1,693,946 research outputs found
A proof of the Riemann hypothesis based on the Koch theorem, on primes in short intervals, and distribution of nontrivial zeros of the Riemann zeta function
Part One: Let define the truncation of the logarithmic integral as First, we
prove which implies that the point of
the truncation depends on x, Next, let . We prove
that is greater than for and tends to
as . Thirdly, we prove Finally, we prove
Part Two: Let define where we proved that the pair of
numbers and in satisfy inequalities
, and the number is approximately a
step function of the variable with a finite amount of deviation, and
proportional to . We obtain much more accurate estimation
of prime numbers, the error range of which is less than
for or less than for
.
Part Three: We show the closeness of and and give the
difference being less than or equal to where
is a constant. Further more, we prove the estimation
. Hence we obtain so
that the Riemann hypothesis is true.
Part Four: Different from former researches on the distribution of primes in
short intervals, we prove a theorem: Let , ,
and which satisfies . Then there are and Comment: 95 page
Effect of isoelectronic doping on honeycomb lattice iridate A_2IrO_3
We have investigated experimentally and theoretically the series
(NaLi)IrO. Contrary to what has been believed so far,
only for the system forms uniform solid solutions. For larger Li
content, as evidenced by powder X-ray diffraction, scanning electron microscopy
and density functional theory calculations, the system shows a miscibility gap
and a phase separation into an ordered NaLiIrO phase with
alternating Na and LiIrO planes, and a Li-rich phase close to pure
LiIrO. For we observe (1) an increase of with Li
doping up to , despite the fact that in pure LiIrO is
smaller than in NaIrO, and (2) a gradual reduction of the
antiferromagnetic ordering temperature and ordered moment. The
previously proposed magnetic quantum phase transition at may
occur in a multiphase region and its nature needs to be re-evaluated.Comment: 8 pages, 9 figures including supplemental informatio
Average prime-pair counting formula
Taking , let denote the number of prime pairs
with . The prime-pair conjecture of Hardy and Littlewood (1923) asserts
that with an explicit constant
. There seems to be no good conjecture for the remainders
that corresponds to
Riemann's formula for . However, there is a heuristic
approximate formula for averages of the remainders which is
supported by numerical results.Comment: 26 pages, 6 figure
A new bound for the smallest with
We reduce the leading term in Lehman's theorem. This improved estimate allows
us to refine the main theorem of Bays and Hudson. Entering Riemann
zeros, we prove that there exists in the interval for which \pi(x)-\li(x) > 3.2 \times 10^{151}. There are at
least successive integers in this interval for which
\pi(x)>\li(x). This interval is strictly a sub-interval of the interval in
Bays and Hudson, and is narrower by a factor of about 12.Comment: Final version, to be published in the International Journal of Number
Theory [copyright World Scientific Publishing
Company][www.worldscinet.com/ijnt
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
and ,
respectively, \mbox{li}(x) is the logarithmic integral, and
are the M\"obius and the Von Mangoldt functions). Negative jumps
in the above functions , and
may potentially occur only at (the set of primes). One
denotes j_p=\mbox{li}(p)-\mbox{li}(p-1) and one investigates the jumps ,
and . In particular, , and
for . Besides, for any odd p \in
\mathcal{\mbox{Ch}}, an infinite set of so-called {\it Chebyshev primes } with
partial list . We establish a few properties of the set
\mathcal{\mbox{Ch}}, give accurate approximations of the jump
and relate the derivation of \mbox{Ch} to the explicit Mangoldt formula for
. In the context of RH, we introduce the so-called {\it Riemann
primes} as champions of the function (or of the function
). 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
The Energetics of Li Off-Centering in KLiTaO; First Principles Calculations
KLiTaO (KLT) solid solutions exhibit a variety of
interesting physical phenomena related to large displacements of Li-ions from
ideal perovskite A-site positions. First-principles calculations for KLT
supercells were used to investigate these phenomena. Lattice dynamics
calculations for KLT exhibit a Li off-centering instability. The energetics of
Li-displacements for isolated Li-ions and for Li-Li pairs up to 4th neighbors
were calculated. Interactions between nearest neighbor Li-ions, in a Li-Li
pair, strongly favor ferroelectric alignment along the pair axis. Such Li-Li
pairs can be considered "seeds" for polar nanoclusters in KLT.
Electrostriction, local oxygen relaxation, coupling to the KT soft-mode, and
interactions with neighboring Li-ions all enhance the polarization from Li
off-centering. Calculated hopping barriers for isolated Li-ions and for nearest
neighbor Li-Li pairs are in good agreement with Arrhenius fits to experimental
dielectric data.Comment: 14 pages including 10 figures. To Physical Review B. Replaced after
corrections due to referees' remark
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