132 research outputs found
The irrationality of some number theoretical series
We prove the irrationality of some factorial series. To do so we combine
methods from elementary and analytic number theory with methods from the theory
of uniform distribution
The exponential sum over squarefree integers
We give an upper bound for the exponential sum over squarefree integers. This
establishes a conjecture by Br\"udern and Perelli
Partitions which are p- and q-core
Let p and q be distinct primes, n an integer with n > p2q2. Then there is no partition of n which is at the same time p- and q-core. Hence there is no irreducible representation of Sn which is of p- and q-defect zero at the same time
Sets with more differences than sums
We show that a random set of integers with density 0 has almost always more
differences than sums. This proves a conjecture by Martin and O'Bryant
An inequality for means with applications
We show that an almost trivial inequality for the first and second mean of a
random variable can be used to give non-trivial improvements on deep results.
As applications we improve on results on lower bounds for the Riemann
zeta-function on the critical line, the determinant of a skew-symmetric matrix
with entries , and on the maximal order of an irreducible character of
the symmetric group
The order of elements in Sylow -subgroups of the symmetric group
Define a random variable by choosing a conjugacy class of the
Sylow -subgroup of by random, and let be the logarithm of
the order of an element in . We show that has bounded variance and
mean order , which differs significantly from the
average order of elements chosen with equal probability
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