \le y < m$, $ m = p_1\cdot p_2 \cdots p_n$)
$ y = \frac{p_1-1}{2} \text{ mod } (p_1) ...

**8**

votes

**2**answers

205 views

### Bounding exponential sum with square roots

It is well known that for each $m\in\mathbb{N}$
$$\lim_{N\to\infty}\frac1N\sum_{n=1}^Ne^{2\pi i\sqrt{nm}}=0$$
My question is whether there is some uniformity in the variable $m$.
More precisely, is it ...

**5**

votes

**3**answers

401 views

### Quantitative and elementary proofs of the Prime Number Theorem

I would like to know two things: one, whether the best quantative bounds in the Prime Number Theorem are still basically those given by the Vinogradov-Korobov zero-free region? and two, whether there ...

**4**

votes

**0**answers

154 views

### Any ways to Simplify Daboussi's Argument for Prime Number Theorem?

One strategy to prove the Prime number theorem involves removing some factors:
$$ \limsup_{x \to \infty} \underbrace{\frac{1}{x}\sum_{n \leq x} \mu(x)}_{\color{red}{A}}\leq
\prod_{p \leq y} \left( 1 ...

**4**

votes

**1**answer

102 views

### Average of Short Character Sum over All Dirichlet Characters Mod n

Cross-posted from M.SE.
Given $a,n$ coprime positive integers, let $L = \{(x,y)\in \mathbb{Z}^2, ax=y(n)\}$ be the lattice of all points satisfying $ax=y\pmod{n}$.
I want to find an ...

**7**

votes

**1**answer

279 views

### Type of a modular form

Let $f$ be an arbitrary weight 1 newform. We know by Serre-Deligne that there is an odd 2-dimensional irreducible Artin representation $\rho$ such that $L_f(s)=L(\rho,s)$.
I was wondering how much ...

**7**

votes

**0**answers

286 views

### rings of modular functions on the upper half plane

Let $\Gamma_1\le SL_2(\mathbb{Z})$ be a noncongruence subgroup of finite index.
Let $\Gamma_2\le SL_2(\mathbb{Z})$ be another subgroup of finite index.
Let $M_0(\Gamma_i)$ denote the ring of modular ...

**7**

votes

**1**answer

206 views

### A characteristic 2 polynomial recursion

Let $c(n)$ in $\mathbb{Z}/2\mathbb{Z}[x]$ be defined by the recursion $$c(n+4)=c(n+3)+(x^4+x^3+x^2+x)c(n)+x^n\cdot(x+x^2),$$ and the initial conditions
$$c(0)=0,\quad c(1)=1,\quad c(2)=x,\quad ...

**10**

votes

**0**answers

263 views

### Progress in Guy's “Unsolved problems in Number Theory”? [on hold]

I often peruse through Guy's book whenever I'm not being boggled down by my research. It crossed my mind today if any of these "unsolved problems" have become indeed solved. I thought about doing a ...

**2**

votes

**0**answers

76 views

### Enumerating factors in intervals

Given $1<a<N-N^{1/\alpha}$ where $\alpha\geq2$, denote the number of distinct factors of $N$ in $[a,a+N^{1/\alpha}]$ as $\sigma_{0,a}(N,\alpha)$ denote ...

**2**

votes

**1**answer

430 views

### Green-Tao theorem for 1-central numbers

This question came to my mind this afternoon while trying to figure out a possible way to tackle de Polignac's conjecture, which states that every even positive integer can be written as the ...

**6**

votes

**0**answers

99 views

### De Bruijn sequence inside De Bruijn sequence

A binary De Bruijn sequence of index $n$ is a circular sequence $S=a_1a_2\ldots a_{2^n}$, with $a_i∈\{0,1\}$, and such that each of the $2^n$ binary $n$-tuples occurs exactly once in $S$.
What is ...

**7**

votes

**0**answers

92 views

### Order of zeros for sparse polynomials mod $p$

It is a fairly well known fact that sparse polynomials $f(x)$ cannot have large order zeros other than at $x=0$. If $f(x)=a_1x^{r_1}+\cdots+a_kx^{r_k}$ then at $c
\neq 0$, $f$ has a zero of order at ...

