L-series and Hurwitz zeta functions associated with the universal formal group
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 1, pp. 133-144.

The properties of the universal Bernoulli polynomials are illustrated and a new class of related L-functions is constructed. A generalization of the Riemann-Hurwitz zeta function is also proposed.

Classification : 11M41, 55N22
Tempesta, Piergiulio 1

1 Departamento de Fisica Teorica II, Universidad Complutense, Ciudad Universitaria, 28040 Madrid, Spain
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Tempesta, Piergiulio. L-series and Hurwitz zeta functions associated with the universal formal group. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 1, pp. 133-144. http://www.numdam.org/item/ASNSP_2010_5_9_1_133_0/

[1] A. Adelberg, Universal higher order Bernoulli numbers and Kummer and related congruences, J. Number Theory 84 (2000), 119–135. | MR | Zbl

[2] G. Almkvist and A. Meurman, Values of Bernoulli polynomials and Hurwitz’s zeta function at rational points, C. R. Math. Acad. Sci. Soc. R. Can. 13 (1991), 104–108. | MR | Zbl

[3] A. Baker, “Combinatorial and Arithmetic Identities Based on Formal Group Laws”, Lecture Notes in Math., Vol. 1298, Springer, 1987, 17–34. | MR

[4] A. Baker, F. Clarke, N. Ray and L. Schwartz, On the Kummer congruences and the stable homotopy of BU, Trans. Amer. Math. Soc. 316 (1989), 385–432. | MR | Zbl

[5] V. M. Bukhshtaber, A. S. Mishchenko and S. P. Novikov, Formal groups and their role in the apparatus of algebraic topology, Uspehi Mat. Nauk 26 (1971), 161–154. | MR | Zbl

[6] K. Bartz and J. Rutkowski, On the von Staudt-Clausen theorem, C. R. Math. Acad. Sci. Soc. R. Can. 15 (1993), 46–48. | MR | Zbl

[7] N. Berline and C. Sabbah (ets.) , “La fonction zêta”, Éditions de l’École polytechnique, 2003. | MR

[8] D. Cvijovic and J. Klinowski, New formulae for the Bernoulli and Euler polynomials at rational arguments, Proc. Amer. Math. Soc. 123 (1995), 1527–1535. | MR | Zbl

[9] F. Clarke, The universal Von Staudt theorems, Trans. Amer. Math. Soc. 315 (1989), 591–603. | MR | Zbl

[10] L. E. Dickson, “History of the Theory of Numbers”, Chelsea Publishing Company, 1971.

[11] B. Doyon, J. Lepowsky and A. Milas, Twisted vertex operators and Bernoulli polynomials, arXiv: math. QA/0311151 (2003). | MR | Zbl

[12] G. V. Dunne and C. Schubert, Bernoulli number identities from quantum field theory, arXiv: math. NT/0406610 (2004).

[13] D. B. Fairlie and A. P. Veselov, Faulhaber and Bernoulli polynomials and solitons, Physica D 152-153 (2001), 47–50. | MR | Zbl

[14] C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139 (2000), 173–199. | MR | Zbl

[15] M. Hazewinkel, “Formal Groups and Applications”, Academic Press, New York, 1978. | MR | Zbl

[16] T. Honda, Formal groups and zeta-functions, Osaka J. Math. 5 (1968), 199–213. | MR | Zbl

[17] K. Ireland and M. Rosen, “A Classical Introduction to Modern Number Theory”, Springer-Verlag, 1982. | MR | Zbl

[18] K. Iwasawa, “Lectures on p-adic L-Functions”, Princeton University Press, Princeton, 1972. | MR | Zbl

[19] S. Marmi and P. Tempesta, Polylogarithms, hyperfunctions and generalized Lipschitz summation formulae, Preprint Scuola Normale Superiore, Centro di Ricerca Matematica “Ennio De Giorgi” 1–2007, arXiv: 0712.1046v1 [math/NT] (2007). | Zbl

[20] S. Marmi and P. Tempesta, On the relation between formal groups, In: “Appell Polynomials and Hyperfunctions, Symmetry and Perturbation Theory”, G. Gaeta, R. Vitolo and S. Walcher (eds.), World Scientific, 2007, 132–139. | MR | Zbl

[21] S. P. Novikov, The methods of algebraic topology from the point of view of cobordism theory, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 885–951, transl. Math. SSR–Izv. 1 (1967), 827–913. | MR

[22] A. Ogg, “Modular Forms and Dirichlet Series”, Mathematics Lecture Note Series, W. A. Benjamin, Inc., 1969. | MR | Zbl

[23] N. Ray, Stirling and Bernoulli numbers for complex oriented homology theory, In: “Algebraic Topology”, G. Carlsson, R. L. Cohen, H. R. Miller and D. C. Ravenel (eds.), Lecture Notes in Math., Vol. 1370, Springer-Verlag, 1986, 362–373, | Zbl

[24] G. C. Rota, “Finite Operator Calculus”, Academic Press, New York, 1975.

[25] J-P. Serre, Courbes elliptiques et groupes formels, Annuaire du Collège de France (1966), 49–58. (Oeuvres, Vol. II, 71, 315–324.)

[26] P. Tempesta, Formal Groups, Bernoulli-type polynomials and L-series, C. R. Math. Acad. Sci. Paris, Ser. I 345 (2007), 303–306. | MR

[27] P. Tempesta, On new Appell sequences of polynomials of Bernoulli and Euler type, J. Math. Anal. Appl. 341 (2008), 1295–1310. | MR | Zbl

[28] P. T. Young, Congruence for Bernoulli, Euler and Stirling Numbers, J. Number Theory 78 (1999), 204–227. | MR