In this note we give the general solution of the functional equation
f(x) f(x+y) = f(y)² f(x−y)² g(y),  x,y∈G,
and all the solutions of
f(x) f(x+y) = f(y)² f(x−y)² g(x) ,  x,y∈G,
with the additional supposition g(x) ≠ 0 for all x∈G. In both cases G denotes an arbitrary group written additively and f,g: G→ℝ are the unknown functions.

We study the homomorphism W???? → WK between the Witt ring of a domain ???? and the Witt ring of its field of fractions K in the case when ???? is not integrally closed. We give sufficient conditions for the noninjectivity of this homomorphism by constructing nonzero elements in the kernel. In particular, when K is an algebraic number field and ???? is a nonmaximal order in K with even conductor, then the ring homomorphism W???? → WK is not injective.

We prove two separation theorems for solutions of conditional Cauchy and Jensen equations.

The author uses the summation of rational series using the properties of the digamma function Ψ(x) and the methods of the residue calculus to evaluate the function H_α(x) for α = 1 and x = a⁻¹(N), N∈ℕ (see Theorem 1) which is called the function generating the generalized harmonic numbers of order 1 (see Definition 1). The relation between the functions H₁(x), x > 0, and Ψ(x) is used to find the approximations of the constant e^6ζ(3) in the form of the infinite product which contains only the numbers e, π and the roots of unity, where ζ(3) is the Apéry constant.

The following theorems are proved:
(1) If α and β≠α are roots of the polynomial x²−Px+Q, where gcd(P,Q)=1, P = α+β is an odd positive integer, then (α+β)ⁿ⁺¹|α^x + β^x if and only if x = (2l+1)(α+β)ⁿ, where l = 0, 1, 2, . . . and then
gcd((α^{(α+β)^n}+β^{(α+β)^n})/(α+β)ⁿ⁺¹, α+β) = 1.
(2) Given integers P,Q with D = P²−4Q ≠ 0,−Q,−2Q,−3Q and ɛ=±1,
every arithmetic progression ax+b, where gcd(a,b)=1 contains an odd integer n₀ such that (D|n₀)=ɛ. The series Σ_n=1^∞1/logP_n^(a)=1, where P_n^(a) is the n-th strong Lucas pseudoprime with parameters P and Q of the form ax+b, where gcd(a,b)=1 such that (D|P_n^(a))=ɛ, is divergent.
(3) Let C_n denote the n-th Carmichael number. From the conjecture of P. Erdős that C(x) > x^1−ɛ for every ɛ>0 and x ≥ x₀(ɛ), where C(x) denotes the number of Carmichael numbers not exceeding x it follows that the series Σ_n=1^∞1/C_n^1-ɛ is divergent for every ɛ>0.

Report of Meeting. The Seventh Katowice–Debrecen Winter Seminaron Functional Equations and Inequalities January 31 – February 2, 2007 Będlewo, Poland.