45 research outputs found

    Integrals Over Polytopes, Multiple Zeta Values and Polylogarithms, and Euler's Constant

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    Let TT be the triangle with vertices (1,0), (0,1), (1,1). We study certain integrals over TT, 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 TT. 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

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    We discuss the summation of certain series defined by counting blocks of digits in the BB-ary expansion of an integer. For example, if s2(n)s_2(n) denotes the sum of the base-2 digits of nn, we show that n1s2(n)/(2n(2n+1))=(γ+log4π)/2\sum_{n \geq 1} s_2(n)/(2n(2n+1)) = (\gamma + \log \frac{4}{\pi})/2. 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

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    In 1845, Bertrand conjectured that for all integers x2x\ge2, there exists at least one prime in (x/2,x](x/2, x]. This was proved by Chebyshev in 1860, and then generalized by Ramanujan in 1919. He showed that for any n1n\ge1, there is a (smallest) prime RnR_n such that π(x)π(x/2)n\pi(x)- \pi(x/2) \ge n for all xRnx \ge R_n. In 2009 Sondow called RnR_n the nnth Ramanujan prime and proved the asymptotic behavior Rnp2nR_n \sim p_{2n} (where pmp_m is the mmth prime). In the present paper, we generalize the interval of interest by introducing a parameter c(0,1)c \in (0,1) and defining the nnth cc-Ramanujan prime as the smallest integer Rc,nR_{c,n} such that for all xRc,nx\ge R_{c,n}, there are at least nn primes in (cx,x](cx,x]. Using consequences of strengthened versions of the Prime Number Theorem, we prove that Rc,nR_{c,n} exists for all nn and all cc, that Rc,npn1cR_{c,n} \sim p_{\frac{n}{1-c}} as nn\to\infty, and that the fraction of primes which are cc-Ramanujan converges to 1c1-c. We then study finer questions related to their distribution among the primes, and see that the cc-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 c=1/2c = 1/2. 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

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    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

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    We consider the construction of Lagrangians that might be suitable for describing the entire pp-adic sector of an adelic open scalar string. These Lagrangians are constructed using the Lagrangian for pp-adic strings with an arbitrary prime number pp. 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

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    Abstract We construct sequences of finite sums ( l ˜ n ) n ≥ 0 (l~n)n0(\tilde{l}_{n})_{n\geq 0} and ( u ˜ n ) n ≥ 0 (u~n)n0(\tilde{u}_{n})_{n\geq 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 2γ2^{\gamma } 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

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    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 ζ\zeta-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

    Birth, growth and computation of pi to ten trillion digits

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