227 research outputs found
Primes from sums of two squares and missing digits
Let be the set of integers missing any three fixed digits from
their decimal expansion. We produce primes in a thin sequence by proving an
asymptotic formula for counting primes of the form , with
.
The proof draws on ideas from the work of Friedlander-Iwaniec on primes of
the form , as well as ideas from the work of Maynard on primes
with restricted digits.Comment: 60 page
Landau-Siegel zeros and their illusory consequences
Updated time
Abstract: Researchers have tried for many years to eliminate the possibility of LandauSiegel zeros—certain exceptional counterexamples to the Generalized Riemann Hypothesis. Often one thinks of these zeros as being a severe nuisance, but there are many situations in which their existence allows one to prove spectacular, though illusory, results. I will review some of this history and some of these results. In the latter portion of the talk I will discuss recent work, joint with H. M. Bui and Alexandru Zaharescu, in which we show that the existence of Landau-Siegel zeros has implications for the behavior of Dirichlet L-functions at the central point
The irrationality of a divisor function series of Erd\H{o}s and Kac
For positive integers and let denote the sum of the
th powers of the divisors of . Erd\H{o}s and Kac asked whether, for every
, the number is
irrational. It is known unconditionally that is irrational if . We prove is irrational.Comment: 28 page
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