**0**

votes

**1**answer

85 views

### Question about sign change of Hecke eigenvalues

I want to write a survey on the subject 'Sign changes for coefficients of symmetric power $L$-functions'. So, I browse the Web and I got some papers. I read it and I gave special interest to the paper ...

**2**

votes

**1**answer

109 views

### A quadrant of residues

Assume that following inequality holds $$\mathsf{w,x,y,z<AB,AC,AD,BC,BD,CD<ABC,ABD,ACD,BCD<wx,wy,wz,xy,xz,yz}$$ with $$\mathsf{gcd(A,B)=gcd(A,C)=gcd(A,D)=gcd(B,C)=gcd(B,D)=gcd(C,D)=1}$$
...

**8**

votes

**1**answer

831 views

### Remark on Fermat's Last Theorem by Darmon, Diamond and Taylor

In their paper, Darmon, Diamond and Taylor remarked the following :
(the previous paragraph of Section 2.2 (p. 55), https://www.math.wisc.edu/~boston/ddt.pdf)
If $\rho : G \rightarrow ...

**8**

votes

**4**answers

460 views

### Lower bounding the multiplicative order of 2 modulo p

For $p$ prime denote by $\mathsf{ord}_p(2)$ the multiplicative order of $2$ modulo $p$.
Does there exist $N > 0$ such that, for ALL primes $p$, $\mathsf{ord}_p(2)$ is at least $\frac{(p-1)}{N}$?
...

**3**

votes

**0**answers

137 views

### A factorial related statement

Is statement $\mathsf{S}$ below in $\mathsf{NP}$ or in $\mathsf{coNP}$?
$$\mathsf{S}:\mathsf{Given}\mbox{ }p,a,s,c\in\Bbb N,\mbox{ }\mathsf{with}\mbox{ }p\mbox{ }\mathsf{a}\mbox{ }\mathsf{prime}\mbox{ ...

**9**

votes

**1**answer

444 views

### A question on Ramanujan's $1/\pi$ formulas

It is known that Ramanujan discovered a number of formulas for $1/\pi$. All of these formulas are of the form $$\frac{1}{\pi}=\sum_{n=0}^{\infty}\frac{(1/2)_n(s)_n(1-s)_n}{(1)_n^3}(a+bn)z^n,$$where ...

**10**

votes

**2**answers

456 views

### Learning the exponents in a sum of two modular roots of unity

$\newcommand{\Z}{\mathbb{Z}}$
Suppose that $n$ is a large and known integer (say, with 100 digits) and that you are given access to a function
$$f(x) = x^a + x^b$$
with unknown exponents $a,b \in ...

**1**

vote

**0**answers

93 views

### Rank of the Jacobian of a family of hyperelliptic curves of genus 2

Assume tha $C$ be the hyperelliptic curve $y^2 = (x-a_1)\cdots (x-a_5)$ of genus $g=2$ and $a_i \in \mathbb{Z}$ and we know that the integers $a_i$ has the form $a_i= d_1^2 - d_i^2$ for some positive ...

**3**

votes

**2**answers

280 views

### Trivial zeroes of the Riemann Zeta function are simple

The trivial zeroes of the Riemann Zeta function are located on $-2\mathbb N^*$ and they are simple. It is not difficult to see that, but the proof I have in mind is using the fact that ...

**4**

votes

**3**answers

376 views

### Bateman-Horn, continued even further

As before, consider the "singular series", which shows up in the Bateman-Horn conjecture: for an irreducible polynomial $f,$ this is equal to
$$
s(f) = \prod_p \frac{1-\frac{n_f(p)}p}{1-\frac1p},
$$
...

**1**

vote

**0**answers

118 views

### System of congruences

I have a system of $n$ congruences.
the generic $m$_th congruence of the system ($m = 1,\dots,n$) is in the form:
$(p_n\#)^2(\displaystyle\sum_{\substack{i=1\\i\neq ...

**3**

votes

**0**answers

127 views

### Does the Bombieri-Lang conjecture imply severe restrictions on rational points on twists of hyperelliptic curves?

According to Silverman, the Bombieri-Lang conjecture implies
that the rational points of surface on general type lie on
finite set of curves, except for a finite set of points.
Let $f$ be univariate ...

