An explicit height bound for the classical modular polynomial

For a prime m, let Phi_m be the classical modular polynomial, and let h(Phi_m) denote its logarithmic height. By specializing a theorem of Cohen, we prove that h(Phi_m) <= 6 m log m + 16 m + 14 sqrt m log m. As a corollary, we find that h(Phi_m) <= 6 m log m + 18 m also holds. A table of h(Phi_m) values is provided for m <= 3607.Comment: Minor correction to the constants in Theorem 1 and Corollary 9. To appear in the Ramanujan Journal. 17 pages

On Plouffe's Ramanujan Identities

Recently, Simon Plouffe has discovered a number of identities for the Riemann zeta function at odd integer values. These identities are obtained numerically and are inspired by a prototypical series for Apery's constant given by Ramanujan: $\zeta(3)=\frac{7\pi^3}{180}-2\sum_{n=1}^\infty\frac{1}{n^3(e^{2\pi n}-1)}$ Such sums follow from a general relation given by Ramanujan, which is rediscovered and proved here using complex analytic techniques. The general relation is used to derive many of Plouffe's identities as corollaries. The resemblance of the general relation to the structure of theta functions and modular forms is briefly sketched.Comment: 19 pages, 3 figures; v4: minor corrections; modified intro; revised concluding statement

An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions

This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein's "An efficient algorithm for computing the Riemann zeta function", to more general series. The algorithm provides a rapid means of evaluating Li_s(z) for general values of complex s and the region of complex z values given by |z^2/(z-1)|<4. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler-Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other; thus, either algorithm can be used to evaluate either function. The Euler-Maclaurin series is a clear performance winner for the Hurwitz zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms are superior to the simple Taylor's series or direct summation. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion of the monodromy group of the polylogarithm is included.Comment: 37 pages, 6 graphs, 14 full-color phase plots. v3: Added discussion of a fast Hurwitz algorithm; expanded development of the monodromy v4:Correction and clarifiction of monodrom

AdS_3 Partition Functions Reconstructed

For pure gravity in AdS_3, Witten has given a recipe for the construction of holomorphically factorizable partition functions of pure gravity theories with central charge c=24k. The partition function was found to be a polynomial in the modular invariant j-function. We show that the partition function can be obtained instead as a modular sum which has a more physical interpretation as a sum over geometries. We express both the j-function and its derivative in terms of such a sum.Comment: 9 page

Sums of products of Ramanujan sums

The Ramanujan sum $c_n(k)$ is defined as the sum of $k$-th powers of the primitive $n$-th roots of unity. We investigate arithmetic functions of $r$ variables defined as certain sums of the products $c_{m_1}(g_1(k))...c_{m_r}(g_r(k))$, where $g_1,..., g_r$ are polynomials with integer coefficients. A modified orthogonality relation of the Ramanujan sums is also derived.Comment: 13 pages, revise

SL(2,Z) Multiplets in N=4 SYM Theory

We discuss the action of SL(2,Z) on local operators in D=4, N=4 SYM theory in the superconformal phase. The modular property of the operator's scaling dimension determines whether the operator transforms as a singlet, or covariantly, as part of a finite or infinite dimensional multiplet under the SL(2,Z) action. As an example, we argue that operators in the Konishi multiplet transform as part of a (p,q) PSL(2,Z) multiplet. We also comment on the non-perturbative local operators dual to the Konishi multiplet.Comment: 14 pages, harvmac; v2: published version with minor change

On fundamental domains and volumes of hyperbolic Coxeter-Weyl groups

We present a simple method for determining the shape of fundamental domains of generalized modular groups related to Weyl groups of hyperbolic Kac-Moody algebras. These domains are given as subsets of certain generalized upper half planes, on which the Weyl groups act via generalized modular transformations. Our construction only requires the Cartan matrix of the underlying finite-dimensional Lie algebra and the associated Coxeter labels as input information. We present a simple formula for determining the volume of these fundamental domains. This allows us to re-produce in a simple manner the known values for these volumes previously obtained by other methods.Comment: v2: to be published in Lett Math Phys (reference added, typo corrected

Seven-branes and Supersymmetry

We re-investigate the construction of half-supersymmetric 7-brane solutions of IIB supergravity. Our method is based on the requirement of having globally well-defined Killing spinors and the inclusion of SL(2,Z)-invariant source terms. In addition to the well-known solutions going back to Greene, Shapere, Vafa and Yau we find new supersymmetric configurations, containing objects whose monodromies are not related to the monodromy of a D7-brane by an SL(2,Z) transformation.Comment: 31 pages, 3 figure

Two-divisibility of the coefficients of certain weakly holomorphic modular forms

We study a canonical basis for spaces of weakly holomorphic modular forms of weights 12, 16, 18, 20, 22, and 26 on the full modular group. We prove a relation between the Fourier coefficients of modular forms in this canonical basis and a generalized Ramanujan tau-function, and use this to prove that these Fourier coefficients are often highly divisible by 2.Comment: Corrected typos. To appear in the Ramanujan Journa

Arithmetical properties of Multiple Ramanujan sums

In the present paper, we introduce a multiple Ramanujan sum for arithmetic functions, which gives a multivariable extension of the generalized Ramanujan sum studied by D. R. Anderson and T. M. Apostol. We then find fundamental arithmetic properties of the multiple Ramanujan sum and study several types of Dirichlet series involving the multiple Ramanujan sum. As an application, we evaluate higher-dimensional determinants of higher-dimensional matrices, the entries of which are given by values of the multiple Ramanujan sum.Comment: 19 page