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-$p$ …

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( …

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 …

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 …

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 …

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 …

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 …

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 …

What is a good upper bound for $$\sum_{d\leq z} \mu(d)\frac{\tau(d)}{d}$$ where $\tau(d)$ is the divisor function?

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 …

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 …

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 …

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 …

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 …

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 …