**2**

votes

**0**answers

63 views

### Listing all Lattice Points in a Box

Let $B := [-1,1]^n$ be an $n$-dimensional box. Moreover, let $v_1,\ldots,v_n \in \mathbb{R}^n$ form a basis of $\mathbb{R}^n$, where the entries of the $v_i$ are explicitly irrational. We can assume ...

**2**

votes

**0**answers

127 views

### $\frac{1}{2}<\sigma<1$, is $f(n) = \Bigl| \,1+ \frac{1}{2^{\sigma + i n}} + \cdots + \frac{1}{n^{\sigma + i n}} \Bigr|$ from $O(\log n)$?

We have $\frac{1}{2} < \sigma < 1$ and
$$
f(n) = \Bigl|\, 1+ \frac{1}{2^{\sigma + i n}} + \cdots + \frac{1}{n^{\sigma + i n}} \Bigr|
$$
. My goal is proving this statement that $|f(n)|$ is ...

**10**

votes

**3**answers

448 views

### Continued Fractions from Digit Streams

let $x=\sum_{i=1}^{\infty}\delta_i2^{-i},\ \delta_i\in\{0,1\}$.
Is there an algorithm that converts the sequence $(\delta_0,\ \delta_1,\ ...)$ of the binary digits of $x$ to the sequence ...

**0**

votes

**0**answers

63 views

### Bounds on sum of reciprocal of logarithm of primes [duplicate]

Are upper/lower bounds known for the following quantity?
$$S(n,a)\stackrel{\triangle}{=}\sum_{p_k \leq n}\frac{1}{(\log p_k)^a}.$$
I am mainly interested in the case, $a=1$. I suppose with the ...

**12**

votes

**3**answers

449 views

### 2-dimensional sublattices with all vectors having very big square (in absolute value)

QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice,
that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form, not
definite, not necessarily unimodular, $n>2$. I want ...

**0**

votes

**0**answers

60 views

### Smallness in modular condition

Given coprime $\mathsf{a_1,a_2\in\Bbb N}$ with $\mathsf{\mathsf{\max(a_1,a_2)\leq2\min(a_1,a_2)}}$, is there pair $\mathsf{(x_1,x_2)\in\Bbb N^2}$ such that
...

**0**

votes

**1**answer

56 views

### For a given $n$, under what condition(s) there exists (at least) two different $c$ and $c′$ such that $X_n^c=X_n^{c'}$

Let $X_n^c=\{\cos\left((4k-c)\frac{\pi}{2n}+\frac{\pi}{4}\right): k=0, 1, \dots, n-1\}$ where $c\in\{0, 1, \ldots, \lfloor\frac{n}{2}\rfloor\}$ and $n$ is any positive integer greater than 3. I want ...

**16**

votes

**2**answers

799 views

### Which algebraic relations are possible between algebraic conjugates?

For which non-constant rational functions $f(x)$ in $\mathbb{Q}(x)$ is there $\alpha$, algebraic over $\mathbb{Q}$, such that $\alpha$ and $f(\alpha) \neq \alpha$ are algebraic conjugates? More ...

**7**

votes

**1**answer

203 views

### Numerically double-checking formula with L-values

I'm working with a special case of Ichino's triple product formula, which for classical holomorphic newforms $f$, $g$ ,$h$ of weights $k$, $m-k$, $m$ (and central characters satisfying $\chi_f \chi_g ...

**2**

votes

**0**answers

159 views

### Morphism of Shimura varieties and differential equations

Is there a way of constructing a morphism between Shimura varieties using differential equations? Maybe, this looks like a completely ridiculous question, so I think that I should explain the context ...

**8**

votes

**2**answers

519 views

### Does the Galois group of a Pisot polynomial contain the alternating group?

Let $n \in \mathbb{N}$, and let $p(X) \in \mathbb{Z}[X]$ be a monic polynomial of degree $n$. Suppose that exactly one complex root of $p$ is of modulus $> 1$, and that the remaining $n-1$ roots of ...

**7**

votes

**1**answer

318 views

### Bounded gaps between primes in arithmetic progressions

Has Zhang's work on bounded gaps between primes been extended to the following theorem?
For any arithmetic progression $an+b,\gcd(a,b)=1$, there is a constant $H$ (depending only on $a$) such that ...

