# Tagged Questions

**7**

votes

**1**answer

420 views

### Measure of a set of irrational numbers

Let $A$ be a set of all irrational numbers $\rho \in (0, 1)$ represented as a continued fraction $\rho=[a_{1}, a_{2},...,a_{n},...],$ such that $a_{n}\leq \text{const}\cdot n^{\epsilon}$ for some ...

**2**

votes

**2**answers

756 views

### Number of 1 in binary representation of n

Let $1(n)$ be the number of digits $1$ in binary representation of number $n$.
For example, $13=1101_2$ so $1(13)=3\\$
Is there explicit form of $\,\,\sum{1(i)x^i} $?
I checked OEIS and didn't find ...

**2**

votes

**1**answer

252 views

### Curves of high genus with many rational points

The seminal theorem of Faltings confirms Mordell's conjecture: that is, curves of genus at least 2 have at most finitely many rational points. The proof of Faltings' theorem is not effective, meaning ...

**-1**

votes

**0**answers

27 views

### A Question about consecutive integers? [migrated]

Assume you have list of consecutive integers of length n, and such that each terms have at least one integer square factor. Then a couple of questions might be asked and i don't know if these has been ...

**0**

votes

**1**answer

168 views

### Small generators of the ideal class group

If $K$ is a number field, a result from Bach tells us that the primes in $K$ of norm smaller that $12 (\log |\mathrm{Discriminant}(K)|)^2$ generate the ideal class group $\mathrm{Cl}_K$. Is there any ...

**5**

votes

**0**answers

182 views

### A sum over zeros of L-functions in the paper “Chebychev's Bias”

Let $\varepsilon>0$ be small and
\begin{align*}
\widetilde{F}_{\varepsilon}(\xi):=\frac{4}{\varepsilon}\sum_{0<\gamma\leq \varepsilon^{-2}}\frac{\sin(\gamma \xi)\sin \frac{\gamma ...

**3**

votes

**1**answer

153 views

### Is this bounded from below?

Let $u_1, u_2, u_3 \in \mathbb{Z}$ such that $u_1^2 + u_2^2 = u_3^2$.
Is $(u_3 + \frac{u_1 + u_2}{\sqrt{2}})^2$ bounded from below?
The irrationality of $\sqrt{2}$ certainly precludes zero, but can ...

**4**

votes

**1**answer

236 views

### Consecutive non-quadratic residues

Inspired by this recent question, I wondered if a similar result is true for quadratic non-residues, namely, if it is true that for every $k \in \mathbb{N}$ there exists a prime $p$ such that exists ...

**2**

votes

**0**answers

59 views

### Bounds for the Tamagawa number of the Jacobian of a hyperelliptic curve

Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$ and let $C$ be a hyperelliptic curve of genus $g$ defined over $K$ with Jacobian $J$.
Suppose that $C$ is given by ...

**6**

votes

**2**answers

637 views

### Ramanujan's tau function

Why was Ramanujan interested in the his tau function before the advent of modular forms? The machinery of modular forms used by Mordel to solve the multiplicative property seems out of context until I ...

**3**

votes

**2**answers

171 views

### equivalence of quadratic forms over finitely generated fields

Over number fields, two quadratic forms are equivalent iff they have the same dimension, signature, discriminant and Hasse invariant.
How is the situation like over finitely generated fields?

**8**

votes

**0**answers

132 views

### Characterization of Volumes of Lattice Cubes

Here is a problem that came up in a conversation with a professor after I made a false assumption about the geometry of $\mathbb{Z}^n$. I do not know if he knew the answer (and told me none of it) and ...

**0**

votes

**0**answers

143 views

### Average order and upper bound of $r_{0}(n)$

Assume Goldbach's conjecture. Then for every integer $n>1$ there exists a non-negative integer $r$ such that $n-r$ and $n+r$ are both primes. For a given $n>1$, the smallest such $r$ will be ...

**3**

votes

**0**answers

141 views

### Uniqueness conjecture for integer triples

The triple $(a,b,c)$ with $a \ge b \ge 1$ is an HT (Hikorski Triple) if
$c = (a b+1)/(a+b)$ is an integer. The triples $(n,1,1)$ are the trivial HTs.
Conjecture: if $(a,b,c)$ and $(p,q,r)$ are both ...

**8**

votes

**0**answers

149 views

### For a given even integer $k >14$ is there always a prime $p$ such that $k \leq p-3$ and $p|B_k$?

Let $k$ be a sufficiently large positive even integer. (I think $k > 14$ should do.) Can one always find a prime $p$ such that $p$ divides the numerator of the $k$-th Bernoulli number $B_k$ and $k ...

**9**

votes

**3**answers

693 views

### Could a perfect squared square be split into two perfect squared squares?

This is a geometric puzzle though it might conceivably
also define a special class of Pythagorean triples.
A perfect squared square PSS is a square (as a plane figure)
partitioned into smaller ...

**4**

votes

**1**answer

321 views

### Balog-Szemeredi-Gowers with dilates of sets

All sets are assumed to be finite subsets of the integers.
The additive energy of two sets $E(A,B)$ is defined as the number of solutions to $a+b=a'+b'$ with $a,a'\in A$ and $b,b'\in B$. The ...