**13**

votes

**1**answer

272 views

### Number of height-limited rational points on a circle

Consider origin-centered circles $C(r)$ of radius $r \le 1$.
I am seeking to learn how many rational points might lie on $C(r)$,
where each rational point coordinate has height $\le h$.
For example, ...

**15**

votes

**2**answers

3k views

### Does the equation $241+2^{2s+1}=m^2$ have a solution?

Let $p$ be a prime congruent to $1$ mod. 8.
If $p= 17$ one has : $p+ 8 = 5 ^2$.
If $p= 41$ one has : $p+ 8 = 7 ^2$.
If $p= 73$ one has : $p+ 8 = 9 ^2$.
If $p= 89$ one has : $p+ 32 = 11 ^2$.
If ...

**1**

vote

**0**answers

92 views

### Primitive triples in a region [duplicate]

Are there at least $cn$ Pythagorean triples and at least $dn$ Primitive Pythagorean triples $(A,B,C)$ with $$2^{\frac n2}<A<2^{\frac n2+1}<2^n<B<C<2^{n+1}$$
with some fixed ...

**6**

votes

**1**answer

93 views

### A case of nested central limits

Consider the random variable $S=(s_0, \dots ,s_{N-1})$, a sequence of signs uniformly distributed on the hypercube $\{-1,1\}^N$. We are interested in $N$ large and prime. The Fourier transform ...

**0**

votes

**2**answers

164 views

### Sharply Estimating Pythagorean Triples [closed]

Given $m,n\in\Bbb N$ with $m<n$, how many pythagorean triples $p^2+r^2=q^2$ satisfy $$m\leq p<r\leq n?$$
Is there a way to give a sharp estimate?

**3**

votes

**0**answers

59 views

### Shifted convolution problem for Coefficients of automorphic forms

The shifted convolution problem for coefficients of modular forms is well studied and many estimates were established for the shifted convolution sums of Hecke eigenvalues. So, one may ask about the ...

**6**

votes

**1**answer

149 views

### Behaviour of $\zeta(1-it)/\zeta(1+it)$?

I am trying to understand the behaviour of
$$\int^\infty_{-\infty}\frac{\xi(1-it)}{\xi(1+it)}h(t)\frac{dt}{t}$$
where $h$ is a Schwartz function on $\mathbb R$, and $\xi(s)$ the completed Riemann zeta ...

**5**

votes

**1**answer

190 views

### Logarithmic weights on number theoretic sums

Suppose we are interested in the sum
$\sum _{n\leq x}a_n.$
The study of the sum
$\sum _{n\leq x}a_n\log (x/n)$
may be easier.
What can one say about the first sum from knowing the behaviour of ...

**22**

votes

**2**answers

2k views

### Mochizuki's “phenomena in number theory” outside the scope of Langlands

(Crossposted from math.stackexchange by suggestion)
On page 12 of Shinichi Mochizuki's "On the Verification of Inter-universal Teichmuller Theory: A Progress Repor", he writes
"The ...

**2**

votes

**1**answer

81 views

### Rate of convergence of ergodic averages related to irrational rotation

Let $\alpha$ be an irrational number, consider the basic dynamical system $T^{n}(0) = \{n \alpha\}$ where $\{.\}$ denotes the fractional part.
Let $a < b$ be two numbers in $[0, 1]$. Then by ...

**6**

votes

**1**answer

175 views

### Order of reduction of infinite order rational point on an Elliptic Curve

Let $E/$ℚ be an elliptic curve and $P$ ∈ $E($ℚ$)$ a rational point of infinite order. Does the reduction of $P$ mod $p$ generate a maximal cyclic subgroup of $E(\mathbb{F}$$p$$)$ for ...

**3**

votes

**2**answers

230 views

+50

### Relative-totient function (2nd attempt)

Let $\Lambda(x,y)$ be the count of totatives of $x$ that are less than or equal to $y$.
I am asking for the following result to be verified, (particularly the final proposal), I have found no ...

**21**

votes

**3**answers

761 views

### On determinants formed by binomial coefficients

Let $q$ be a number. Let us consider the $q^2-1$-th line of the Pascal triangle (i.e. numbers ${{q^2-1} \choose i}$, $i=0,1,...q^2-1$). We have $q^2$ numbers.
Let us form naively a $q \times q$ ...

