A k-decomposition (G1, . . . , Gk) of a graph G is a partition of its edge set to form k spanning subgraphs G1, . . . , Gk. The classical theorem of Nordhaus and Gaddum bounds χ(G1) + χ(G2) and χ(G1)χ(G2) over all 2-decompositions of Kn. For a graph parameter p, let p(k;G) denote the maximum of ∑k i=1 p(Gi) over all k-decompositions of the graph G. The clique number ω, chromatic number χ, list ...

We obtain upper bounds for mixed exponential sums of the type S(χ, f, pm) = ∑pm x=1 χ(x)epm (ax n+bx) where pm is a prime power with m ≥ 2 and χ is a multiplicative character (mod pm). If χ is primitive or p (a, b) then we obtain |S(χ, f, pm)| ≤ 2np 2 3 . If χ is of conductor p and p (a, b) then we get the stronger bound |S(χ, f, pm)| ≤ npm/2.

Given a non-principal Dirichlet character χ (mod q), an important problem in number theory is to obtain good estimates for the size of L(1, χ). The best bounds known give that q−ǫ ≪ǫ |L(1, χ)| ≪ log q, while assuming the Generalized Riemann Hypothesis, J.E. Littlewood showed that 1/ log log q ≪ |L(1, χ)| ≪ log log q. Littlewood’s result reflects the true range of the size of |L(1, χ)| as it is ...

10. Dirichlet characters and Dirichlet L-functions Definition 10.1. Let m ∈ N. A Dirichlet character modulo m is a function χ : N→ C that satisfies the following three conditions. (1) χ is periodic with period m, i.e. if a ≡ b (mod m) then χ(a) = χ(b). (2) χ is completely multiplicative, i.e. χ(ab) = χ(a)χ(b) for all a, b ∈ N. (3) χ(a) = 0 if and only if gcd(a,m) > 1. For m ∈ N we let (Z/mZ)× d...

Vertex-colorings, edge-colorings and total-colorings of the Sierpiński gasket graphs Sn, the Sierpiński graphs S(n, k), graphs S (n, k), and graphs S(n, k) are considered. In particular, χ′′(Sn), χ (S(n, k)), χ(S(n, k)), χ(S(n, k)), χ(S(n, k)), and χ(S(n, k)) are determined.

We had fixed the notation ζ f = e 2πi f , and τ (χ) = f k=1 χ(k)ζ k f , if χ has conductor f. We want to show: Theorem 0.1. Let χ be a Dirichlet character of conductor f > 1. Then for χ even, we have L(1, χ) = −

and Applied Analysis 3 2. Symmetry of Power Sum and the Generalized Bernoulli Polynomials Let χ be the Dirichlet character with conductor d ∈ N. From 1.3 , we note that ∫ X χ x edx t ∑d−1 i 0 χ i e it edt − 1 ∞ ∑ n 0 Bn,χ t n! , 2.1 where Bn,χ x are the nth generalized Bernoulli numbers attached to χ. Now, we also see that the generalized Bernoulli polynomials attached to χ are given by ∫ X χ (...

Let M be a compact manifold with a metric g and with a fixed spin structure χ. Let λ + 1 (g) be the first non-negative eigenvalue of the Dirac operator on (M, g, χ). We set τ (M, χ) := sup inf λ + 1 (g) where the infimum runs over all metrics g of volume 1 in a conformal class [g 0 ] on M and where the supremum runs over all conformal classes [g 0 ] on M. Let (M # , χ #) be obtained from (M, χ)...