What are the fertile areas of research in Analytic Number Theory? My professor once told me that Analytic Number Theory was "dead," which at the time was something of a disappointment, and which I struggled to agree with. Surely any subject may appear inferrtile in periods when researchers do not study the subject? I was just wondering, is this a generally accepted view? What reasons would be behind such opinion, and what are the fertile directions of research in this field?

These and these are some sample and well known open problems in Analytic Number Theory. Also look at this to have a flavor of the most recent 'gossip' in Analytic Number Theory world.