124 research outputs found
Some bounds and limits in the theory of Riemann's zeta function
For any real a>0 we determine the supremum of the real \sigma\ such that
\zeta(\sigma+it) = a for some real t. For 0 1 the
results turn out to be quite different.}
We also determine the supremum E of the real parts of the `turning points',
that is points \sigma+it where a curve Im \zeta(\sigma+it) = 0 has a vertical
tangent. This supremum E (also considered by Titchmarsh) coincides with the
supremum of the real \sigma\ such that \zeta'(\sigma+it) = 0 for some real t.
We find a surprising connection between the three indicated problems:
\zeta(s) = 1, \zeta'(s) = 0 and turning points of \zeta(s). The almost extremal
values for these three problems appear to be located at approximately the same
height.Comment: 28 pages 1 figur
Algorithms for determining integer complexity
We present three algorithms to compute the complexity of all
natural numbers . The first of them is a brute force algorithm,
computing all these complexities in time and space . The
main problem of this algorithm is the time needed for the computation. In 2008
there appeared three independent solutions to this problem: V. V. Srinivas and
B. R. Shankar [11], M. N. Fuller [7], and J. Arias de Reyna and J. van de Lune
[3]. All three are very similar. Only [11] gives an estimation of the
performance of its algorithm, proving that the algorithm computes the
complexities in time , where . The other two algorithms, presented in [7] and
[3], were very similar but both superior to the one in [11]. In Section 2 we
present a version of these algorithms and in Section 4 it is shown that they
run in time and space . (Here ).
In Section 2 we present the algorithm of [7] and [3]. The main advantage of
this algorithm with respect to that in [11] is the definition of kMax in
Section 2.7. This explains the difference in performance from
to .
In Section 3 we present a detailed description a space-improved algorithm of
Fuller and in Section 5 we prove that it runs in time and space
, where and
.Comment: 21 pages. v2: We improved the computations to get a better bound for
$\alpha
On the exact location of the non-trivial zeros of Riemann's zeta function
In this paper we introduce the real valued real analytic function kappa(t)
implicitly defined by exp(2 pi i kappa(t)) = -exp(-2 i theta(t)) *
(zeta'(1/2-it)/zeta'(1/2+it)) and kappa(0)=-1/2. (where theta(t) is the
function appearing in the known formula zeta(1/2+it)= Z(t) * e^{-i theta(t)}).
By studying the equation kappa(t) = n (without making any unproved hypotheses),
we will show that (and how) this function is closely related to the (exact)
position of the zeros of Riemann's zeta(s) and zeta'(s). Assuming the Riemann
hypothesis and the simplicity of the zeros of zeta(s), it will follow that the
ordinate of the zero 1/2 + i gamma_n of zeta(s) will be the unique solution to
the equation kappa(t) = n.Comment: 28 pages, 9 figures. Added a referenc
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