**0**

votes

**0**answers

36 views

### Maximum norm of discrete Fourier transform [duplicate]

I have considerable numerical evidence that
for all $0\leq k\leq{{n-1}\over 2}$ ($n$ odd) there exists a subset $
S_k$ of {1,2,...,n} of cardinality $k$
such that the modulus square of ...

**0**

votes

**1**answer

168 views

### A three variable linear diophantine promise problem

Given $a,b,c,s\in\Bbb N$ such that $(a,b,c)=1$ with promise that we have at most one triple $x,y,z\in\Bbb N$ such that $ax+by+cz=s$, what is a good algorithm that runs in $O(\log(abcs))$ time to find ...

**2**

votes

**2**answers

201 views

### Decomposition and valuation rings

I am reading Algebraic Number Theory by A. Fröhlich, M. J. Taylor, it first introduced the theorem:
$(K,u)$ be a field and its absolute value, $(K_u,\bar u)$ be its completion and absolute value ...

**2**

votes

**2**answers

87 views

### Relation between number of non-negative and positive integers points in simplices

I asked this question on math.SE before, but did get not get an answer. Therefore I hope it is ok to post it here on this site..
Let $q \in \mathbb{R}_+$ and $0 < w_1 \leq w_2 \leq \ldots w_d \in ...

**6**

votes

**1**answer

176 views

### Can a product of conjugates be a Pisot number again?

Let $p(X) \in \mathbb{Z}[X]$ be an irreducible polynomial, and let $\alpha_1 \dots, \alpha_n$ be its roots in $\mathbb{C}$. Suppose that $\alpha_1$ is a Pisot number, that is, $\alpha_1 \in ...

**2**

votes

**0**answers

149 views

### Rank of the Jacobian of twists of hyperelliptic curves

Suppose that a hyperelliptic curve $C$ of genus $g \geq 4$ is given by the equation
$$\displaystyle C: y^2 = a_0 x^{2g+2} + a_1 x^{2g+1} + \cdots + a_{2g+2} = f(x).$$
The Jacobian variety $J(C)$ of ...

**1**

vote

**1**answer

118 views

### Question about mean square estimate for sums of Dirichlet coefficients of Symmetric Power $L$-functions

I have a question related to Coefficients of Symmetric power $L$-functions and I would be grateful if you could answer it.
Let $\lambda_{Sym^rf}(n)$ be the $n$th Dirichlet coefficient of ...

**2**

votes

**0**answers

90 views

### Possible argument against Height bound hypothesis

From this paper.
$f(x,y)$ is polynomial with integer coefficients.
$s(f)$ is its size, the sum of the logarithms of the absolute
values of the nonzero coefficients, defined on p. 6. From p. 7.
...

**2**

votes

**1**answer

185 views

### Trace of a Product of Finitely Many Matrices with Cosine Entry

Can someone help me prove the following identity?
$$
\mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix}
2\cos\frac{2j\pi}{n} & -m \\
1 & 0
\end{pmatrix}\right)=
\begin{cases}
2 & ...

**2**

votes

**1**answer

166 views

### Euler's totient function relative function

For the $\sigma$ function, the ratio $\sigma(m)/m$ is known as the abundancy index. Is there any special name for $\phi(m)/m$ with $\phi$ the Euler's totient function ?

**0**

votes

**0**answers

68 views

### Is this factoring algorithm sufficiently efficient for some integers of special kind?

Basically the question is if $m$ is factored over the integers,
can it be relatively efficiently factored over $\mathbb{Z}[\sqrt{n}]$
where $n$ is not factored, but might be of special form?
Suppose ...

**4**

votes

**0**answers

87 views

### Integer sum of distinct reciprocals with no integer subset sum

Question
$\def\nn{\mathbb{N}}$
For any $n \in \nn^+$, is there a finite set $S \subset \nn^+$ such that $\sum_{k \in S} \frac{1}{k} = n$ but $\sum_{k \in T} \frac{1}{k} \notin \nn^+$ for any $T ...

**7**

votes

**1**answer

254 views

### Definition of p-adic modular forms

I have been reading Hida's book "p-Adic automorphism forms on Shimura varieties" and I don't understand a point.
He first describes p-adic modular forms of tame level N as functions on the Igusa ...