**-2**

votes

**0**answers

81 views

### Does Selberg's central limit theorem have any implication on the quantity $\alpha_{n}$? [closed]

In my previous question About Goldbach's conjecture, I define a quantity denoted by $\alpha_{n}$ which measures the quality of the approximation of the quantity denoted by $N_{2}(n)$ by ...

**5**

votes

**0**answers

108 views

### On factorization algorithms for $\mathcal{O}[x]$

We know that $\mathsf{LLL}$ algorithm provides factorization procedure that runs in poly time for polynomials in $\Bbb Z[x]$ that are primitive.
What other rings $\mathcal{O}$ can we use instead of ...

**4**

votes

**1**answer

99 views

### Zeta zeros standard normal distribution about $\vartheta (\gamma_n)$

Asked at MSE here without response.
I realise that this resembles Odlyzko's famous nearest neighbours plot, and was wondering whether this is simply a manifestation of the same phenomenon.
That ...

**6**

votes

**3**answers

644 views

### Do there exist elliptic curves over schemes which have all primes as residue characteristics?

It's well known that there are no elliptic curves over Spec $\mathbb{Z}$, but it's unclear (to me at least) if the proof generalizes.
My question is: If $S$ is a connected scheme such that has every ...

**6**

votes

**1**answer

282 views

### Is the axiom schema of replacement used in algebraic number theory (or more generally outside logic)

Here's a precise question. Does Wiles' proof of FLT run just fine in the set theory that logicians would perhaps call "Zermelo + choice" -- i.e. drop the axiom schema of replacement but assume the ...

**2**

votes

**1**answer

258 views

### Greatest number of coprime numbers between two numbers

We know that from prime number theorem that the number of primes below $n$ is approximately $$\frac{n}{\log_en}.$$
$\star$ Given $n,m$, what is the largest list of pairwise coprime numbers that one ...

**11**

votes

**1**answer

258 views

### Connection between Bernoulli numbers and Riemann-Siegel theta function?

I have come across a strange approximation for the Riemann-Siegel theta function involving the Bernoulli numbers - namely that
$$\frac{1}{2} \log \left| B_{2 n}\right|\approx \vartheta (2n)\ ,\quad n ...

**1**

vote

**1**answer

70 views

### Given Sequence of Numbers find number of combinations [closed]

I have the sequence of numbers $1,2,4,8,16,\ldots$. This is an infinite sequence. So my problem is that if I have any positive integer value, $x$, what are the possible ways that I can write $x$ as ...

**4**

votes

**0**answers

45 views

### Question on some coverings of the euclidean space

Let $L$ be a maximal integral lattice in the euclidean $(\mathbf R^{8m},q)$ (thus the associated bilinear form $b(u,v)=q(u+v)-q(u)-q(v)$, once restricted to $L$, takes values in $2\mathbf Z$ and has ...

**0**

votes

**1**answer

97 views

### Computability of prime difference function

Consider the following function $f: \omega\to \{0,1\}$:
Set $f(n) = 1$ if for all $k\in \omega$ there are prime numbers $p,q > k$ such that $n = p-q$; and set $f(n) = 0$ otherwise.
(Trivially, if ...

**4**

votes

**2**answers

295 views

### Approximating integers with prime quotients [closed]

Is this statement true for all positive integers $n\in\mathbb{N}$?
For all $\varepsilon >0$ there are prime numbers $p,q$ such that $|\frac{p}{q} - n| < \varepsilon$.

**5**

votes

**1**answer

154 views

### Lattice points on the boundary of an ellipse

How many points of the integer lattice ${\mathbb Z}^2$ can an axis-parallel ellipse of radius $r$ contain on its boundary? (that is, we consider ${\mathbb Z}^2$ as lying in ${\mathbb R}^2$). ...

**3**

votes

**2**answers

201 views

### Minimal expression of 0 as a sum of kth powers in a finite field

Let $l=\min\{s\in \mathbb{N}|0\in s\cdot (\mathbb{F}_{p^n}^\times)^k\}$. Is any information known about this number already as a function of $k$? Any reference would be greatly appreciated!

**8**

votes

**1**answer

176 views

### An effective version of Kronecker's approximation theorem and its variations

Let $\theta_1,\dots,\theta_n$ be algebraic numbers of degree $\leq D$ and Weil height $\leq H$ such that $1,\theta_1,\dots,\theta_n$ are $\mathbb Q$-linearly independent. An effective version of ...