The famous paper An elementary proof of the prime nuber theorem, A. Selberg, Annals of Mathematics 50 (1949), pp.303-313

The English translation of the paper M. Riesz Sur l’hypothese de Riemann, Acta Mathematica, pp.185-190 v.40 (1916),

originally published in French.

I was really impressed when I read about the electromechanical device constructed by B. van der Pol in the forties of last century which plotted Riemann zeta function on critical line, here is the original image (zeros are marked by ticks "|" under t axis):

The paper appeared as An electro-mechanical investigation of the Riemann zeta function in the critical strip", Bulletin of the AMS

The pdf file of the famous paper "A Proof that Euler Missed... Apery's Proof of the Irrationality of zeta(3)" by Alfred van der Poorten.

Classical paper G.H. Hardy and J.E. Littlewood Contributions to the theory of the Riemann zeta function and the theory of prime distribution Acta Mathematica 41 (1918) p.119 as a pdf (9 MB) or djvu (0.6 MB) file. To display djvu files the DjVu Browser is needed and it is available for free from here.

Two famous papers written by Skewes where he obtained estimation of vaule of such x that

for the first time: Skewes, S. On the Difference [pi(x)-li(x)] J. London Math. Soc. 8, 277-283, 1933 as pdf (3.8 MB) or as djvu (only 150 kB) (the paper just before it "Statement of a problem in quantum mechanics" was written by P.A.M. Dirac) and the second paper Skewes, S. On the Difference [pi(x)-li(x)]. II. Proc. London Math. Soc. 5, 48-70, 1955 as pdf (11 MB) or as djvu (400 kB).

E.C. Titchmarsh, The zeros of the Riemann zeta-function,

Here is the pdf file (13 MB) of the paper "The Roots of Trigonometric Integrals" Duke Math. J. 17, 197-226, 1950, written over fifty years ago by N.G. de Bruijn which opened a new way of proving or disproving The Riemann Hypothesis. It leads to the notion of de Bruijn - Newman constant , which should be less or equal to zero if RH is true (and reverse: if RH is true then is less or equal 0). As the best lower bound is now > - 2.7x10

You can download the djvu file of the above paper The Roots of Trigonometric Integrals which is only 0.3 MB in size (instead of 13 MB for pdf !!!). To display djvu files the DjVu Browser is needed and it is available for free from here.