1,823 research outputs found
Multiplicative Congruences with Variables from Short Intervals
Recently, several bounds have been obtained on the number of solutions to
congruences of the type modulo a prime with variables
from some short intervals. Here, for almost all and all and also for a
fixed and almost all , we derive stronger bounds. We also use similar
ideas to show that for almost all primes, one can always find an element of a
large order in any rather short interval
Equidistribution of cusp forms in the level aspect
Let f traverse a sequence of classical holomorphic newforms of fixed weight
and increasing squarefree level q tending to infinity. We prove that the
pushforward of the mass of f to the modular curve of level 1 equidistributes
with respect to the Poincar\'{e} measure.
Our result answers affirmatively the squarefree level case of a conjecture
spelled out by Kowalski, Michel and Vanderkam (2002) in the spirit of a
conjecture of Rudnick and Sarnak (1994).
Our proof follows the strategy of Holowinsky and Soundararajan (2008) who
show that newforms of level 1 and large weight have equidistributed mass. The
new ingredients required to treat forms of fixed weight and large level are an
adaptation of Holowinsky's reduction of the problem to one of bounding shifted
sums of Fourier coefficients (which on the surface makes sense only in the
large weight limit), an evaluation of the p-adic integral needed to extend
Watson's formula to the case of three newforms where the level of one divides
but need not equal the common squarefree level of the other two, and some
additional technical work in the problematic case that the level has many small
prime factors.Comment: 24 pages; slightly expanded, nearly accepted for
Almost all primes have a multiple of small Hamming weight
Recent results of Bourgain and Shparlinski imply that for almost all primes
there is a multiple that can be written in binary as with or ,
respectively. We show that (corresponding to Hamming weight )
suffices.
We also prove there are infinitely many primes with a multiplicative
subgroup , for some
, of size , where the sum-product set
does not cover completely
Constructing Carmichael numbers through improved subset-product algorithms
We have constructed a Carmichael number with 10,333,229,505 prime factors,
and have also constructed Carmichael numbers with k prime factors for every k
between 3 and 19,565,220. These computations are the product of implementations
of two new algorithms for the subset product problem that exploit the
non-uniform distribution of primes p with the property that p-1 divides a
highly composite \Lambda.Comment: Table 1 fixed; previously the last 30 digits and number of digits
were calculated incorrectl
Evasive Properties of Sparse Graphs and Some Linear Equations in Primes
We give an unconditional version of a conditional, on the Extended Riemann
Hypothesis, result of L. Babai, A. Banerjee, R. Kulkarni and V. Naik (2010) on
the evasiveness of sparse graphs.Comment: This version corrects a mistake made in the previous version, which
was pointed out to the author by Laszlo Baba
Galois cohomology of a number field is Koszul
We prove that the Milnor ring of any (one-dimensional) local or global field
K modulo a prime number l is a Koszul algebra over Z/l. Under mild assumptions
that are only needed in the case l=2, we also prove various module Koszulity
properties of this algebra. This provides evidence in support of Koszulity
conjectures that were proposed in our previous papers. The proofs are based on
the Class Field Theory and computations with quadratic commutative Groebner
bases (commutative PBW-bases).Comment: LaTeX 2e, 25 pages; v.2: minor grammatic changes; v.3: classical
references added, remark inserted in subsection 1.6, details of arguments
added in subsections 1.4, 1.7 and sections 5 and 6; v.4: still more misprints
corrected, acknowledgement updated, a sentence inserted in section 4, a
reference added -- this is intended as the final versio
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