# Volume 38

### 1. On congruences for certain sums of E. Lehmer's type

Let n > 1 be an odd natural number and let r (1 < r < n) be a natural number relatively prime to n. Denote by χn the principal character modulo n. In Section 3 we prove some new congruences for the sums T r,k (n) = n r ] i=1 (χn(i) i k) (mod n s+1) for s ∈ {0, 1, 2}, for all divisors r of […]

### 2. Remarks on the impossibility of a Siegel-Shidlovskii like theorem for G-functions

The Siegel-Shidlovskii Theorem states that the transcendence degree of the field generated over Q(z) by E-functions solutions of a differential system of order 1 is the same as the transcendence degree of the field generated over Q by the evaluation of these E-functions at non-zero algebraic points […]

### 3. Algebraic independence results on the generating Lambert series of the powers of a fixed integer

In this paper, the algebraic independence of values of the functionG d (z) := h≥0 z d h /(1 − z d h), d > 1 a fixed integer, at non-zero algebraic points in the unit disk is studied. Whereas the case of multiplicatively independent points has been resolved some time ago, a particularly […]

### 4. Effective irrationality measures for quotients of logarithms of rational numbers

We establish uniform irrationality measures for the quotients of the logarithms of two rational numbers which are very close to 1. Our proof is based on a refinement in the theory of linear forms in logarithms which goes back to a paper of Shorey.

### 5. Baker's Explicit abc-Conjecture and Waring's problem

The conjecture of Masser-Oesterlé, popularly known as abc-conjecture has many consequences. We show that Waring's problem is a consequence of an explicit version of abc−conjecture due to Baker.