16,611 research outputs found
Non-vanishing of Dirichlet series without Euler products
We give a new proof that the Riemann zeta function is nonzero in the
half-plane . A novel feature of this proof is
that it makes no use of the Euler product for .Comment: 13 pages; some minor edits of the previous versio
Supersymmetry, the Cosmological Constant and a Theory of Quantum Gravity in Our Universe
There are many theories of quantum gravity, depending on asymptotic boundary
conditions, and the amount of supersymmetry. The cosmological constant is one
of the fundamental parameters that characterize different theories. If it is
positive, supersymmetry must be broken. A heuristic calculation shows that a
cosmological constant of the observed size predicts superpartners in the TeV
range. This mechanism for SUSY breaking also puts important constraints on low
energy particle physics models. This essay was submitted to the Gravity
Research Foundation Competition and is based on a longer article, which will be
submitted in the near future
Optimal primitive sets with restricted primes
A set of natural numbers is primitive if no element of the set divides
another. Erd\H{o}s conjectured that if S is any primitive set, then \sum_{n\in
S} 1/(n log n) \le \sum_{n\in \P} 1/(p log p), where \P denotes the set of
primes. In this paper, we make progress towards this conjecture by restricting
the setting to smaller sets of primes. Let P denote any subset of \P, and let
N(P) denote the set of natural numbers all of whose prime factors are in P. We
say that P is Erd\H{o}s-best among primitive subsets of N(P) if the inequality
\sum_{n\in S} 1/(n log n) \le \sum_{n\in P} 1/(p log p) holds for every
primitive set S contained in N(P). We show that if the sum of the reciprocals
of the elements of P is small enough, then P is Erd\H{o}s-best among primitive
subsets of N(P). As an application, we prove that the set of twin primes
exceeding 3 is Erd\H{o}s-best among the corresponding primitive sets. This
problem turns out to be related to a similar problem involving multiplicative
weights. For any real number t>1, we say that P is t-best among primitive
subsets of N(P) if the inequality \sum_{n\in S} n^{-t} \le \sum_{n\in P} p^{-t}
holds for every primitive set S contained in N(P). We show that if the sum on
the right-hand side of this inequality is small enough, then P is t-best among
primitive subsets of N(P).Comment: 10 page
Sums and Products with Smooth Numbers
We estimate the sizes of the sumset A + A and the productset A A in
the special case that A = S (x, y), the set of positive integers n less than or
equal to x, free of prime factors exceeding y.Comment: 12 page
Sato--Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height
We obtain asymptotic formulae for the number of primes for which the
reduction modulo of the elliptic curve \E_{a,b} : Y^2 = X^3 + aX + b
satisfies certain ``natural'' properties, on average over integers and
with and , where and are small relative to .
Specifically, we investigate behavior with respect to the Sato--Tate
conjecture, cyclicity, and divisibility of the number of points by a fixed
integer
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