5,930 research outputs found
The twin prime conjecture
This is an exposition of recent developments in the theory of bounded
differences between primes. Readers are expected to be beginners of analytic
number theory.
The present text is a substantially improved and augmented version of the one
that I had prepared for my talk which I delivered at the Annual Meeting of the
Mathematical Society of Japan on 15 March 2014.Comment: 17 page
The Hyperbolic Lattice Point Count in Infinite Volume with Applications to Sieves
We develop novel techniques using abstract operator theory to obtain
asymptotic formulae for lattice counting problems on infinite-volume hyperbolic
manifolds, with error terms which are uniform as the lattice moves through
"congruence" subgroups.
We give the following application to the theory of affine linear sieves. In
the spirit of Fermat, consider the problem of primes in the sum of two squares,
f(c,d)=c^2+d^2, but restrict (c,d) to the orbit O = (0,1).Gamma, where Gamma is
an infinite-index non-elementary finitely-generated subgroup of SL(2,Z). Assume
that the Reimann surface Gamma\H^2 has a cusp at infinity. We show that the set
of values f(O) contains infinitely many integers having at most R prime factors
for any R>4/(delta-theta), where theta>1/2 is the spectral gap and delta<1 is
the Hausdorff dimension of the limit set of Gamma. If delta>149/150, then we
can take theta=5/6, giving R=25. The limit of this method is R=9 for
delta-theta>4/9. This is the same number of prime factors as attained in Brun's
original attack on the twin prime conjecture.Comment: 33 pages, 1 figure, minor corrections. To appear, Duke Math
Divisors of the Euler and Carmichael functions
We study the distribution of divisors of Euler's totient function and
Carmichael's function. In particular, we estimate how often the values of these
functions have "dense" divisors.Comment: v.3, 11 pages. To appear in Acta Arithmetica. Very small corrections
and changes suggested by the referee. Added abstract, keywords, MS
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