Showing papers on "Riemann zeta function published in 2006"
TL;DR: In this article, the authors start to hunt the motive, restricting their attention to a subclass of graphs in four dimensional scalar field theory which give scheme independent contributions to the above functions.
Abstract: The appearance of multiple zeta values in anomalous dimensions and β-functions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization group functions. In this paper we start to hunt the motive, restricting our attention to a subclass of graphs in four dimensional scalar field theory which give scheme independent contributions to the above functions.
273 citations
150 citations
TL;DR: In this article, by applying the Mellin transformation to these generating functions, they obtained integral representations of the new twisted (h, q ) -zeta function and twisted ( h, q )-L-function, which interpolate the twisted (H, q) -Bernoulli numbers and generalized twisted ( H, q ), q ) numbers at non-positive integers, respectively.
148 citations
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TL;DR: In this article, an upper bound for the moments of the Riemann zeta-function on the critical line was obtained, which is nearly as sharp as the conjectured asymptotic formulae for these moments.
Abstract: Assuming the Riemann Hypothesis we obtain an upper bound for the moments of the Riemann zeta-function on the critical line. Our bound is nearly as sharp as the conjectured asymptotic formulae for these moments. The method extends to moments in families of $L$-functions.
143 citations
TL;DR: The main object of as discussed by the authors is to further investigate the generalized Apostol-Bernoulli polynomials of higher order, which were introduced and studied recently by Luo and Srivastava.
Abstract: The main object of this paper is to further investigate the generalized Apostol–Bernoulli polynomials of higher order, which were introduced and studied recently by Luo and Srivastava [2005, Journal of Mathematical Analysis and Applications, 308, 290–302; 2006, Computers and Mathematics with Applications, 51, 631–642]. Here, we first derive an explicit representation of these generalized Apostol–Bernoulli polynomials of higher order in terms of a generalization of the Hurwitz–Lerch Zeta function and then proceed to establish a functional relationship between the generalized Apostol–Bernoulli polynomials of rational arguments and the Hurwitz (or generalized) Zeta function. Our results would provide extensions of those given earlier by (for example) Apostol [1951, Pacific Journal of Mathematics, 1, 161–167] and Srivastava [2000, Mathematical Proceedings of the Cambridge Philosophical Society, 129, 77–84].
111 citations
TL;DR: In this article, the authors present a systematic investigation of expansion and transformation formulas for several general families of the Hurwitz-Lerch Zeta functions, making use of fractional calculus.
Abstract: By making use of fractional calculus, the authors present a systematic investigation of expansion and transformation formulas for several general families of the Hurwitz–Lerch Zeta functions. Relevant connections of the results discussed here with those obtained in earlier works are also indicated precisely.
105 citations
TL;DR: A number of potentially useful integral representations for the familiar Mathieu a-series as well as for its alternating version are presented and are shown to yield sharp bounding inequalities involving the Mathieu and alternating MathieuA-series.
93 citations
TL;DR: In this paper, a twisted q-partial zeta function and some twisted two-variable q-L-functions that interpolate q-Bernoulli numbers, β fixme (q), and Bernoulli polynomials, β petertodd (x, q), respectively, at negative integers were constructed.
Abstract: In this paper, we construct a twisted q-partial zeta function and some twisted two-variable q-L-functions that interpolate q-Bernoulli numbers, β
(q), and Bernoulli polynomials, β
(x, q), respectively, at negative integers. Using these functions, we prove the existence of a p-adic interpolation function that interpolates the q-generalized polynomials β
(x, q) at negative integers. Consequently, we define a p-adic twisted q-L-function which is a solution of a question of Kim et al.
89 citations
01 Jan 2006
82 citations
TL;DR: In this paper, a multidimensional cut-off technique for calculating the Casimir energy of massless scalar fields in d-dimensional rectangular spaces with q large dimensions and d − q dimensions of length L was developed.
Abstract: Quantum fluctuations of massless scalar fields represented by quantum fluctuations of the quasiparticle vacuum in a zero-temperature dilute Bose–Einstein condensate may well provide the first experimental arena for measuring the Casimir force of a field other than the electromagnetic field. This would constitute a real Casimir force measurement—due to quantum fluctuations—in contrast to thermal fluctuation effects. We develop a multidimensional cut-off technique for calculating the Casimir energy of massless scalar fields in d-dimensional rectangular spaces with q large dimensions and d − q dimensions of length L and generalize the technique to arbitrary lengths. We explicitly evaluate the multidimensional remainder and express it in a form that converges exponentially fast. Together with the compact analytical formulae we derive, the numerical results are exact and easy to obtain. Most importantly, we show that the division between analytical and remainder is not arbitrary but has a natural physical interpretation. The analytical part can be viewed as the sum of individual parallel plate energies and the remainder as an interaction energy. In a separate procedure, via results from number theory, we express some odd-dimensional homogeneous Epstein zeta functions as products of one-dimensional sums plus a tiny remainder and calculate from them the Casimir energy via zeta function regularization.
