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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.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1522 SHIGERU KANEMITSU, HAILONG LI, AND NIANLIANG WANG equation. We have come upon this standpoint by viewing Szmidt, Urbanowicz and Zagier's formula as a form of a Lambert series.At the time when we wrote the first version of the present paper, we thought we would need the functional equation for the Hurwitz-Lerch L-function introduced by Morita [31] (for this, cf. the proof of Theorem 3), but it has turned out that we need that of the Dirichlet L-function and the complete character sum (for applications to short-interval sums). Thus we are led to give formulas for the special values of the relevant -function (cf. (2.5)) at positive integer arguments, which is essential in the case of imprimitive characters, whence those for the Dirichlet L-functions with imprimitive characters. This part is also based on Riemann, Hecke and Bochner's correspondence.Secondly, we will state some concrete formulas for the short-interval character sums with imprimitive characters for convenience of reference. This is because, to treat such sums, most of the authors refer to [7] or [18] but not [35], [38] or more recently [34]; cf. [37]for examples of such redundancies. Most of the literature contains formulas for primitive characters but one sometimes needs formulas for imprimitive characters as in the case of the analytic-number-theoretic application of discrete mean squares (cf. e.g. [22]) in which one needs a formula valid for all characters in order to use the orthogonality. In [38] and [34] the authors' intension was to apply the results to the quadratic fields and therefore all the characters they need are primitive Kronecker characters. Since Szmidt, Urbanowicz and Zagier's formula is valid for all characters, we are led to trying to generalize Yamamoto's method to this case. We note that Yamamoto's method allows us to treat shortinterval character sums with Clausen function weight (log sin integrals, cf. [30]) as well, which enables us to obtain class number relations for real quadrat...