**10**

votes

**1**answer

324 views

### Local Langlands for $GL(2,\mathbf{C})$ and reducible principal series

My naive picture of the local Langlands correspondence for $GL(2,\mathbf{C})$ is this. The Weil group of $\mathbf{C}$ is canonically $\mathbf{C}^\times$. On the Galois side then we're looking at ...

**4**

votes

**1**answer

168 views

### Theta series for the Leech lattice

The Leech lattice, found by John Leech in 1965, is a fascinating combinatorial object and gives the best possible lattice sphere packing in $\mathbb{R}^{24}$. This result was proved by Cohn and Kumar ...

**4**

votes

**2**answers

171 views

### Asymptotics of special square-free numbers

What is the asymptotic number of square-free numbers less than $x$ with exactly $k$ prime divisors?

**0**

votes

**0**answers

44 views

### Bounding a sum of functions defined on effective divisors

The following is from a PhD thesis "Algebraic Circle Method" by Thibaut Pugin, which I am currently reading.
Let $k$ be a finite field of order $q$.
Let $X \subseteq \mathbb{A}^{n+1}_k$ be the ...

**1**

vote

**0**answers

113 views

### Ratio of periods for elliptic curves in an isogeny class

Let $E \to E^\prime $ be an isogeny of elliptic curves defined over $\mathbb{Q}$. Then what is the definition of ratio of periods for elliptic curves in the isogeny class and how to calculate the ...

**3**

votes

**0**answers

157 views

### Why Whittaker functions are useful?

Whittaker functions appears in Langlands program. Recently, it is shown that some Whittaker functions can be obtained by integrating a function related to decoration over a geometric crystal in ...

**0**

votes

**0**answers

69 views

### Minimize the length of two disjoint segments in the string with given property

You are given a string s of size n, consisting of characters A and B only. You have to find minimum sum of size of the two disjoint segments of the string s such that number of A's in them are >= z.
...

**22**

votes

**2**answers

742 views

### An Interesting Optimization Problem

You are given n non-negative integers $a_1, a_2 ,, a_n$. In a single operation, you take any two integers out of these integers and replace them with a new integer having value equal to difference ...

**2**

votes

**1**answer

420 views

### Need help publishing a mathematical proof? [closed]

Im not a mathematician with profession, but I know a group working on a Beal's Conjecture for years. They think that they have found a proof but the problem is that they don't know how to publish ...

**1**

vote

**0**answers

66 views

### partial sums of fractional parts

I'd like to bound the partial sum: $S(p,q,\alpha):=\sum\limits^{p+q-1}_{k=[\alpha(p+q)]}\{-\frac{qk}{p+q}\}-\sum\limits^{p+2q-1}_{[\alpha(p+2q)]}\{-\frac{qk}{p+2q}\}$.
Here $p,q$ are naturals, ...

**3**

votes

**0**answers

111 views

### Question related to the number of rational curves on a hypersurface

The following is from a PhD thesis "Algebraic Circle Method" by Thibaut Pugin, which I am currently reading.
Let $k$ be a finite field of order $q$.
Let $\mathfrak{X} \subseteq \mathbb{P}^n_k$ be ...

**1**

vote

**0**answers

160 views

### Ideals with norm in arithmetic progression

Let K/Q be a number field extention. Is there an asymptotic formular for the numer of ideals $\sum\limits_{\substack{N(A)\leq x\\N(A)\equiv k(q)}}1$,where $(k,q)=1$ and $A$ runs over ideals in $O_K$. ...

**6**

votes

**1**answer

199 views

### Question about a certain class of primes

I've come across a set of primes in a problem I'm working on, and I'm wondering if there's more information available about them. I'm guessing not much, particularly since the question of infinitude ...

**9**

votes

**2**answers

456 views

### What is a sieve and why are sieves useful?

I have been trying to understand what is exactly a sieve and why sieves are useful.
I have read Wikipedia articles about sieve theory but they don't provide a definition of what is a sieve or why they ...

**4**

votes

**0**answers

118 views

### Continuity of the Hilbert pairing

I would like to know if the Kummer pairing (or the analogue of the Hilbert Symbol) for a one dimensional group defined over the ring of integers of a higher-dimensional local field is continuous (with ...

**7**

votes

**2**answers

369 views

### Is realness of number fields exponentially bounded?

In a given non-real algebraic number field (say, given by an irreducible polynomial over $\mathbb{Q}$) is there a complexity bound on the summands $x,\dots,t$ that make $-1$ a sum of squares? So ...

**4**

votes

**0**answers

146 views

### Degree of Kummer extensions of number fields

Let $K$ be a number field and $a\in K^*$ of infinite order in $K^*$. How do I show that
$$[K(\sqrt[n]{a},\zeta_n):K]\geq C\cdot n\cdot\varphi(n)$$
holds for all positive integers $n$, with a positive ...

**0**

votes

**0**answers

55 views

### Growth of sums of multiplicative functions over Squarefrees

When one looks at the quotient of Euler products $$\prod_p\frac{\sum_{\alpha=0}^{\infty}f(p^{\alpha})p^{-\alpha s}}{1+f(p)p^{-s}}$$
with $|f|\leq 1$, it is observed that the resulting expression ...

