Efficient prime counting and the Chebyshev primes

The function \epsilon(x)=\mbox{li}(x)-\pi(x) is known to be positive up to the (very large) Skewes' number. Besides, according to Robin's work, the functions \epsilon_{\theta}(x)=\mbox{li}[\theta(x)]-\pi(x) and \epsilon_{\psi}(x)=\mbox{li}[\psi(x)]-\pi(x) are positive if and only if Riemann hypothesis (RH) holds (the first and the second Chebyshev function are $\theta(x)=\sum_{p \le x} \log p$ and $\psi(x)=\sum_{n=1}^x \Lambda(n)$, respectively, \mbox{li}(x) is the logarithmic integral, $\mu(n)$ and $\Lambda(n)$ are the M\"obius and the Von Mangoldt functions). Negative jumps in the above functions $\epsilon$, $\epsilon_{\theta}$ and $\epsilon_{\psi}$ may potentially occur only at $x+1 \in \mathcal{P}$ (the set of primes). One denotes j_p=\mbox{li}(p)-\mbox{li}(p-1) and one investigates the jumps $j_p$, $j_{\theta(p)}$ and $j_{\psi(p)}$. In particular, $j_p<1$, and $j_{\theta(p)}>1$ for $p<10^{11}$. Besides, $j_{\psi(p)}<1$ for any odd p \in \mathcal{\mbox{Ch}}, an infinite set of so-called {\it Chebyshev primes } with partial list $\{109, 113, 139, 181, 197, 199, 241, 271, 281, 283, 293, 313, 317, 443, 449, 461, 463, \ldots\}$. We establish a few properties of the set \mathcal{\mbox{Ch}}, give accurate approximations of the jump $j_{\psi(p)}$ and relate the derivation of \mbox{Ch} to the explicit Mangoldt formula for $\psi(x)$. In the context of RH, we introduce the so-called {\it Riemann primes} as champions of the function $\psi(p_n^l)-p_n^l$ (or of the function $\theta(p_n^l)-p_n^l$ ). Finally, we find a {\it good} prime counting function S_N(x)=\sum_{n=1}^N \frac{\mu(n)}{n}\mbox{li}[\psi(x)^{1/n}], that is found to be much better than the standard Riemann prime counting function.Comment: 15 pages section 2.2 added, new sequences added, Fig. 2 and 3 are ne

Chebyshev's bias and generalized Riemann hypothesis

It is well known that $li(x)>\pi(x)$ (i) up to the (very large) Skewes' number $x_1 \sim 1.40 \times 10^{316}$ \cite{Bays00}. But, according to a Littlewood's theorem, there exist infinitely many $x$ that violate the inequality, due to the specific distribution of non-trivial zeros $\gamma$ of the Riemann zeta function $\zeta(s)$, encoded by the equation $li(x)-\pi(x)\approx \frac{\sqrt{x}}{\log x}[1+2 \sum_{\gamma}\frac{\sin (\gamma \log x)}{\gamma}]$ (1). If Riemann hypothesis (RH) holds, (i) may be replaced by the equivalent statement $li[\psi(x)]>\pi(x)$ (ii) due to Robin \cite{Robin84}. A statement similar to (i) was found by Chebyshev that $\pi(x;4,3)-\pi(x;4,1)>0$ (iii) holds for any $x<26861$ \cite{Rubin94} (the notation $\pi(x;k,l)$ means the number of primes up to $x$ and congruent to $l\mod k$). The {\it Chebyshev's bias}(iii) is related to the generalized Riemann hypothesis (GRH) and occurs with a logarithmic density $\approx 0.9959$ \cite{Rubin94}. In this paper, we reformulate the Chebyshev's bias for a general modulus $q$ as the inequality $B(x;q,R)-B(x;q,N)>0$ (iv), where $B(x;k,l)=li[\phi(k)*\psi(x;k,l)]-\phi(k)*\pi(x;k,l)$ is a counting function introduced in Robin's paper \cite{Robin84} and $R$ resp. $N$) is a quadratic residue modulo $q$ (resp. a non-quadratic residue). We investigate numerically the case $q=4$ and a few prime moduli $p$. Then, we proove that (iv) is equivalent to GRH for the modulus $q$.Comment: 9 page

Electrode thickness measurement of a Si(Li) detector for the SIXA array

Cathode electrodes of the Si(Li) detector elements of the SIXA X-ray spectrometer array are formed by gold-palladium alloy contact layers. The equivalent thickness of gold in one element was measured by observing the characteristic L-shell X-rays of gold excited by monochromatised synchrotron radiation with photon energies above the L3 absorption edge of gold. The results obtained at 4 different photon energies below the L2 edge yield an average value of 22.4(35) nm which is consistent with the earlier result extracted from detection efficiency measurements. PACS: 29.40.Wk; 85.30.De; 07.85.Nc; 95.55.Ka Keywords: Si(Li) detectors, X-ray spectrometers, X-ray fluorescence, detector calibration, gold electrodes, synchrotron radiationComment: 10 pages, 4 PostScript figures, uses elsart.sty, submitted to Nucl. Instrum. Meth.

The superwind mass-loss rate of the metal-poor carbon star LI-LMC 1813 in the LMC cluster KMHK 1603

LI-LMC 1813 is a dust-enshrouded Asymptotic Giant Branch (AGB) star, located in the small open cluster KMHK 1603 near the rim of the Large Magellanic Cloud (LMC). Optical and infrared photometry between 0.5 and 60 micron is obtained to constrain the spectral energy distribution of LI-LMC 1813. Near-infrared spectra unambiguously show it to be a carbon star. Modelling with the radiation transfer code Dusty yields accurate values for the bolometric luminosity, L=1.5 x 10^4 Lsun, and mass-loss rate, Mdot=3.7(+/-1.2) x 10^-5 Msun/yr. On the basis of colour-magnitude diagrams, the age of the cluster KMHK 1603 is estimated to be t=0.9-1.0 Gyr, which implies a Zero-Age Main Sequence mass for LI-LMC 1813 of M(ZAMS)=2.2+/-0.1 Msun. This makes LI-LMC 1813 arguably the object with the most accurately and reliably determined (circum)stellar parameters amongst all carbon stars in the superwind phase.Comment: Accepted for publication in MNRAS (better quality figure 1 on request from jacco

Towards symmetric scheme for superdense coding between multiparties

Recently Liu, Long, Tong and Li [Phys. Rev. A 65, 022304 (2002)] have proposed a scheme for superdense coding between multiparties. This scheme seems to be highly asymmetric in the sense that only one sender effectively exploits entanglement. We show that this scheme can be modified in order to allow more senders to benefit of the entanglement enhanced information transmission.Comment: 6 page