TL;DR: In this article, the authors apply a classical series identity involving the digamma function with a view to deriving series representations for a number of known mathematical constants, and several closely related consequences and results are also considered.
TL;DR: In this article, the functional determinant of a radially separable partial differential operator was derived in various dimensions, and the results agreed with what one would obtain using the angular momentum cutoff method based on radial WKB.
Abstract: We derive simple new expressions, in various dimensions, for the functional determinant of a radially separable partial differential operator, thereby generalizing the one-dimensional result of Gel'fand and Yaglom to higher dimensions. We use the zeta function formalism, and the results agree with what one would obtain using the angular momentum cutoff method based on radial WKB. The final expression is numerically equal to an alternative expression derived in a Feynman diagrammatic approach, but is considerably simpler.
TL;DR: In this paper, the Mellin transformation formula was applied to generate q-analogue Dirichlet L-function and two-variable q-L-function, and the relations between these sums and Dedekind sums were given.
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TL;DR: A deterministic polynomial time algorithm is presented for computing the zeta function of an arbitrary variety of fixed dimension over a finite field of small characteristic and an efficient method is found for Computing the order of the group of rational points on the Jacobian of a smooth geometrically connected projective curve.
Abstract: We present a deterministic polynomial time algorithm for computing the zeta function of an arbitrary variety of fixed dimension over a finite field of small characteristic. One consequence of this result is an efficient method for computing the order of the group of rational points on the Jacobian of a smooth geometrically connected projective curve over a finite field of small characteristic.
TL;DR: In this paper, the authors investigated the surface energy-momentum tensor for a massive scalar field with general curvature coupling parameter obeying the Robin boundary conditions on two codimension one-parallel branes in a (D+1)-dimensional background spacetime.
Abstract: We investigate the vacuum expectation value of the surface energy-momentum tensor for a massive scalar field with general curvature coupling parameter obeying the Robin boundary conditions on two codimension one parallel branes in a (D+1)-dimensional background spacetime $AdS_{D_{1}+1}\times \Sigma $ with a warped internal space $\Sigma $. These vacuum densities correspond to a gravitational source of the cosmological constant type for both subspaces of the branes. Using the generalized zeta function technique in combination with contour integral representations, the surface energies on the branes are presented in the form of the sum of single brane and second brane induced parts. For the geometry of a single brane both regions, on the left and on the right of the brane, are considered. At the physical point the corresponding zeta functions contain pole and finite contributions. For an infinitely thin brane taking these regions together, in odd spatial dimensions the pole parts cancel and the total zeta function is finite. The renormalization procedure for the surface energies and the structure of the corresponding counterterms are discussed. The parts in the surface densities generated by the presence of the second brane are finite for all nonzero values of the interbrane separation and are investigated in various asymptotic regions of the parameters. In particular, it is shown that for large distances between the branes the induced surface densities give rise to an exponentially suppressed cosmological constant on the brane. The total energy of the vacuum including the bulk and boundary contributions is evaluated by the zeta function technique and the energy balance between separate parts is discussed.
TL;DR: The Stieltjes constants γ k (a ) are the expansion coefficients in the Laurent series for the Hurwitz zeta function about s = 1 as mentioned in this paper, and they can be expressed as asymptotic, summatory, and other exact expressions for these constants.
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TL;DR: In this article, it was shown that the asymptotics of the zeta function determines an ideal strictly larger than Ω(n, √ n) on which the Dixmier trace may be defined.
Abstract: We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the trace of the heat semigroup. We prove our results in a general semi-finite von Neumann algebra. We find for p>1 that the asymptotics of the zeta function determines an ideal strictly larger than {\mathcal L}^{p,\infty} on which the Dixmier trace may be defined. We also establish stronger versions of other results on Dixmier traces and zeta functions.
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TL;DR: In this article, a p-adic algorithm for computing the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology is presented.
