Mean values of L-functions and Dedekind sums

TextFor arbitrary non-negative integers a1,…,ada1,…,ad and m1,…,mdm1,…,md, we introduce and investigate the mean value of the productχ¯1(a1)…χ¯d(ad)L(m1+1,χ1)…L(md+1,χd), such that m1,…,mdm1,…,md have the same parity and χi(−1)=(−1)mi+1χi(−1)=(−1)mi+1, i=1,…,di=1,…,d. Using recent results of the authors on Dedekind reciprocity law we give explicit formulae for this mean. Our studies recover and improve the previous works of Walum, Louboutin, Liu and Zhang.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=FG2aZBD3VS8.