For a character χ of finite group G, the co-degree is c (1)=[G:kerχ] χ(1). Let p be prime and let e positive integer. In this paper, we first show that if G p-solvable such e+1 ∤χ (1), for every irreducible then p-length not greater than e. Next, study groups satisfying condition 2 does divide co-degrees their characters.

This upper bound is used to calculate several chromatic numbers of strong products. It is shown in particular that for k ≥ 2, χ(C5 2 ×C5 2 ×C2k+1) = 10 + d 5 ke, and for k ≥ 2 and n ≥ 1, χ(C2k+1 2 ×Kn) = k n + d2 e. That the general upper bound cannot be improved for graphs which are not χ–critical is demonstrated by two infinite series of graphs. The paper is concluded with an application to g...

Let χ be a primitive, real and even Dirichlet character with conductor q, and let s be a positive real number. An old result of H. Davenport is that the cycle sums Sν(s, χ) = ∑(ν+1)q−1 n=νq+1 χ(n) ns , ν = 0, 1, 2, . . . , are all positive at s = 1, and this has the immediate important consequence of the positivity of L(1, χ). We extend Davenport’s idea to show that in fact for ν ≥ 1, Sν(s, χ) ...

We obtain formulae and estimates for character sums of the type S(χ, f, pm) = ∑pm x=1 χ(f(x)), where p m is a prime power with m ≥ 2, χ is a multiplicative character (mod pm), and f = f1/f2 is a rational function over Z. In particular, if p is odd, d = deg(f1) + deg(f2) and d∗ = max(deg(f1), deg(f2)) then we obtain |S(χ, f, pm)| ≤ (d− 1)pm(1− 1 d∗ ) for any non constant f (mod p) and primitive ...