Remark on a Paper of Yu on Heilbronn's Exponential Sum

✍ Scribed by Jan-Christoph Puchta

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 77 KB
• Volume: 87
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.2000.2593

✦ Synopsis

We show that S h (a)= p n=1 e(an hp Âp 2 )< <(h, p&1) 11Â16 p 7Â8 , sharpening a result of Yu.

2001 Academic Press

Let p be a prime, e(x)=e 2?ix , and define the exponential sum S h (a) by S h (a)= : p n=1 e \ an hp p 2 + .

For a long time it was an unsolved problem whether S 1 (a)=o( p) uniformly in a. In 1996 Heath-Brown [1] proved that S 1 (a)< <p 11Â12 . Further Heath-Brown proved S h (a)< <(h, p&1) 5Â4 p 11Â12 (unpublished). Yu [3] sharpened this to S h (a)< <(h, p&1) p 11Â12 . Recently, Heath-Brown and Konyagin [2] improved the bound for S 1 (a) to S 1 (a)< <p 7Â8 . Their results together with the method of [3] give the bound S h (a)< <(h, p&1) p 7Â8 . The aim of this note is to improve the dependence on h further. We will prove the following theorem.

Theorem 1. We have S h (a)< <( p&1, h) 11Â16 p 7Â8 . Note that we may assume p |3 a, since otherwise Weil's estimate gives S h (a)< <hp 1Â2 , which together with the trivial bound S h (a) p implies our theorem. Further we may assume h | p&1.

Our proof follows the lines of [3]; the improvement comes from the fact that we will use a nontrivial bound for the occurring sum on the characters