Notes sur la fonction ζ de Riemann, 3

✍ Scribed by Luis Báez-Duarte; Michel Balazard; Bernard Landreau; Eric Saias

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 170 KB
• Volume: 149
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1999.1861

✦ Synopsis

a la me moire de gian-carlo rota Let (t) denote the fractional part of t, H the Hilbert space L 2 (0, + ), B the subspace of H of functions f (t)= n k=1 c k (% k Ât), where n # N, c k # C, and 0<% k 1 for 1 k n, / the characteristic function of ]0, 1] and D(*) the distance in H between / and B * , the subspace of B of functions f such that all % k *. A well-known result of B. Nyman and A. Beurling implies that the Riemann hypothesis is equivalent to the statement lim * Ä 0 D(*)=0. We prove here that inf 0<*<1 D(*) -log(2Â*)>0, and we conjecture that lim * Ä 0 D(*) -log(1Â*)= -2+#&log(4?), where # denotes Euler's constant. This conjecture is supported by numerical experiments. 2000 Academic Press 1. INTRODUCTION Il re sulte facilement des travaux de Nyman [8] et Beurling [4] que l'hypotheÁ se de Riemann e quivaut aÁ la proprie te d'approximation suivante.