25 research outputs found
Square-full primitive roots
No full text We use character sum estimates to give some bounds on the least square-full primitive root modulo a prime. In particular, we show that there is a square-full primitive root mod [Formula: see text] less than [Formula: see text]. </jats:p
Twin progress in number theory
No full textThere are many jokes of the form "X's are like buses: you wait ages for one
and then n show up at once." There appear to be many admissible values of
{X,n}: {Ashes series,2}, {efficiency dividends, (m, where m > ∞)}.
Normally, when X is progress on a problem in number theory, n is a non-
negative integer, strictly less than unity. Therefore it was with ebullience that,
on the same day, I read of the proof of a centuries-old problem, and of admirable,
and completely unexpected, progress towards a millenia-old problem. This short
note attempts to explain the two problems and to give a brief outline of the
methods used to tackle them.Copyright Information: http://www.sherpa.ac.uk/romeo/issn/0311-0729/author can archive pre-print (ie pre-refereeing); author cannot archive post-print (ie final draft post-refereeing); subject to two year embargo, author can archive publisher's version/PD
A wishlist for Diophantine quintuples (Analytic Number Theory and Related Areas)
Get PDFWe are concerned with Diophantine quintuples, that is, sets {a, b, c, d, e} of distinct positive integers the product of any two of which is one less than a perfect square. We present some recent progress in the area and give a series of 9 wishes for future research on the topic
An improved upper bound for the argument of the Riemann zeta-function on the critical line II
No full textThis paper concerns the function , where is the argument of the Riemann zeta-function along the critical line. The main result is that \begin{equation*} |S(T)| \leq 0.112\log T + 0.278\log \log T + 2.510, \end{equation*}
which holds for all .Supported by ARC Grant DE120100173
An improved upper bound for the error in the zero-counting formulae for Dirichlet L-functions and Dedekind zeta-functions
Get PDFThis paper contains new explicit upper bounds for the number of zeroes of Dirichlet L-functions and Dedekind zeta-functions in rectangles.Supported by Australian Research Council DECRA Grant DE12010017
Improved bounds on Brun's constant
Get PDFBrun’s constant is B = Pp∈P2p−1 + (p + 2)−1, where the summation is over all twin primes. We improve the unconditional bounds on Brun’s constant to 1.840503 < B < 2.288490, which are about 13% tighter
Searching for Diophantine quintuples
Get PDFWe consider Diophantine quintuples {a,b,c,d,e}. These are sets of 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 5.441⋅10²⁶ Diophantine quintuples
Topics in explicit number theory
No full textThis doctoral thesis is made up of a collection of papers of the author. All of these papers are already published. The structure of this thesis is as follows. After the introduction, there are four chapters.
In Chapters 2 and 3, we obtain explicit bounds for the number of non-trivial zeros of the Riemann zeta function and Dedekind zeta functions, which improve previous results of Trudgian. This improvement is based on ideas from previous works of Bennett et al., Kadiri and Ng and Trudgian. This is a joint work with Quanli Shen and Peng-Jie Wong.
In Chapter 4, we apply explicit results from transcendental number theory due to Bugeaud, Laurent and Matveev to completely solve a Diophantine inequality involving the Fibonacci numbers and to study a particular case of the Terai-Shinsho conjecture.
In Chapter 5, using elementary methods and explicit bounds for primes in arithmetic progressions due to Bennett et al. we study two Diophantine equations which involve multiplicative functions
