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Best textbooks for analytic number theory?

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It's simply impossible to study modular forms without any complex analysis. Generally, a lot of analytic number theory depends on complex analysis.

Elliptic curves can be studied either via algebraic geometry or (over C) via complex analysis.

So my recommendation would be a book that's not actually a book on analytic number theory per se: Freitag's Complex Analysis. This discusses modular forms, elliptic curves (via elliptic functions) and some other analytic NT such as the functional equation for the Riemann zeta function, while also giving you the necessary background from complex analysis.
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You don’t indicate your background in either topic yet.

The book by Diamond and Shurman aims to explain the modularity theorem linking the topics you listed.  To relate them, the purely analytic theory of elliptic curves over C is inadequate. You need to develop things algebro-geometrically too.
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Serre's "A Course in Arithmetic" is pretty much a standard for algebraic and analytic number theory. I know it goes into modular forms but I've only read through the algebraic portion so far so I'm not too sure if it includes anything on elliptic curves. It was published in 1973 so it's in the right time period for there to be mention of elliptic curves, though probably not with the depth of research we have after Wiles and others.
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There a chapter in Iwaniec&Kowalski on the topic of modular forms (chap 14). Elliptic curves are also discussed.

I don’t think it’s a good book for an introductory course but it’s a standard reference.

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