45 research outputs found
Integrals Over Polytopes, Multiple Zeta Values and Polylogarithms, and Euler's Constant
Let be the triangle with vertices (1,0), (0,1), (1,1). We study certain
integrals over , one of which was computed by Euler. We give expressions for
them both as a linear combination of multiple zeta values, and as a polynomial
in single zeta values. We obtain asymptotic expansions of the integrals, and of
sums of certain multiple zeta values with constant weight. We also give related
expressions for Euler's constant. In the final section, we evaluate more
general integrals -- one is a Chen (Drinfeld-Kontsevich) iterated integral --
over some polytopes that are higher-dimensional analogs of . This leads to a
relation between certain multiple polylogarithm values and multiple zeta
values.Comment: 19 pages, to appear in Mat Zametki. Ver 2.: Added Remark 3 on a Chen
(Drinfeld-Kontsevich) iterated integral; simplified Proposition 2; gave
reference for (19); corrected [16]; fixed typ
Summation of Series Defined by Counting Blocks of Digits
We discuss the summation of certain series defined by counting blocks of
digits in the -ary expansion of an integer. For example, if denotes
the sum of the base-2 digits of , we show that . We recover this previous
result of Sondow in math.NT/0508042 and provide several generalizations.Comment: 12 pages, Introduction expanded, references added, accepted by J.
Number Theor
Generalized Ramanujan Primes
In 1845, Bertrand conjectured that for all integers , there exists at
least one prime in . This was proved by Chebyshev in 1860, and then
generalized by Ramanujan in 1919. He showed that for any , there is a
(smallest) prime such that for all .
In 2009 Sondow called the th Ramanujan prime and proved the asymptotic
behavior (where is the th prime). In the present
paper, we generalize the interval of interest by introducing a parameter and defining the th -Ramanujan prime as the smallest integer
such that for all , there are at least primes in
. Using consequences of strengthened versions of the Prime Number
Theorem, we prove that exists for all and all , that as , and that the fraction of primes which
are -Ramanujan converges to . We then study finer questions related to
their distribution among the primes, and see that the -Ramanujan primes
display striking behavior, deviating significantly from a probabilistic model
based on biased coin flipping; this was first observed by Sondow, Nicholson,
and Noe in the case . This model is related to the Cramer model, which
correctly predicts many properties of primes on large scales, but has been
shown to fail in some instances on smaller scales.Comment: 13 pages, 2 tables, to appear in the CANT 2011 Conference
Proceedings. This is version 2.0. Changes: fixed typos, added references to
OEIS sequences, and cited Shevelev's preprin
Special Values of Generalized Polylogarithms
We study values of generalized polylogarithms at various points and
relationships among them. Polylogarithms of small weight at the points 1/2 and
-1 are completely investigated. We formulate a conjecture about the structure
of the linear space generated by values of generalized polylogarithms.Comment: 32 page
Nonlocal Dynamics of p-Adic Strings
We consider the construction of Lagrangians that might be suitable for
describing the entire -adic sector of an adelic open scalar string. These
Lagrangians are constructed using the Lagrangian for -adic strings with an
arbitrary prime number . They contain space-time nonlocality because of the
d'Alembertian in argument of the Riemann zeta function. We present a brief
review and some new results.Comment: 8 page
Monotone and fast computation of Euler’s constant
Abstract We construct sequences of finite sums ( l ˜ n ) n ≥ 0 and ( u ˜ n ) n ≥ 0 converging increasingly and decreasingly, respectively, to the Euler-Mascheroni constant γ at the geometric rate 1/2. Such sequences are easy to compute and satisfy complete monotonicity-type properties. As a consequence, we obtain an infinite product representation for 2 γ converging in a monotone and fast way at the same time. We use a probabilistic approach based on a differentiation formula for the gamma process
Graphene and non-Abelian quantization
In this article we employ a simple nonrelativistic model to describe the low
energy excitation of graphene. The model is based on a deformation of the
Heisenberg algebra which makes the commutator of momenta proportional to the
pseudo-spin. We solve the Landau problem for the resulting Hamiltonian which
reduces, in the large mass limit while keeping fixed the Fermi velocity, to the
usual linear one employed to describe these excitations as massless Dirac
fermions. This model, extended to negative mass, allows to reproduce the
leading terms in the low energy expansion of the dispersion relation for both
nearest and next-to-nearest neighbor interactions. Taking into account the
contributions of both Dirac points, the resulting Hall conductivity, evaluated
with a -function approach, is consistent with the anomalous integer
quantum Hall effect found in graphene. Moreover, when considered in first order
perturbation theory, it is shown that the next-to-leading term in the
interaction between nearest neighbor produces no modifications in the spectrum
of the model while an electric field perpendicular to the magnetic field
produces just a rigid shift of this spectrum.
PACS: 03.65.-w, 81.05.ue, 73.43.-fComment: 23 pages, 4 figures. Version to appear in the Journal of Physics A.
The title has been changed into "Graphene and non-Abelian quantization". The
motivation and presentation of the paper has been changed. An appendix and
Section 6 on the evaluation of the Hall conductivity have been added.
References adde