An Asymptotic Formula for a Trigonometric Sum of Vinogradov

✍ Scribed by Todd Cochrane; J.C Peral

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 189 KB
• Volume: 91
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.2001.2679

✦ Synopsis

We obtain a representation formula for the trigonometric sum f (m, n)

and deduce from it the upper bound f(m, n) < (4/p 2 ) m log m+ (4/p 2 )(c -log(p/2)+2C G ) m+O(m/log m), where C G is the supremum of the function G(t) :=; . k=1 log |2 sin pkt|/(4k 2 -1), over the set of irrationals. The coefficients on both the main term and the second term are shown to be best possible. This improves earlier bounds for f(m, n). It is conjectured that C G =G(2) % 0.236. We also obtain the following asymptotic formula: If a is a real algebraic integer of degree 2 with 0 < a < 1, then for any rational approximation n/m of a with 0 < n < m we have f(m, n)=(4/p 2 ) m log m+(4/p 2 )(c -log(p/2)+2G(a)) m+ O a (|a -n m | 1/2 m log m).