Abstract: In this paper we present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since all known cases, e.g. hyperelliptic, superelliptic and C_{ab} curves, can be transformed to fit the nondegenerate case. For curves with a fixed Newton polytope, the property of being nondegenerate is generic, so that the algorithm works for almost all curves with given Newton polytope. For a genus g curve over F_{p^n}, the expected running time is O(n^3g^6 + n^2g^{6.5}), whereas the space complexity amounts to O(n^3g^4), assuming p is fixed.
TL;DR: A review of the classical setting of real quadratic fields can be found in this paper, where the Bruhat-Tits tree and the Dedekind sums are discussed.
Abstract: 1. A review of the classical setting 2. Elliptic units for real quadratic fields 2.1. p-adic measures 2.2. Double integrals 2.3. Splitting a two-cocycle 2.4. The main conjecture 2.5. Modular symbols and Dedekind sums 2.6. Measures and the Bruhat-Tits tree 2.7. Indefinite integrals 2.8. The action of complex conjugation and of Up 3. Special values of zeta functions 3.1. The zeta function 3.2. Values at negative integers 3.3. The p-adic valuation 3.4. The Brumer-Stark conjecture 3.5. Connection with the Gross-Stark conjecture 4. A Kronecker limit formula 4.1. Measures associated to Eisenstein series 4.2. Construction of the p-adic L-function 4.3. An explicit splitting of a two-cocycle 4.4. Generalized Dedekind sums 4.5. Measures on Zp × Zp 4.6. A partial modular symbol of measures on Zp × Zp 4.7. From Zp × Zp to X 4.8. The measures μ and Γ-invariance
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TL;DR: In this paper, the authors used the self-similar tilings constructed by the second author in "Canonical self-affine tilings by iterated function systems" to define a generating function for the geometry of a selfsimilar set in Euclidean space.
Abstract: We use the self-similar tilings constructed by the second author in "Canonical self-affine tilings by iterated function systems" to define a generating function for the geometry of a self-similar set in Euclidean space. This tubular zeta function encodes scaling and curvature properties related to the complement of the fractal set, and the associated system of mappings. This allows one to obtain the complex dimensions of the self-similar tiling as the poles of the tubular zeta function and hence develop a tube formula for self-similar tilings in \$\mathbb{R}^d$. The resulting power series in $\epsilon$ is a fractal extension of Steiner's classical tube formula for convex bodies $K \ci \bRd$. Our sum has coefficients related to the curvatures of the tiling, and contains terms for each integer $i=0,1,...,d-1$, just as Steiner's does. However, our formula also contains terms for each complex dimension. This provides further justification for the term "complex dimension". It also extends several aspects of the theory of fractal strings to higher dimensions and sheds new light on the tube formula for fractals strings obtained in "Fractal Geometry and Complex Dimensions" by the first author and Machiel van Frankenhuijsen.
TL;DR: In this article, the authors considered fractional generalization of finite temperature Klein-Gordon (KG) field and vector potential in covariant gauge and static temporal gauge and showed that quantization of the fractional derivative fields can be carried out by using the Parisi-Wu stochastic quantization.
Abstract: This paper considers fractional generalization of finite temperature Klein–Gordon (KG) field and vector potential in covariant gauge and static temporal gauge. Fractional derivative quantum field at positive temperature can be regarded as a collection of infinite number of fractional thermal oscillators. Generalized Riemann zeta function regularization and heat kernel techniques are used to obtain the high temperature expansion of free energy associated with the fractional KG field. We also show that quantization of the fractional derivative fields can be carried out by using the Parisi–Wu stochastic quantization.
TL;DR: In this paper, the functional determinant of a radially separable partial differential operator was derived in various dimensions, and the results agreed with what one would obtain using the angular momentum cutoff method based on radial WKB.
Abstract: We derive simple new expressions, in various dimensions, for the functional determinant of a radially separable partial differential operator, thereby generalizing the one-dimensional result of Gel'fand and Yaglom to higher dimensions. We use the zeta function formalism, and the results agree with what one would obtain using the angular momentum cutoff method based on radial WKB. The final expression is numerically equal to an alternative expression derived in a Feynman diagrammatic approach, but is considerably simpler.
TL;DR: In this paper, the authors consider some parametrized classes of multiple sums first studied by Euler and show that identities between meromorphic functions of one or more variables in many cases account for reduction formulae for these sums.
TL;DR: In this article, the Riemann hypothesis holds for all truncation integrals with truncation parameter T ⩾ 1, with at most two exceptional real zeros, which occur exactly for those y > 4 π e − γ = 7.0555 +.