**2**

votes

**1**answer

316 views

### Quadratic Diophantine equation in $\mathbb Z[T]$

I am trying to solve the following quadratic diophantine equation in $\mathbb Z[T]$:
$$((T+1)X+TY-1-Z)((T+1)X+TY-1+Z)=24XY$$
One has the following trivial solutions:
$(X,Y,Z)=(0,Y,\pm(1-TY))$, ...

**6**

votes

**1**answer

209 views

### A question about $(0,1]$-valued multiplicative functions

Suppose $f:\mathbb{N}\to [0,1]$ is a multiplicative function (i.e. $f(nm)=f(n)f(m)$ whenever $m$ and $n$ are coprime). Suppose $f$ has non-zero mean, which means
$$
\lim_{N\to\infty}\frac{1}{N} ...

**2**

votes

**2**answers

176 views

### Does the Divisor Function $\sigma(n)$ have analogues for other Fuchsian groups?

I have been reading about the divisor function $\sigma = 1 \ast 1$ and proved an elementary identity:
$$ \Big[\sum_{d|n} \sigma_0(d)\Big]^2 = \sum_{d|n} \sigma_0(d)^3$$
Here $\sigma_0 = \sum_{d|n} ...

**2**

votes

**1**answer

106 views

### Unique extension of the absolute value

Let $(K,u)$ be a complete valued field, $u$ be its discrete absolute value (corresponds to a discrete valuation on $K$), then:
($\ast)$ Let $E/K$ is a finite separable field extension, then the ...

**5**

votes

**1**answer

193 views

### Hyperelliptic curves with fixed genus and many rational points

It is a famous theorem of Faltings, previously a conjecture by Mordell, that any algebraic curve of genus at least $2$ defined over the rational numbers have at most finitely many rational points. A ...

**2**

votes

**0**answers

247 views

### A variant frobenius problem

From Sylvester's theorem we know that using only coins of sizes $a,b$, we can change exactly $\frac{(a-1)(b-1)}2$ different big coins up to $(a-1)(b-1)$.
Denote sets ...

**4**

votes

**2**answers

113 views

### Expected Cardinality of the First n Coefficients of a Continued Fraction

Is there anything known about the asymptotic expected growth of $card\{a_0,a_1,...,a_n\}$, where the $a_i$ are the first $n$ coefficient of a continued fraction $[a_0; a_1, ... ]$?

**4**

votes

**0**answers

137 views

### For which rings $A$ is $A-\{0\}$ Diophantine over $A$?

Let $A$ be a commutative ring. Recall that we call $S\subset A$ a Diophantine subset of $A$ if there exists a polynomial $P(t;x_1,\ldots,x_n)$ with coefficients in $A$ such that:
$$
t_0 \in S ~~ ...

**3**

votes

**1**answer

239 views

### On Heath-Brown's “Prime twins and Siegel zeros”

With a graduate student, I'm going through the paper (Proc. London Math. Soc. (3) 47 (1983), no. 2, 193–224.) We are quite baffled by the proof of Lemma 3 on p. 198.
Here's the background and ...

**6**

votes

**0**answers

171 views

### Criteria for irreducibility using the location of complex roots

I would like to see criteria for the irreducibility of a polynomial over $\mathbb{Z}$ based (mainly) on the location of the roots of the polynomial in the complex plane. An example of such a criterion ...

**2**

votes

**1**answer

184 views

### Finite sequences which happen to be the sequence of orders of elements of a simple group

Let $n\in \Bbb N$. Any finite group $G$ with $|G|=n$ has a solution for the Diophantine equation
$$\sum_{d|n}x_d\phi(d)=n~~~~~~~~~~~~~~~~~~~~~~~(1)$$
where $\phi$ is the Euler's totient function, $d$ ...

**-4**

votes

**1**answer

202 views

### Gauss-Wantzel theorem, Fermat primes and solvability of S_n [closed]

Gauss-Wantzel theorem asserts that a polygon with $n$ sides is constructible if and only if $n$ is a product of a power of $2$ and distinct prime Fermat numbers, where the Fermat number of index $k$ ...

**4**

votes

**0**answers

167 views

### Additivity of upper densities with respect to arithmetic progressions of integers

Let $\mathsf{d}^\star$ be the asymptotic upper density, defined on the power set of positive integers $\mathbf{N}^+$, so that
$$
\mathsf{d}^\star\colon \mathcal{P}(\mathbf{N}^+) \to\mathbf{R}\colon ...