39 research outputs found
An improved upper bound for the argument of the Riemann zeta-function on the critical line
No full textBounds on the number of Diophantine quintuples
No full textWe consider Diophantine quintuples {a,b,c,d,e}{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 2.3⋅10292.3⋅1029 Diophantine quintuples
An improved upper bound for the argument of the Riemann zeta-function on the critical line
No full textThis paper concerns the function S(t), the argument of the Riemann
zeta-function along the critical line. Improving on the method of Backlund,
and taking into account the refinements of Rosser and McCurley it is
proved that for sufficiently large t,
|S(t)| ≤ 0.1013 log t. Theorem 2 makes the above result explicit, viz. it enables one to select values
of a and b such that, for t>t0,
|S(t)| ≤ a + b log t
Improvements to Turing’s method
No full textThis article improves the estimate of the size of the definite integral of
S
(
t
)
S(t)
, the argument of the Riemann zeta-function. The primary application of this improvement is Turing’s Method for the Riemann zeta-function. Analogous improvements are given for the arguments of Dirichlet
L
L
-functions and of Dedekind zeta-functions.</p
A new upper bound for |ζ(1+it)|
No full textIt is known that ζ(1+ it) 1. This paper provides a new explicit estimate ζ(1+ it) ≤ 3/4 log t, for t≥ 3. This gives the best upper bound on ζ (1+ it) for t≤ 10 2̇ 105
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 Log-Free Zero-Density Estimate and Small Gaps in Coefficients of L-Functions
No full textLet L(s, π × π ′) be the Rankin–Selberg L-function attached to automorphic representa-tions π and π ′. Let π ̃ and π ̃ ′ denote the contragredient representations associated to π and π ′. Under the assumption of certain upper bounds for coefficients of the log-arithmic derivatives of L(s, π × π̃) and L(s, π ′ × π ̃ ′), we prove a log-free zero-density estimate for L(s, π × π ′) which generalizes a result due to Fogels in the context of Dirichlet L-functions. We then employ this log-free estimate in studying the distribu-tion of the Fourier coefficients of an automorphic representation π. As an application, we examine the nonlacunarity of the Fourier coefficients bf (p) of a modular newform f(z) =∑∞n=1 bf (n) e2πinz of weight k, level N, and character χ. More precisely, for f(z) and a prime p, set jf (p):=maxx;x>p Jf (p, x), where Jf (p, x): = #{prime q; aπ (q) = 0 for all p< q ≤ x}. We prove that jf (p) f,θ pθ for some 0< θ < 1.
Further results on Gram's Law
No full textThis thesis shows that Gram's Law and the Rosser Rule (methods for locating zeroes of the Riemann zeta-function) fail in a positive proportion of cases. A weaker version of Gram's Law is shown to be true in a positive proportion of cases. Also included are theorems on Turing's Method and its extensions to Dirichlet L-functions and Dedekind zeta-functions
