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Jan van de Lune

Publications and source records attributed to Jan van de Lune.

5 recordsLinked to original sources
arXiv2014

A note on the real part of the Riemann zeta-function

We consider the real part $\Re(ζ(s))$ of the Riemann zeta-function $ζ(s)$ in the half-plane $\Re(s) \ge 1$. We show how to compute accurately the constant $σ_0 = 1.19\ldots$ which is defined to be the supremum of $σ$ such that $\Re(ζ(σ+it))$ can be negative (or zero) for some real $t$. We also consider intervals where $\Re(ζ(1+it)) \le 0$ and show that they are rare. The first occurs for $t$ approximately 682112.9, and has length about 0.05. We list the first fifty such intervals.

math.NT↗
arXiv2013

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.

math.NT↗
arXiv2012

On the sign of the real part of the Riemann zeta-function

We consider the distribution of $\argζ(σ+it)$ on fixed lines $σ> \frac12$, and in particular the density \[d(σ) = \lim_{T \rightarrow +\infty} \frac{1}{2T} |\{t \in [-T,+T]: |\argζ(σ+it)| > π/2\}|\,,\] and the closely related density \[d_{-}(σ) = \lim_{T \rightarrow +\infty} \frac{1}{2T} |\{t \in [-T,+T]: \Reζ(σ+it) < 0\}|\,.\] Using classical results of Bohr and Jessen, we obtain an explicit expression for the characteristic function $ψ_σ(x)$ associated with $\argζ(σ+it)$. We give explicit expressions for $d(σ)$ and $d_{-}(σ)$ in terms of $ψ_σ(x)$. Finally, we give a practical algorithm for evaluating these expressions to obtain accurate numerical values of $d(σ)$ and $d_{-}(σ)$.

math.NT↗
arXiv2010

A glimpse inside the mathematical kitchen

We prove the inequality sum_{k=1}^infty (-1)^{k+1} r^k cos(k*phi) (k+2)^{-1} < sum_{k=1}^infty(-1)^{k+1} r^k (k+2)^{-1} for 0 < r <= 1 and 0 < phi < pi. For the case r = 1 we give two proofs. The first one is by means of a general numerical technique (maximal slope principle) for proving inequalities between elementary functions. The second proof is fully analytical. Finally we prove a general rearrangement theorem and apply it to the remaining case 0 < r < 1. Some of these inequalities are needed for obtaining general sharp bounds for the errors committed when applying the Riemann-Siegel expansion of Riemann's zeta function.

math.CA↗