Upper Bounds for |l(1, Χ)|
نویسندگان
چکیده
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 known that there exist characters χ± for which L(1, χ+) ≍ log log q and L(1, χ−) ≍ 1/ log log q. In this paper we focus on sharpening the upper bounds known for |L(1, χ)|; in particular, we wish to determine constants c (as small as possible) for which the bound |L(1, χ)| ≤ (c+o(1)) log q holds. To set this in context, observe that ifX is such that ∑
منابع مشابه
On the use of explicit bounds on residues of Dedekind zeta functions taking into account the behavior of small primes
Lately, explicit upper bounds on |L(1, χ)| (for primitive Dirichlet characters χ) taking into account the behaviors of χ on a given finite set of primes have been obtained. This yields explicit upper bounds on residues of Dedekind zeta functions of abelian number fields taking into account the behavior of small primes, and it as been explained how such bounds yield improvements on lower bounds ...
متن کاملUpper Bounds on a Two-term Exponential Sum
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.
متن کاملSubconvexity Bounds for Automorphic L–functions
We break the convexity bound in the t–aspect for L–functions attached to cuspforms f for GL2(k) over arbitrary number fields k. The argument uses asymptotics with error term with a power saving, for second integral moments over spectral families of twists L(s, f ⊗χ) by grossencharacters χ, from our previous paper [Di-Ga]. §0. Introduction In many instances, for cuspidal automorphic forms f on r...
متن کاملNordhaus-Gaddum-type Theorems for decompositions into many parts
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 ...
متن کاملUpper Bounds on L-functions at the Edge of the Critical Strip
In this paper, we are concerned with establishing bounds for L(1) where L(s) is a general L-function, and specifically, we shall be most interested in the case where no good bound for the size of the coefficients of the Lfunction is known. In this case, results are available due to Iwaniec [9], [10], and Molteni [16], but this type of investigation is still in its infancy, and the limitations o...
متن کامل