The point of this note is to prove that the secrecy function attains its maximum at y = 1 on all known extremal even unimodular lattices. This is a special case of a conjecture by Belfiore and Solé. Further, we will give a very simple method to verify or disprove the conjecture on any given unimodular lattice.

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We define modified Li coefficients, called τ -Li coefficients for a very broad class S (σ 0 , σ 1 ) of L-functions that contains the Selberg class, the class of all automorphic L-functions and the Rankin-Selberg L-functions, as well as products of suitable shifts of those functions. We prove the generalized Li criterion for zero-free regions of functions belonging to the class S (σ 0 , σ 1 ), derive an arithmetic formula for the computation of τ -Li coefficients and conduct numerical investigation of τ -Li coefficients for a certain product of shifts of the Riemann zeta function. .fi (A.-M. Ernvall-Hytönen), almasa@pmf.unsa.ba (A. Odžak), lejlas@pmf.unsa.ba (L. Smajlović), medina.susic@pmf.unsa.ba (M. Sušić).

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Let I denote an imaginary quadratic field or the field Q of rational numbers and Z I its ring of intergers. We shall prove an explicit Baker type lower bound for Z I -linear form of the numbers 1, e α 1 , ..., e αm , m ≥ 2,where α 0 = 0, α 1 , ..., α m , are m + 1 different numbers from the field I. Our work gives gives some improvements to the existing explicit versions of of Baker's work about exponential values at rational points. In particilar, dependences on m are improved.

Let l(n) be the number of lines through at least two points of an n × n rectangular grid. We prove recursive and asymptotic formulas for it using respectively combinatorial and number theoretic methods. We also study the ratio l(n)/l(n−1). All this originates from Mustonen's experimental results.

In this paper, the probability of Eve the Eavesdropper's correct decision is considered both in the Gaussian and Rayleigh fading wiretap channels when using lattice codes for the transmission.First, it is proved that the secrecy function determining Eve's performance attains its maximum at y = 1 on all known extremal even unimodular lattices. This is a special case of a conjecture by Belfiore and Solé. Further, a very simple method to verify or disprove the conjecture on any given unimodular lattice is given.Second, preliminary analysis on the behavior of Eve's probability of correct decision in the fast fading wiretap channel is provided. More specifically, we compute the truncated inverse norm power sum factors in Eve's probability expression. The analysis reveals a performance-secrecy-complexity tradeoff: relaxing on the legitimate user's performance can significantly increase the security of transmission. The confusion experienced by the eavesdropper may be further increased by using skewed lattices, but at the cost of increased complexity.

The purpose of the article is to estimate the mean square of a squareroot length exponential sum of Fourier coefficients of a holomorphic cusp form. 1≤n≤M a(n)e(nα), where α is a real number, have been widely studied. See e.g. Wilton [11] and Jutila [9]. Short sums M ≤n≤M +∆ a(n)e(nα), where ∆ ≪ M 3/4 have been studied for instance in [3] and [4]. However, it seems that very short sums, in particular, sums with ∆ ≍ M 1/2 seem to be extremely difficult to treat, even though this is an important special case. According to the results in [1] and the computer data in [2], it is plausible to believe the correct upper bound to be M ≤n≤M + √ M a(n)e(nα) ≪ M 1/4+ε .