On Robin's criterion for the Riemann Hypothesis

Robin's criterion states that the Riemann Hypothesis (RH) is true if and only if Robin's inequality sum_{d|n}d=5041, where gamma denotes the Euler(-Mascheroni) constant. We show by elementary methods that if n>=37 does not satisfy Robin's criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, moreover, that n must be divisible by a fifth power >1. As a consequence we infer that RH holds true if and only if every natural number divisible by a fifth power >1 satisfies Robin's inequality.Comment: 15 pages. Corrected version. In the first version in Theorem 5 (main result) it was falsely asserted that n must be superabundant, invalidating the proof. An alternative proof is provided in this version, some typos have been corrected, and the presentation has been improve

Periods of modular forms on Γ0(N)\Gamma_0(N) and products of Jacobi theta functions

Generalizing a result of [15] for modular forms of level one, we give a closed formula for the sum of all Hecke eigenforms on $\Gamma_0(N)$, multiplied by their odd period polynomials in two variables, as a single product of Jacobi theta series for any squarefree level $N$. We also show that for $N=2$,3 and 5 this formula completely determines the Fourier expansions of all Hecke eigenforms of all weights on $\Gamma_0(N)$

On the special values of certain L-series related to half-integral weight modular forms

Let h be a cuspidal Hecke eigenform of half-integral weight, and En/2+1/2 be Cohen’s Eisenstein series of weight n/2+1/2. For a Dirichlet character χ we define a certain linear combination R(χ)(s, h,En/+1/2) of the Rankin-Selberg convolution products of h and En/2+1/2 twisted by Dirichlet characters related with χ. We then prove a certain algebraicity result for R(χ)(l, h,En/2+1/2) with l integers

Directed Graph Representation of Half-Rate Additive Codes over GF(4)

We show that (n,2^n) additive codes over GF(4) can be represented as directed graphs. This generalizes earlier results on self-dual additive codes over GF(4), which correspond to undirected graphs. Graph representation reduces the complexity of code classification, and enables us to classify additive (n,2^n) codes over GF(4) of length up to 7. From this we also derive classifications of isodual and formally self-dual codes. We introduce new constructions of circulant and bordered circulant directed graph codes, and show that these codes will always be isodual. A computer search of all such codes of length up to 26 reveals that these constructions produce many codes of high minimum distance. In particular, we find new near-extremal formally self-dual codes of length 11 and 13, and isodual codes of length 24, 25, and 26 with better minimum distance than the best known self-dual codes.Comment: Presented at International Workshop on Coding and Cryptography (WCC 2009), 10-15 May 2009, Ullensvang, Norway. (14 pages, 2 figures

Peripheral Blood Signatures of Lead Exposure

BACKGROUND: Current evidence indicates that even low-level lead (Pb) exposure can have detrimental effects, especially in children. We tested the hypothesis that Pb exposure alters gene expression patterns in peripheral blood cells and that these changes reflect dose-specific alterations in the activity of particular pathways. METHODOLOGY/PRINCIPAL FINDING: Using Affymetrix Mouse Genome 430 2.0 arrays, we examined gene expression changes in the peripheral blood of female Balb/c mice following exposure to per os lead acetate trihydrate or plain drinking water for two weeks and after a two-week recovery period. Data sets were RMA-normalized and dose-specific signatures were generated using established methods of supervised classification and binary regression. Pathway activity was analyzed using the ScoreSignatures module from GenePattern. CONCLUSIONS/SIGNIFICANCE: The low-level Pb signature was 93% sensitive and 100% specific in classifying samples a leave-one-out crossvalidation. The high-level Pb signature demonstrated 100% sensitivity and specificity in the leave-one-out crossvalidation. These two signatures exhibited dose-specificity in their ability to predict Pb exposure and had little overlap in terms of constituent genes. The signatures also seemed to reflect current levels of Pb exposure rather than past exposure. Finally, the two doses showed differential activation of cellular pathways. Low-level Pb exposure increased activity of the interferon-gamma pathway, whereas high-level Pb exposure increased activity of the E2F1 pathway

A morphometric analysis of algal response to low dose, short-term heavy metal exposure

Three algae, Melosira granulata, Fragilaria capucina , and Anacystis cyanea , collected as part of a natural phytoplankton assemblage were found to differ in their cytological responses to low dose short-term exposure to copper and lead. In general, all were more sensitive to copper than to lead. Fragilaria was more sensitive to both metals than the other species examined. Most immediate changes in relative volume categories can be ascribed to changes in vacuole volume that are most likely the result of changes in membrane permeability. There was some degree of accommodation in all three species at 24 hours. These results are discussed in view of the natural environment of the algae, as well as in relationship to previous studies.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41729/1/709_2005_Article_BF01281533.pd