1
vote
0answers
343 views
+150
A problem in number theorem with a number of the base p
First we define a function $f(x,p)$, with $x$ a natural number and with $p$ a prime number.
$f(x,p)$ stands for the location where the digit "$p-1$" first appears in the base-$p$ …
11
votes
1answer
225 views
On the Hasse-Weil L-function of $P^n$
So let us start with the "simplest" scheme over $Spec(\mathbf{Z})$ namely $X_0=Spec(\mathbf{Z})$. Then the (reciprocal) Weil zeta function of $X_0$ at a prime $p$ is given by $Z_p( …
3
votes
1answer
91 views
upper bounds for the ranks of the minus parts of modular jacobians
Let $p$ be a prime, and $J^-(p)$ be the maximal quotient of the Jacobian of the modular curve $X_0(p)$ on which the involution acts by $-1$.
Is anything known or conjectured about …
3
votes
3answers
197 views
Canonical form of symmetric integer matrix M
Let $M$, $N$ be a symmetric matrix over a ring $R$.
$M$ and $N$ are said to be equivalent if there exist an invertible
matrix $U$ (over the same ring $R$) such that $N=U M U^T$ ($U …
10
votes
5answers
4k views
On the series 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + …
It is well-known that
A: The series of the reciprocals of the primes diverges
My question is whether property A is in some sense a truth strongly tied to the nature of the prime …
9
votes
0answers
256 views
Möbius Randomness of the Rudin-Shapiro Sequence
The Rudin-Shapiro sequence (also known as the Golay-Rudin-Shapiro sequence) is defined as follows.
Let $a_n = \sum \epsilon_i\epsilon_{i+1}$ where $\epsilon_1,\epsilon_2,\dots$ ar …
13
votes
4answers
650 views
Primes with more ones than zeroes in their Binary expansion
This question is also motivated by the developement around my old MO question about Mobius randomness. It is also motivated by Joe O'Rourke's question on finding primes in sparse s …
17
votes
4answers
477 views
Does $(x^2 - 1)(y^2 - 1) = c z^4$ have a rational point, with z non-zero, for any given rational c?
I need this result for something else. It seems fairly hard, but I may be missing something obvious.
Just one non-trivial solution for any given $c$ would be fine (for my applicat …
6
votes
3answers
199 views
Special divisor function summation
What is a good upper bound for
$$\sum_{d\leq z} \mu(d)\frac{\tau(d)}{d}$$
where $\tau(d)$ is the divisor function?
10
votes
4answers
289 views
non-trivial zeros of partial zeta functions
Let $N,a\in\mathbf{Z}_{\geq 1}$. Define a partial $\zeta$-function as
$$
\zeta(s;N,a):=\sum_{\substack{n\geq 1\newline n\equiv a\pmod{N}}} \frac{1}{n^s}
$$
where $Re(s)>1$. Let $\o …
3
votes
3answers
106 views
sum of fractional parts (nx_i),x_i are irrational
Let $x_1,x_2,...,x_k$ be irrational number,is it always true that:
$\liminf_{n\rightarrow\infty} \sum_{i=1}^k (nx_i)=0$ (where $(x)$ denotes the fractional part of $x$)
If not,wha …
4
votes
3answers
425 views
Congruences between Fermat quotients
This a repost of a question I asked at Stack Exchange:
http://math.stackexchange.com/questions/35264/congruences-for-fermat-quotients
I didn't get a complete answer to my quest …
2
votes
1answer
147 views
Is there a Dirichlet Unitary Unit Theorem?
Dirichlet's unit theorem computes the group of units of the algebraic numbers of a number field. There are a few generalisations for orders available.
Assume the order has an invo …
2
votes
1answer
177 views
Solving polynomial equations in radicals provided all roots are rational
This question is related to this question of Joseph O'Rourke and this question of mine.
Question. Let $f$ be a polynomial with integer coefficients. Suppose that all roots of …
6
votes
1answer
181 views
Complexity of finding a rational root of a polynomial
This is inspired by this question. Let $f(x)=a_nx^n+...+a_0$ be a polynomial with rational coefficients. The sandard procedure of finding a rational root $p/q$ involves checking a …

