2,642 research outputs found
A variant of the Euclid-Mullin sequence containing every prime
We consider a generalization of Euclid's proof of the infinitude of primes
and show that it leads to variants of the Euclid-Mullin sequence that provably
contain every prime number.Comment: 5 pages, submitte
Finite connected components of the aliquot graph
Conditional on a strong form of the Goldbach conjecture, we determine all
finite connected components of the aliquot graph containing a number less than
, as well as those containing an amicable pair below or one of
the known perfect or sociable cycles below . Along the way we develop
a fast algorithm for computing the inverse image of an even number under the
sum-of-proper-divisors function.Comment: 10 pages, to appear in Mathematics of Computatio
A converse theorem without root numbers
We answer a challenge posed in (Math. Ann. 363 (2015), no. 1-2, 423-454) by
proving a version of Weil's converse theorem that assumes a functional equation
for character twists but allows their root numbers to vary arbitrarily.Comment: 10 pages, to appear in Mathematik
L-functions as distributions
We define an axiomatic class of L-functions extending the Selberg class. We
show in particular that one can recast the traditional conditions of an Euler
product, analytic continuation and functional equation in terms of
distributional identities akin to Weil's explicit formula. The generality of
our approach enables some new applications; for instance, we show that the
L-function of any cuspidal automorphic representation of GL_3(A_Q) has
infinitely many zeros of odd order.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/s00208-015-1178-
A note on Maass forms of icosahedral type
Using ideas of Ramakrishnan, we consider the icosahedral analogue of the
theorems of Sarnak and Brumley on Hecke-Maass newforms with Fourier
coefficients in a quadratic order. Although we are unable to conclude the
existence of an associated Galois representation in this case, we show that one
can deduce some implications of such an association, including weak automorphy
of all symmetric powers and the value distribution of Fourier coefficients
predicted by the Chebotarev density theorem.Comment: 9 pages, to appear in Mathematische Zeitschrif
Rapid computation of -functions attached to Maass forms
Let be a degree- -function associated to a Maass cusp form. We
explore an algorithm that evaluates values of on the critical line in
time . We use this algorithm to rigorously compute an
abundance of consecutive zeros and investigate their distribution
Squarefree smooth numbers and Euclidean prime generators
We show that for each prime p > 7, every residue mod p can be represented by
a squarefree number with largest prime factor at most p. We give two
applications to recursive prime generators akin to the one Euclid used to prove
the infinitude of primes.Comment: 8 pages, to appear in Proceedings of the AM
- …