Euler numbers congruences and Dirichlet L-functions

TextIn this paper we apply Yamamoto's Theorem [Y. Yamamoto, Dirichlet series with periodic coefficients, in: Proc. Intern. Sympos. “Algebraic Number Theory”, Kyoto, 1976, JSPS, Tokyo, 1977, pp. 275–289] to find the residue modulo a prime power of the linear combination of Dirichlet L  -function values L(s,χ)L(s,χ) at positive integral arguments s such that s and χ   are of the same parity, in terms of Euler numbers, whereby we obtain the finite expressions for short interval character sums. The results obtained generalize the previous results pertaining to the congruences modulo a prime power of the class numbers as the special case of s=1s=1.VideoFor a video summary of this paper, please visit http://www.youtube.com/watch?v=_KAv4FCdVUs.