Existence of Ramanujan primes for GL(3)
نویسنده
چکیده
Let π be a cusp form on GL(n)/Q , i.e., a cuspidal automophic representation of GL(n,A ), where A denotes the adele ring of Q . We will say that a prime p is a Ramanujan prime for π iff the corresponding πp is tempered. The local component πp will necessarily be unramified for almost all p, determined by an unordered n-tuple {α1,p, α2,p, . . . , αn,p} of non-zero complex numbers, often represented by the corresponding diagonal matrix Ap(π) in GL(n,C ), unique up to permutation of the diagonal entries. The L-factor of π at p is given by
منابع مشابه
Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes
We study the parallel properties of the Ramanujan primes and a symmetric counterpart, the Labos primes. Further, we study all primes with these properties (generalized Ramanujan and Labos primes) and construct two kinds of sieves for them. Finally, we give a further natural generalization of these constructions and pose some conjectures and open problems.
متن کاملRamanujan Primes: Bounds, Runs, Twins, and Gaps
The nth Ramanujan prime is the smallest positive integer Rn such that if x ≥ Rn, then the interval ( 1 2x, x ] contains at least n primes. We sharpen Laishram’s theorem that Rn < p3n by proving that the maximum of Rn/p3n is R5/p15 = 41/47. We give statistics on the length of the longest run of Ramanujan primes among all primes p < 10n, for n ≤ 9. We prove that if an upper twin prime is Ramanuja...
متن کاملZeta Functions of Complexes Arising from Pgl(3) Ming-hsuan Kang and Wen-ching
In this paper we obtain a closed form expression of the zeta function Z(X Γ , u) of a finite quotient X Γ = Γ\P GL 3 (F)/P GL 3 (O F) of the Bruhat-Tits building of P GL 3 over a nonar-chimedean local field F. Analogous to a graph zeta function, Z(X Γ , u) is a rational function and it satisfies the Riemann hypothesis if and only if X Γ is a Ramanujan complex.
متن کاملExpander Graphs and Gaps between Primes∗
The explicit construction of infinite families of d-regular graphs which are Ramanujan is known only in the case d−1 is a prime power. In this paper, we consider the case when d− 1 is not a prime power. The main result is that by perturbing known Ramanujan graphs and using results about gaps between consecutive primes, we are able to construct infinite families of “almost” Ramanujan graphs for ...
متن کاملExpander graphs and gaps between primes * Sebastian
The explicit construction of infinite families of d-regular graphs which are Ramanujan is known only in the case d 1 is a prime power. In this paper, we consider the case when d 1 is not a prime power. The main result is that by perturbing known Ramanujan graphs and using results about gaps between consecutive primes, we are able to construct infinite families of ‘‘almost’’ Ramanujan graphs for...
متن کامل