Bounds on the number of Diophantine quintuples

We consider Diophantine quintuples $\{a, b, c, d, e\}$. 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 $1.9\cdot 10^{29}$ Diophantine quintuples.Comment: 16 page

The sum of the unitary divisor function

This article establishes a new upper bound on the function $\sigma^{*}(n)$, the sum of all coprime divisors of $n$. 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 $\tau$-function are improved through the use of some elementary considerations.Comment: 4 pages; to appear in J. Math. Inequa

Searching for Diophantine quintuples

We consider Diophantine quintuples $\{a, b, c, d, e\}$. 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 $1.18\cdot 10^{27}$ Diophantine quintuples.Comment: 15 page

Linear relations of zeroes of the zeta-function

This article considers linear relations between the non-trivial zeroes of the Riemann zeta-function. The main application is an alternative disproof to Mertens' conjecture. We show that $\limsup M(x)x^{-1/2} \geq 1.6383$ and that $\liminf M(x)x^{-1/2}\leq -1.6383$.Comment: 12 pages, 2 figures, 2 tables. Version 2: some typos corrected. To appear in Math. Com

Linnik's approximation to Goldbach's conjecture, and other problems

We examine the problem of writing every sufficiently large even number as the sum of two primes and at most $K$ powers of 2. We outline an approach that only just falls short of improving the current bounds on $K$. Finally, we improve the estimates in other Waring--Goldbach problems.Comment: Second version: 10 pages, another problem adde