\(q\)-analogue of Riemann’s \(\zeta\)-function and \(q\)-Euler numbers. (English) Zbl 0675.12010

N. Koblitz [J. Number Theory 14, 332–339 (1982; Zbl 0501.12020)] has given a \(q\)-analogue of the \(p\)-adic \(L\)-function, and asked whether there is a corresponding complex analytic \(q\)-\(L\)-function. The author answers this by giving
\[ L_q(s,\chi) = \frac{2-s}{s-1}(q-1)\sum^{\infty}_{n=1}\frac{q^n\chi(n)}{[n]^{s- 1}}+\sum_{n=1}\frac{q^n\chi (n)}{[n]^s}, \]
where \([n]=(1-q^n)/(1-q)\). This correctly interpolates the \(q\)-Bernoulli numbers at negative integers. As an application the author proves Kummer congruences for \(q\)-Euler numbers.


11S40 Zeta functions and \(L\)-functions
11B68 Bernoulli and Euler numbers and polynomials
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)


Zbl 0501.12020
Full Text: DOI


[1] Askey, R., The q-gamma and q-beta functions, Appl. anal., 8, 125-141, (1978) · Zbl 0398.33001
[2] Carlitz, L., q-Bernoulli numbers and polynomials, Duke math. J., 15, 987-1000, (1948) · Zbl 0032.00304
[3] Carlitz, L., q-Bernoulli and Eulerian numbers, Trans. amer. soc., 76, 332-350, (1954) · Zbl 0058.01204
[4] Koblitz, N., A new proof of certain formulas for p-adic L-functions, Duke math. J., 46, 455-468, (1979) · Zbl 0409.12028
[5] Koblitz, N., On Carlitz’s q-Bernoulli numbers, J. number theory, 14, 332-339, (1982) · Zbl 0501.12020
[6] Ribet, K., Fonctions lp-adiques et théorie d’Iwasawa, (1979), Publ. Math Orsay, Cours rédigé par Ph. Satgé
[7] Shiratani, K., On Euler numbers, Memo. facu. sci. kyushu univ., 26, 119-138, (1972) · Zbl 0243.12009
[8] Shiratani, K.; Yamamoto, S., On a p-adic interpolation function for the Euler numbers and its derivatives, Memo. facu. sci. kyushu univ., 39, 113-125, (1985) · Zbl 0574.12017
[9] Sinnott, W., On the μ-invariant of the γ-transform of a rational function, Invent. math., 75, 273-282, (1984) · Zbl 0531.12004
[10] Tsumura, H., On a p-adic interpolation of the generalized Euler numbers and its applications, Tokyo J. math., 10, 281-293, (1987) · Zbl 0641.12007
[11] Washington, L., Introduction to cyclotomic fields, () · Zbl 0966.11047
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