On some Extremal Problems of Landau

Révész, Szilárd

Serdica Mathematical Journal, Tome 33 (2007) no. 1, pp. 125-162

Résumé

The prime number theorem with error term presents itself as '(x) = ∫2x [dt/ logt] + O ( x e- K logL x). In 1909, Edmund Landau provided a systematic analysis of the proof seeking better values of L and K. At a key point of his 1899 proof de la Vallée Poussin made use of the nonnegative trigonometric polynomial 2/3 (1+cos x)2 = 1+4/3 cosx +1/3 cos2x. Landau considered more general positive definite nonnegative cosine polynomials 1+a1cos x+… + ancos nx ≥ 0, with a1> 1,ak ≥ 0 (k = 1,…,n), and deduced the above error term with L = 1/2 and any K 1/(2V(a))½, where V(a): = (a1+a2+…+ an)/(( (a1)½-1)2). Thus the extremal problem of finding V: = minV(a) over all admissible coefficients, i.e. polynomials, arises. The question was further studied by Landau and later on by many other eminent mathematicians. The present work surveys these works as well as current questions and ramifications of the theme, starting with a long unnoticed, but rather valuable Bulgarian publication of Lubomir Chakalov.

Détail

Keywords: Prime Number Formula, Positive Trigonometric Polynomials, Positive Definite Functions, Extremal Problems, Borel Measures, Convexity, Duality