the riemann zeta function is a refinement of eulers product formula. the idea is to define a complex analytic function whose definition is entirely determined by the distribution of primes in the integers, and then to use complex analysis on that function to try to understand the prime distribution better.
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i.e. the primes determine the function, so anything you can learn about the function should illuminate the distribution of primes.
the well known harmonic series summation 1/n for all positive integers, can be factored formally as the infinite product, over all primes p, of the series summation 1/p^r for all r. this is eulers product.
if we sum this geometric series we get 1/[1 - (1/p)], hence tha hrmonic series "equals" the product of these terms obver all primes.
now although the ahrmonic series diverges, if we raise each tewrm to a power s greater than 1, we have a convergent series, summation 1/n^s, correspondiung via eualers product, to the product over all p of the factors
1/[1 - p^(-s)]. thus someone, probably riemann had the brilliant idea that if we define a function to be zeta(s) = summation 1/n^s =
product 1/[1 - p^(-s)], we get an interesting function, and of mcourse the most informnation is obtained by using compelkx variables s.
then one topic of interest is the rate at which this functioin goes to infinity as s approaches 1, i.e. as the zeta function approaches the harmonic series.
then someone (riemann?) also observed that this function, defiend above for re(s) > 1, could actually be analytically continued, i.e. extended, to all s as a function with isolated zeroes and poles. then another topic of interest is to determine its zewroes.
indeed riemann tried to show that the zeroes were all on the line re(s) = 1/2, and to derive from this a very precise estimate of the number of primes below any given magnitude.
riemanns paper has been translated several tiems into english and i recommend you get hold of a copy and read it, whatever the result in comporehension.
