Good book for complex analysis

[Maths Absorber]

Hi friends,

I am looking for a good book on complex analysis. My goal is to add some new techniques to my problem solving arsenal, learn some elegant non-intuitive proofs and finally build up to knowing enough about the Zeta function to understand it's relationship with prime numbers.

 

[mathwonk]

here are some remarks i wrote down after reading the best asource on this tipioc, namely riemann's works:

it seems that first euler noticed that, assuming the factorization of natural numbers into primes, that speaking purely formally, one has the sum of all the natural numbers n equal to the product of the infinitely many factors

(1+2 + 2^2 + 2^3 +...)(1+3+3^2+3^3+...)(...)(1+p+p^2+p^3+...)(... ..).

To try to render this meaningful, one can instead consider reciprocals, since now

(1+1/2+1/2^2+...) = 1/(1 - [1/2]),..., (1+1/p+1/p^2+...) = 1/[1 - 1/p], ...

so now at least all the factors are finite, and we may hope that the product of the factors

1/[1 - 1/p] for all primes p, equals the sum of the reciprocals of the natural numbers

1 + 1/2 + 1/3 + 1/4 +... however still both these infinite expressions diverge.

then euler thought to use exponents. indeed he knew the sum of the squares of the reciprocals of natural numbers equalled pi^2/6, etc...,

so he looked at the equation PRODUCT (over all primes p) of 1/[1 - (1/p)^s]

= SUM over all natural numbers n, of 1/n^s. he considered it principally for integers s.

then perhaps legendre looked at this expression as a function of a real variable s, and noted it converges for any s > 1.

then riemann took it up and naturally for him asked for its fullest range of definition. thus he considered complex values of s, obtaining a complex meromorphic function called the zeta function.

i.e. for riemann a complex holomorphic function is determined globally by its values in any region, and moreover, his overriding point of view was that a meromorphic complex function is best understood by its zeroes and poles.

hence here is a function which is wholly determined by an expression involving only prime numbers, so its behavior reflects the nature, i.e. location, of the prime numbers, but which is best understood by considering its zeroes and poles (of which it has none for Re(s) > 1). thus the curious fact that s = 1 is a point where the original formula makes no sense, is replaced by the intersting fact that s =1 is the unique pole of the global extension of this function. hence the other crucial points must be the zeroes of the function, none of which are visible until Riemann's extension of the function.

now the problem riemann considered was that of determining roughly the number of prime numbers that are less than a given number x = pi(x).

according to edwards, euler's observations on the rate of divergence of the series 1+ 1/2 + 1/3 + 1/5 +...+1/p+... can be phrased as saying the density of primes is 1/log(x).

Then Gauss conjectured that the number pi(x) was well approximated by Li(x) = the integral from 0 to x of dt/ln(t), (up to a small constant).

now riemann's paper attempted to improve this approximation for pi(x) as follows:

Using his zeta function, and various manipulations of integral expressions for it, Riemann claimed to show that in fact Gauss's estimate Li(x) approximates more closely the number

Li(x) (roughly) = pi(x) + (1/2)pi(x^[1/2]) + (1/3)pi(x^[1/3]) + ...

i.e. not just the number of primes less than x but also 1/2 the number of squares of primes, plus 1/3 the number of cubes of primes, ...

hence he says that Li(x) overestimates pi(x) by a term of order of magnitude x^(1/2).

By solving the approximate equation above for pi(x) he obtains a formula approximating pi(x) (roughly) =

Li(x) - (1/2)Li(x^[1/2]) - (1/3)Li(x^[1/3]) - (1/5)Li(x^[1/5]) + (1/6)Li(x^[1/6]) - +...

where in this sum, only square - free integers appear, and the sign is determined by the number of prime factors, minus if an odd number, or plus if an even number.

Empirically this approximation is indeed superior to Gauss's, but Riemann's argument for the error expression between this formula and pi(x), i.e. for his claimed theoretical accuracy of his approximation, is apparently what needs the Riemann hypothesis on just where the zeroes of the zeta function are located for a rigorous proof.

 

[Ssnow]

"A second course in Complex Analysis" of William A. Veech, Dover , the last part of the book contains techniques about Zeta Riemann function and number theory.