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Averages of Real Character Sums

✍ Scribed by M.V. Armon

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 150 KB
Volume: 77
Category: Article
ISSN: 0022-314X
DOI: 10.1006/jnth.1999.2383

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✦ Synopsis

Two mean-value estimates are proved for sums of real characters /(n) defined by Kronecker's symbol (DÂn), where D ranges over Q, the set of quadratic discriminants.

1999 Academic Press

1. INTRODUCTION AND STATEMENT OF RESULTS

Every real nonprincipal character / D (n) can be represented by the Kronecker symbol (DÂn) for n>0, where D is taken from the set Q of quadratic discriminants:

Each D # Q is uniquely expressible as dm 2 , where d # F, the set of fundamental discriminants,

so that / D is induced by the primitive character / d , where / d (n)=(dÂn) for n>0. For fixed positive n, the values of (DÂn) coincide with the values of / n (D), where / n is a real character modulo n or 4n. Since / n (D) is defined for all D, this observation will allow us to sum (DÂn) over all D for a fixed n>0. Details and further information on the Kronecker symbol can be found in [3 5, 7, 10].

The first theorem is an extension to Kronecker symbols of a theorem of Montgomery and Vaughan [9] that bounds an average of sums of Legendre symbols. Its proof requires a few modifications of Montgomery and Vaughan's proof and is simplified somewhat by a result of .

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