Search Results

As a math amateur, I am finding the study reduced residue systems relative a primorial a very interesting way to understand the distribution of primes. … Is it that research in reduced residue systems relative primorial is now focused on highly technical topics? …

On page 109, the author generalizes the reduced residue system modulo $v$ where $v=pq$, $p$ and $q$ are primes. …

I made this sieve for prime numbers, which I briefly describe: We consider $\quad p=r+modulus \cdot k \quad$ with $\quad modulus=p_1*p_2* \cdots *p_m$ and then we choose an appropriate reduced residue … Example if $\quad modulus=30\quad $ then$\quad \phi (30) =8 \quad $ and $\quad Remainder = \{-23, -19, -17, -13, -11, -7, -1, 1\}$ is a reduced residue system $\quad \pmod {30}$. …

For instance, the recent progress on finding large gaps between primes (see e.g. these two papers) relies on finding somewhat large gaps inside the reduced residue system of a primorial. … Residue class rings ${\bf Z}/q{\bf Z}$ are often referred to by other names, such as cyclic groups or reductions of the integers modulo $q$, and the reduced residue system might be referred to as the group …

If $g$ is a primitive root mod $p$, then consider the following coloring of the reduced residue system: $x\mapsto c$ if $x\equiv g^{ny+c}\pmod{p}$ for some $y$ and $0\leq c \lt n$ . …

For $n=1$ it is the reduced special fiber $\mathcal M_1=\mathcal M_{\mathrm{red}}$, thus a smooth scheme over the residue field $\kappa(E)$. … However, among other things, their geometric description implies that the reduced special fiber $\mathcal N_{\mathrm{red}}$ is in fact... not smooth! …

Then we let $\Sigma=\Sigma(G,T_0)\subset X^*(T_0)_F$ be the root system ($T_0$ the split component of $T$). Let $P$ be a standard $F$-parabolic subgroup (i.e. … $\Sigma_{\mathrm{red}}(\bar{P})$) for the set of roots (resp. reduced roots) appearing in the Lie algebra $\bar{\mathfrak{n}}:=\mathrm{Lie}(\bar{N})$. …

Let $\Phi$ be the root system of $G$ (which is $A_r$), and $\Phi^+$ be the positive roots. Let $W\colon G \to \mathbf{C}$ be the spherical Whittaker function on $(\pi_z, V_z)$. … Why does the limit of the spherical Whittaker functions, as the cardinality of the residue field goes to infinity, converge to the character of the corresponding highest weight representation of the dual …

part of the statement below: Let $X$ be a proper smooth adic space over $\mathrm{Spa}(k, \mathcal O_{k})$ (where $k$ is some complete discretely valued field extension of $\mathbb Q_p$ with perfect residue … Let $(\mathcal E, \nabla, \mathrm{Fil}^\bullet)$ be a filtered $\mathcal O_X$ module with integrable connection, giving rise to a ${\mathbb B} _ {\mathrm{dR}}^+$ local system $\mathbb M$. …

I have a simple Fourier transform problem, originating from mathematical physics (system of linear PDEs), which reduces to taking the integral $$ I(\alpha)\equiv\int_{-\infty}^\infty e^{ikr} \cfrac{\alpha …

Hence, we may replace $A$ with $A'$ to reduce to the case that the residual extension is purely inseparable. … of reduced points when $X_0$ and $Y_0$ are curves. …

only definition which comes to mind at a good place (namely, cohomology over $\mathfrak{o}_v$ with coefficients in the abelian scheme Neron model at $v$) vanishes due to Lang's theorem over the finite residue … Using this geometric interpretation and a concrete description of finite etale $R$-algebras (and finiteness of the residue fields of the local factor rings $\mathscr{O}_v$), the vanishing is proved on …

This is already a well-known phenomenon for $p$-adic fields $K$ (complete discretely valued with perfectly residue field), where letting $C=\widehat{\overline{K}}$ it is the difference between $K$-vector …

You can see how this phenomenon requires the setting of a non-reduced root system to happen. … At the hyperspecial points, the group would be a quasisplit $SU_3$ over the residue field. …

Divide a given set A of integers into residue classes mod p, so there are a_j many members of A equal to j mod p. … So there is a solution to the system of equations above, not just for odd primes p, but also for odd divisors d of (n+1) (and for odd d dividing n). …