**2**

votes

**2**answers

212 views

### When does a dyadic prime ramify in a relative quadratic extension?

In a quadratic extension $\mathbb{Q}(\sqrt{d})$of $\mathbb{Q}$ it is clear that 2 ramifies if and only if $d\equiv 2,3\mod 4$ (easy to see if you compute the discriminant). But if I take a relative ...

**6**

votes

**0**answers

151 views

### Inequality regarding sum of gaussian on lattices

When S is a subset of an inner product space, let d(S) denote ${\sum\limits_{s \in S} e^{- \langle s,s \rangle}}$
Suppose L is a discrete additive subgroup of $\mathbb{R^n}$, M is a subgroup of L, ...

**1**

vote

**0**answers

174 views

### An inequality about Goldbach conjecture

Let $N$ a large natural number, let $\forall n\leq N,\, R_{2}\left(n\right)=\underset{p_{1}+p_{2}=n}{\sum}\log\left(p_{1}\right)\log\left(p_{2}\right)$ and let $S\left(\alpha\right)=\underset{p\leq ...

**0**

votes

**0**answers

58 views

### analytic continuation related to Chebyshev functions

Let $\psi$ be the Chebyshev function. I would like to prove that the function
$\sum_{n\ge0}(\sum_{k=0}^n\binom{n}{k}e^{\psi(k)})w^n$ can be analyticaly continued on an open set of $\mathbb C$ ...

**1**

vote

**2**answers

217 views

### Infinite play with tape, or covering the integers with prime arithmetic progressions

It is possible that a more technical version of this question has been
asked and answered in the literature. If so, then a reference is much
appreciated. I will phrase it in terms of colored tapes ...

**0**

votes

**0**answers

44 views

### Estimates for the number of lattices associated with a ternary form with short minimal vectors

Suppose that $F(x,y,z) \in \mathbb{Z}[x,y,z]$ is a ternary form, i.e. it is homogeneous, of degree $d$. Consider a solution $(a,b,c)$ to the congruence
$$\displaystyle F(x,y,z) \equiv 0 \pmod{m}$$
...

**1**

vote

**1**answer

143 views

### Problem on the digits of $n!$

let $m$ be a natural number, does always exist a $N\in \mathbb{N}$ such that $m$ or more "$0$" digits (excluding the terminal ones) appears amongs the decimal digits of $n!$ if $n\ge N$?

**2**

votes

**1**answer

70 views

### Principally split primes with factors in arbitrarily small angular sectors

I wonder if the following is known:
let $n$ be a (square-free) positive integer. Is there ever/always a sequence of prime numbers $p$ that can be written in the form $$p = x^2 + ny^2,$$ where
$x, y$ ...

**1**

vote

**1**answer

201 views

### On $x^3-y^2=1728 \text{ unit}$ in number fields

Consider solution of
$$x^3-y^2=1728 \text{ unit} \qquad (1)$$
in a number field.
This is related to the discriminant of elliptic curve
in terms of $c_4,c_6$.
Via elliptic curves it might have ...

**3**

votes

**2**answers

200 views

### Gelfand pair and double coset decomposition

Let $F$ be a non-Archimedean local field with ring of integers $O$, $\pi$ be a uniformizer. Let $\tilde{G}$ be a connected algebraic group over $F$ and splits over $F$, fix a split maximal torus ...

**2**

votes

**0**answers

124 views

### Number of rational curves on varieties over finite fields

Let $k$ be a finite field. Let $P_r$ be set of degree $r$ binary forms
over $k$.
We define
$$
\mathcal{M}_r = \{ (x_0, ..., x_n) \in P_r^{n+1} : x_0^d + ... + x_n^d = 0 \text{ and the }x_i \text{'s ...

**1**

vote

**1**answer

137 views

### Relations between Multizeta Values

I am studying Multizeta values at the moment and I found that at weight 5, the basis is given by ζ(5) and ζ(3)ζ(2) in the literature. Solving all shuffle and stuffle relations using mathematica I, ...

**3**

votes

**0**answers

127 views

### An estimate for dividing n^2 by each of the primes up to and including n, and then summing the results [closed]

I know that the asymptotic for the sum of all the primes up to n is $n^2/2\log n$. But I'm trying to find the formula (an estimate) for when $n^2$ is divided by each of the primes up to $n$, in turn ...

**1**

vote

**0**answers

152 views

### $\mathfrak{q}$-ideal class bound

Let $K$ be a number field, $\mathcal{O}_K$ be its ring of integers. Let $\mathfrak{q}$ be a nonzero ideal in $\mathcal{O}_K$.
The $\mathfrak{q}$-ideal class group consists of equivalence classes of ...

**6**

votes

**1**answer

154 views

### Is (Z,+,0,1,P2,P3) decidable?

Is Presburger arithmetic, augmented with two unary predicates P2, P3, for powers of 2 and powers of 3 respectively, decidable?
I know that adding just one of P2, P3 to Presburger keeps it decidable, ...