TL;DR: In this article, a Beurling generalized number system is constructed with integer counting function, whose prime counting function satisfies the oscillation estimate, and whose zeta function has infinitely many zeros on the curve σ=1−a/logt, t≥2, and no zero to the right of this curve, where a is chosen so that a>(4/e)(1−θ).
Abstract: A Beurling generalized number system is constructed having integer counting function N
B
(x) = κ
x +O(x
θ) with κ>0 and 1/2 <θ <1, whose prime counting function satisfies the oscillation estimate π
B
(x) =li(x) + Ω(xexp(-c
)), and whose zeta function has infinitely many zeros on the curve σ=1−a/logt, t≥2, and no zero to the right of this curve, where a is chosen so that a>(4/e)(1−θ). The construction uses elements of classical analytic number theory and probability.
TL;DR: The case of t = 0, that is T (s, 0, u), is called the Euler-Zagier double zeta function [10] as mentioned in this paper.
Abstract: The case of t = 0, that is T (s, 0, u), is called the Euler–Zagier double zeta function [10]. The values T (a, b, c) for a, b, c ∈ N were first investigated by Tornheim [7] in 1950 and later Mordell [5] in 1958. Tornheim [7, Theorem 7] showed that T (a, b, c) can be expressed as a polynomial in {ζ(j) | 2 ≤ j ≤ a+ b+ c} with rational coefficients when a + b + c is odd, and that the same is true for T (2r, 2r, 2r) and T (2r− 1, 2r, 2r+1) [7, Theorem 8], but he did not give the coefficients. Mordell [5, Theorem III] proved that T (2r, 2r, 2r) = krπ 6r for some rational number kr. In 1985 Subbarao and Sitaramachandrarao [6, Theorem 4.1] explicitly determined T (2p, 2q, 2r)+T (2q, 2r, 2p)+T (2r, 2p, 2q) (p, q, r ∈ N). Then, by taking p = q = r, they gave an explicit formula for T (2r, 2r, 2r) (r ∈ N) [6, Remark 3.1]. In 1996 Huard, Williams and Zhang [3, Theorems 1–3] determined T (r, 0, N−r) (r ∈ N, N ∈ 2N+1, 1 ≤ r ≤ N−2), T (p, q,N − p − q) (p, q ∈ N ∪ {0}, N ∈ 2N + 1, 1 ≤ p + q ≤ N − 1, 0 ≤ p, q ≤ N − 2) and T (r, r, r) (r ∈ N). In 2002 Tsumura [8, Theorem 1]
TL;DR: In this article, a connection between the theory of spherical designs and the question of minima of Epstein's zeta function was made, and it was shown that a Euclidean lattice, all layers of which hold a 4-design, achieves a local minimum of the Epstein's zero function at least at any real s>n/2.
Abstract: We set up a connection between the theory of spherical designs and the question of minima of Epstein's zeta function. More precisely, we prove that a Euclidean lattice, all layers of which hold a 4-design, achieves a local minimum of the Epstein's zeta function, at least at any real s>n/2. We deduce from this a new proof of Sarnak and Strombergsson's theorem asserting that the root lattices D4 and E8, as well as the Leech lattice, achieve a strict local minimum of the Epstein's zeta function at any s>0. Furthermore, our criterion enables us to extend their theorem to all the so-called extremal modular lattices(up to certain restrictions) using a theorem of Bachoc and Venkov, and to other classical families of lattices (e.g. the Barnes-Wall lattices).
TL;DR: In this paper, the moments of the derivative of the Riemann ζ function on the critical line were investigated and a conjecture was formulated for Hardy's Z-function on the unit circle.
Abstract: We investigate the moments of the derivative, on the unit circle, of characteristic polynomials of random unitary matrices and use this to formulate a conjecture for the moments of the derivative of the Riemann ζ function on the critical line. We do the same for the analogue of Hardy’s Z-function, the characteristic polynomial multiplied by a suitable factor to make it real on the unit circle. Our formulae are expressed in terms of a determinant of a matrix whose entries involve the I-Bessel function and, alternately, by a combinatorial sum.
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TL;DR: In this paper, the authors formalized a distance which gives a measure of the pretentiousness of multiplicative number theory, and as one consequence obtained a curious inequality for the zeta-function.
Abstract: We note how several central results in multiplicative number theory may be rephrased naturally in terms of multiplicative functions $f$ that pretend to be another multiplicative function $g$. We formalize a `distance' which gives a measure of such {\sl pretentiousness}, and as one consequence obtain a curious inequality for the zeta-function.