Finite expressions for higher derivatives of the Dirichlet L-function and the Deninger R-function

K Chakraborty (1) , S Kanemitsu (2) , T Kuzumaki (3)

1 Harish-Chandra Research Institute
2 University ok Kinki
3 Gifu University

We show the equivalence of the finite expression of Deninger's `R-function' at the rational arguments and the Kronecker limit formula on the line of our past study on the Gauss formula for the digamma function and the Dirichlet class number formula. Here the Gauss formula and the class number formula will be replaced by its analogue for the `R-function' and by the Kronecker limit formula or rather a closed form for the derivative of the Dirichlet L-function respectively. We also make a systematic study of the `$R_k$-function' by appealing to the Lipschitz-Lerch transcendent in which there is the vector space structure built in of these special functions.

Mots clés: Kronecker limit formula; Generalized Euler constant; Lipschitz-Lerch transcendent; Euler digamma function

Domaines: Mathématiques [math]

K Chakraborty, S Kanemitsu, T Kuzumaki. Finite expressions for higher derivatives of the Dirichlet L-function and the Deninger R-function. Hardy-Ramanujan Journal, 2009, Volume 32 - 2009, pp.38-53. ⟨10.46298/hrj.2009.166⟩. ⟨hal-01112359⟩