I am an undergraduate student of mathematics and recently took an introductory course in analytic number theory, where the instructor roughly followed Apostol's first text on the subject. I have now started reading Davenport's 'Multiplicative Number Theory'. Without going into too much detail, I found that, while the results used some common techniques in their proofs, they were otherwise quite independent. Since I am relatively inexperienced, I found that quite strange, considering that most subjects I have read so far (real and complex analysis, abstract algebra, measure theory, functional analysis, algebraic topology) seem to have a coherent development, rather than simply be a collection of problems that have been solved using roughly similar machinery.

I was wondering if there is an overarching idea behind the study of analytic number theory (classical, as well as sieve methods), any specific open problems that motivated past research in the field, and if there is any textbook that treats it from that perspective, rather than just a collection of interesting problems. For instance, although I know very little about it, my professor once told me about the Langlands Program and said that the current goal of several mathematicians working in areas of algebraic number theory and automorphic forms to resolve the conjectures of that program.

Also, unlike many other areas, I couldn't effectively use the approach many of my professors seem to recommend, of reading the theorem and attempting to prove it by myself. I am inclined to believe that it is my own shortcomings that prevent this approach, but if it is a part of a wider trend, and I am reading it "wrong", for want of a better phrase, I would like to know the same, and would like to know how exactly such a subject is to be most efficiently learnt.

I haven't quite managed to phrase the question as well as I had hoped to, so, if you have any replies to the title in itself, any motivation towards a coherent understanding of the subject, then, I would be much obliged to you for your input/advice.

I wasn't quite sure what tag to use, and so, chose what I thought was most appropriate. I hope this will not be an issue.

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