Summary
We show that Kummer's conjectured asymptotic estimate for the size of the first factor of the class number of a cyclotomic field is untrue under the assumption of two well-known and widely believed conjectures of analytic number theory.
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Ankeny, N.C., Chowla, S.: The class number of the cyclotomic field. Proc. Natl. Acad. Sci. USA35, 529–532 (1949)
Bombieri, E., Friedlander, J.B., Iwaniec, H.: Primes in arithmetic progressions to large moduli. Acta Math.156, 203–251 (1986); II. Math. Ann.277, 361–393 (1987); III. J. Am. Math. Soc.2, 215–224 (1989)
Chen, J.: On the representation of a large even integer as the sum of a prime and the product of at most two primes. Sci. Sinica16, 157–176 (1973)
Davenport, H.: Multiplicative number theory. (2nd. Edn.) New York: Springer 1980
Elliott, P.D.T.A., Halberstam, H.: A conjecture in prime number theory. Symp. Math.4, 59–72 (1968–69)
Erdös, P., Nicolas, J.L.: On functions connected with prime divisors of an integer, Number Theory and Applications, NATO ASI series, Mollin, R.A. (ed.) pp. 381–391, 1989
Friedlander, J.B., Granville, A.: Limitations to the equi-distribution of primes. I Ann. Math.129, 363–382 (1989)
Fung, G., Granville, A., Williams, H.C.: Computations of the first factor of the class number of cyclotomic fields. (preprint)
Halberstam, H., Richert H.E.: Sieve methods. New York: Academic Press 1974
Hardy, G., Littlewood, J.E.: Some problems of ‘partitio numerorum’, III. On the expression of a number as a sum of primes. Acta Math.44, 1–70 (1923)
Hasse, H.: Über die Klassenzahl abelscher Zahlkörper, Berlin: Akademic-Verlag 1952
Hensley, D., Richards, I.: Primes in intervals. Acta Arithm.25, 375–391 (1974)
Kummer, E.E.: Mémoire sur la théorie des nombres complexes composés de racines de l'unité et des nombres entiers. J. Math. Pures Appl.16, 377–498 (1851); Collected Works, Vol. I., p. 459
Maier, H.: Small differences between prime numbers. Mich. Math J.35, 323–344 (1988)
Masley, J.M., Montgomery, H.L.: Cyclotomic Fields with unique factorization. J. Reine Angew. Math.286/287, 248–256 (1976)
Montgomery, H.L.: Topics in multiplicative number theory. (Lect. Notes Math., Vol. 227). New York: Springer 1971
Montgomery, H.L., Vaughan, R.C.: The large sieve, Mathematika20, 119–134 (1973)
Pajunen, S.: Computation of the growth of the first factor for prime cyclotomic fields. BIT16, 85–87 (1976)
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Granville, A. On the size of the first factor of the class number of a cyclotomic field. Invent Math 100, 321–338 (1990). https://doi.org/10.1007/BF01231189
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DOI: https://doi.org/10.1007/BF01231189