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Register a new userGroups of prime degree and the Bateman-Horn Conjecture
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As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${rm PSL}_n(q)$ is prime. We present heuristic arguments and computational evidence based on the Bateman-Horn Conjecture to support a conjecture that for each prime $nge 3$ there are infinitely many primes of this form, even if one restricts to prime values of $q$. Similar arguments and results apply to the parameters of the simple groups ${rm PSL}_n(q)$, ${rm PSU}_n(q)$ and ${rm PSp}_{2n}(q)$ which arise in the work of Dixon and Zalesskii on linear groups of prime degree.
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We confirm the Hanna Neumann conjecture for topologically finitely generated closed subgroups $U$ and $W$ of a nonsolvable Demushkin group $G$. Namely, we show that begin{equation*} sum_{g in U backslash G/W} bar d(U cap gWg^{-1}) leq bar d(U) bar d(W) end{equation*} where $bar d(K) = max{d(K) - 1, 0}$ and $d(K)$ is the least cardinality of a topological generating set for the group $K$.
The formal degree conjecture and the root number conjecture are verified with respect to supercuspidal representations of $Sp_{2n}(F)$ and $L$-parameters associated with tamely ramified extension $K/F$ of degree $2n$. The supercuspidal representation is constructed as a compact induction from an irreducible unitary representation of the hyper special compact group $Sp_{2n}(O_F)$, which is explicitly constructed, based upon the general theory developed by the author, by $K$ and certain character $theta$ of the multiplicative group $K^{times}$. $L$-parameter is constructed by the data ${K,theta}$ by means of the local Langlands correspondence of tori and Langlands-Schelstad procedure.
We establish cancellation in short sums of certain special trace functions over $mathbb{F}_q[u]$ below the P{o}lya-Vinogradov range, with savings approaching square-root cancellation as $q$ grows. This is used to resolve the $mathbb{F}_q[u]$-analog of Chowlas conjecture on cancellation in M{o}bius sums over polynomial sequences, and of the Bateman-Horn conjecture in degree $2$, for some values of $q$. A final application is to sums of trace functions over primes in $mathbb{F}_q[u]$.
We prove the Ramanujan-Petersson conjecture for Maass forms of the group $SL(2,Z)$, with the help of automorphic distribution theory: this is an alternative to classical automorphic function theory, in which the plane takes the place usually ascribed to the hyperbolic half-plane.
The primary purpose of this paper is to report on the successful enumeration in Magma of representatives of the $195,826,352$ conjugacy classes of transitive subgroups of the symmetric group $S_{48}$ of degree 48. In addition, we have determined that 25707 of these groups are minimal transitive and that 713 of them are elusive. The minimal transitive examples have been used to enumerate the vertex-transitive graphs of degree $48$, of which there are $1,538,868,366$, all but $0.1625%$ of which arise as Cayley graphs. We have also found that the largest number of elements required to generate any of these groups is 10, and we have used this fact to improve previous general bounds of the third author on the number of elements required to generate an arbitrary transitive permutation group of a given degree. The details of the proof of this improved bound will be published by the third author as a separate paper
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