279 research outputs found
Bounds on the number of Diophantine quintuples
We consider Diophantine quintuples . These are sets of
distinct positive integers, the product of any two elements of which is one
less than a perfect square. It is conjectured that there are no Diophantine
quintuples; we improve on current estimates to show that there are at most
Diophantine quintuples.Comment: 16 page
A modest improvement on the function
This paper contains a small improvement to the explicit bounds on the growth
of the function . It is shown how more substantial improvements are
possible if one has better explicit bounds on the growth of
.Comment: 10 page
The sum of the unitary divisor function
This article establishes a new upper bound on the function ,
the sum of all coprime divisors of . The article concludes with two
questions concerning this function.Comment: 6 pages, to appear in Publ. Inst. Math. (Beograd) (N.S.
A short extension of two of Spira's results
Two inequalities concerning the symmetry of the zeta-function and the
Ramanujan -function are improved through the use of some elementary
considerations.Comment: 4 pages; to appear in J. Math. Inequa
Improvements to Turing's Method
This paper refines the argument of Lehman by reducing the size of the
constants in Turing's method. This improvement is given in Theorem 1 and scope
for further improvements is also given. Analogous improvements to Dirichlet
L-functions and Dedekind zeta-functions are also included.Comment: 21 pages, third edition: expanded to include sections on Dirichlet
L-functions and Dedekind zeta-function
Between the conjectures of P\'{o}lya and Tur\'{a}n
This paper is concerned with the constancy in the sign of , where the Liouville
function. The non-positivity of is the P\'{o}lya conjecture, and the
non-negativity of is the Tur\'{a}n conjecture --- both of which are
false. By constructing an auxiliary function, evidence is provided that is the best contender for constancy in sign. The core of this
paper is the conjecture that for all :
this has been verified for .Comment: 5 page
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