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 53 (1947) pp. 976-981

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  \pi(x)<li(x)
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 Proceedings of the Royal Society of London, volume 151, pages 234-255, 1935. The djvu file is only 0.2 MB instead of  8 MB for pdf file.

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 Lambda, which should be less or equal to zero if RH is true (and reverse: if RH is true then Lambda is less or equal 0). As the best lower bound is now  Lambda > - 2.7x10-9  we can say that RH is true up to epsilonwhere currently epsilon< 2.7x10-9. It lead Odlyzko to say that  "if RH is true, it is barely true", see his paper
An improved bound for the de Bruijn-Newman constant, which appeared in Numerical Algorithms, 25 (2000), pp. 293-303
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.