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We shall resurrect the instinctive direction of B. Riemann on his posthumous fragment on the limit values of elliptic modular functions à la C.G.J. Jacobi, Fundamenta Nova. In the spirit of Riemann who considered the odd part, we shall realize the situation where there is no singularity occurring in taking the radial limits, thus streamlining and elucidating the recent investigation by Arias de Reyna. By the new Dirichlet-Abel theorem (which should be within reach of Riemann), we may directly sum the series in question, which allows us to condense Arias de Reyna's paper into a few pages.

Abstract. In this paper we shall establish the counterpart of Szmidt, Urbanowicz and Zagier's formula in the sense of the Hecker correspondence. The motivation is the derivation of the values of the Riemann zeta-function at positive even integral arguments from the partial fraction expansion for the hyperbolic cotangent function (or the cotangent function). Since the last is equivalent to the functional equation, we may view their elegant formula as one for the Lambert series, and comparing the Laurent coefficients, we may give a functional equational approach to the short-interval character sums with polynomial weight.In view of the importance of these short-interval character sums, we assemble some handy formulations for them that are derived from Szmidt, Urbanowicz and Zagier's formula and Yamamoto's method, which also gives the conjugate sums. We shall also state the formula for the values of the Dirichlet L-function with imprimitive characters.