Prime numbers, Diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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7
votes
1answer
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
2answers
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
2answers
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
3answers
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
1answer
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
1answer
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
1answer
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
2answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
2answers
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
0answers
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
2answers
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
0answers
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
0answers
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
2answers
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
0answers
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
0answers
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
0answers
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
2answers
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
0answers
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
1answer
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
1answer
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
1answer
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
2answers
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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, ...