The Hooley-Huxley contour method for problems in number fields III : frobenian functions
Journal de Théorie des Nombres de Bordeaux, Volume 13 (2001) no. 1, pp. 65-76.

In this paper we study finite valued multiplicative functions defined on ideals of a number field and whose values on the prime ideals depend only on the Frobenius class of the primes in some Galois extension. In particular we give asymptotic results when the ideals are restricted to “small regions”. Special cases concern Ramanujan's tau function in small intervals and relative norms in “small regions” of elements from a full module of the Galois extension.

On se donne une fonction multiplicative définie sur l’ensemble des idéaux d’un corps de nombres. On suppose que les valeurs prises par cette fonction sur les idéaux premiers ne dépendent que de la classe de Frobenius des idéaux premiers dans une certaine extension galoisienne. Dans ce texte, nous donnons une estimation asymptotique du nombre d’idéaux d’un corps de nombres lorsqu’ils varient dans un “ petit domaine ”. Nous nous intéressons particulièrement aux cas de la fonction τ de Ramanujan dans de petits intervalles, ainsi qu’à la fonction norme relative pour des éléments d’un module d’une extension galoisienne variant dans de petits domaines.

@article{JTNB_2001__13_1_65_0,
     author = {Coleman, Mark D.},
     title = {The Hooley-Huxley contour method for problems in number fields III : frobenian functions},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {1},
     year = {2001},
     pages = {65-76},
     zbl = {1067.11071},
     mrnumber = {1838070},
     language = {en},
     url={jtnb.centre-mersenne.org/item/JTNB_2001__13_1_65_0/}
}
Coleman, Mark D. The Hooley-Huxley contour method for problems in number fields III : frobenian functions. Journal de Théorie des Nombres de Bordeaux, Volume 13 (2001) no. 1, pp. 65-76. https://jtnb.centre-mersenne.org/item/JTNB_2001__13_1_65_0/

[1] M.D. Coleman, The Hooley-Huxley contour method for problems in number fields I: Arithmetic Functions. J. Number Theory 74, (1999), 250-277. | MR 1671657 | Zbl 0978.11052

[2] M.D. Coleman, The Hooley-Huxley contour method for problems in number fields II: Factorization and Divisiblity. Submitted to J. Number Theory.

[3] P. Deligne, J.-P. Serre, Formes modulaires de poids 1. Ann. scient. Ec. Norm. Sup. (4) 7 (1974), 507-530. | Numdam | MR 379379 | Zbl 0321.10026

[4] R.W.K. Odoni, On the norms of algebraic integers. Mathematika 22 (1975), 71-80. | MR 424757 | Zbl 0313.12007

[5] R.W.K. Odoni, Representations of algebraic integers by binary quadratic forms and norm forms of full modules of extension fields. J. Number Theory 10 (1978), 324-333. | MR 506642 | Zbl 0411.12010

[6] R.W.K. Odoni, The distribution of integral and prime-integral values of systems of full-norm polynomials and affine-decomposable polynomials. Mathematika 26 (1979), 80-87. | MR 557130 | Zbl 0444.12008

[7] R.W.K. Odoni, Notes on the method of Frobenian functions, with applications to the coefficents of modular forms. In: Elementary and analytic theory of numbers, Banach Center Publications, vol. 17, Polish Scientific Publishers, Warsaw 1985, pp. 371-403. | MR 840484 | Zbl 0596.10040

[8] R.W.K. Odoni, On the distribution of norms of ideals in given ray-classes and the theory of central ray-class fields. Acta Arith. 52 (1989), 373-397. | MR 1030089 | Zbl 0694.12007

[9] K. Ramachandra, Some problems of analytic number theory. Acta Arith. 31 (1976), 313-324. | MR 424723 | Zbl 0291.10034