Haran, Shai Riesz potentials and explicit sums in arithmetic. (English) Zbl 0788.11055 Invent. Math. 101, No. 3, 697-703 (1990). The paper originates with Weil’s formulas relating zeros of zeta functions of number fields and the distribution of prime ideals. This is shown to be connected with adelic Riesz potentials. For the motivation behind the paper, the reader is referred to S. Haran [Proc. Symp., Durham 1989, Lond. Math. Soc. Lect. Notes Ser. 153, 257-270 (1991; Zbl 0744.11042)]. The paper gives a finite expression for the contribution at infinity and a uniform way of expressing the contribution for a finite and infinite prime. Reviewer: A.A.Terras (La Jolla) Cited in 1 ReviewCited in 29 Documents MSC: 11R42 Zeta functions and \(L\)-functions of number fields Keywords:Weil explicit formulas; zeros of zeta functions; distribution of prime ideals; adelic Riesz potentials Citations:Zbl 0744.11042 PDF BibTeX XML Cite \textit{S. Haran}, Invent. Math. 101, No. 3, 697--703 (1990; Zbl 0788.11055) Full Text: DOI EuDML OpenURL References: [1] Berg, C., Forst, G.: Potential theory on locally compact abelian groups. Berlin-Heidelberg-New York: Springer 1975 · Zbl 0308.31001 [2] Feller, W.: An Introduction to Probability Theory and its Applications. Vol. 2, New York: John Wiley and Sons 1970 · Zbl 0077.12201 [3] Haran, S.: Index theory, Potential Theory, and the Riemann hypothesis. In: Proc. LMS Symp. on ?L-functions and Arithmetic?, Durham, 1989 (to appear) · Zbl 0744.11042 [4] Haran, S.: Analytic Potential Theory over thep-Adics (Preprint) · Zbl 0847.31006 [5] Landkof, N.S.: Foundations of Modern Potential Theory. Berlin-Heidelberg-New York: Springer 1972 · Zbl 0253.31001 [6] Taibleson, M.H.: Fourier Analysis on Local Fields, Princeton University Press 1975 · Zbl 0319.42011 [7] Weil, A.: Sur les courbes algébriques et les variétés qui s’en déduisent. Paris: Hermann (1948) · Zbl 0036.16001 [8] Weil, A.: Sur les formules explicites de la théorie des nombres, premiers. Proc. R. Physiogr. Soc. Lund21, 252-265 (1952) · Zbl 0049.03205 [9] Weil, A.: Sur les formules explicites de la théorie des nombres. Izv. Mat. Nauk.36, 3-18 (1972) · Zbl 0245.12010 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.