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

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.

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.

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     title = {The {Hooley-Huxley} contour method for problems in number fields {III} : frobenian functions},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {65--76},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {1},
     year = {2001},
     mrnumber = {1838070},
     zbl = {1067.11071},
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     url = {http://archive.numdam.org/item/JTNB_2001__13_1_65_0/}
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Coleman, Mark D. The Hooley-Huxley contour method for problems in number fields III : frobenian functions. Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 65-76. http://archive.numdam.org/item/JTNB_2001__13_1_65_0/

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