You might want to look at the paper "Le théorème des nombres premiers et la transformation de Fourier" by Jean-Benoît Bost, available at http://www.math.polytechnique.fr/xups/xups02-01.pdf It gives a proof of PNT freely using harmonic analysis and some basics of distributions, but as little complex analysis as possible. In particular, section 4 proves a statement that is equivalent to the non-vanishing of zeta(1+it) using only real analysis. Unfortunately, for my taste there still is too much complex reasoning in the preceding sections of the paper.

(I'm sure you know that the brevity of Don Zagier's proof of PNT can't be beaten, provided you accept complex analysis.)

You might want to look at the paper "Le théorème des nombres premiers et la transformation de Fourier" by Jean-Benoît Bost, available at https://www.cmls.polytechnique.fr/xups/xups02-01.pdf It gives a proof of PNT freely using harmonic analysis and some basics of distributions, but as little complex analysis as possible. In particular, section 4 proves a statement that is equivalent to the non-vanishing of zeta(1+it) using only real analysis. Unfortunately, for my taste there still is too much complex reasoning in the preceding sections of the paper.

(I'm sure you know that the brevity of Don Zagier's proof of PNT can't be beaten, provided you accept complex analysis.)

You might want to look at the paper "Le théorème des nombres premiers et la transformation de Fourier" by Jean-Bonoit Bost, available at http://www.math.polytechnique.fr/xups/xups02-01.pdf It gives a proof of PNT freely using harmonic analysis and some basics of distributions, but as little complex analysis as possible. In particular, section 4 proves a statement that is equivalent to the non-vanishing of zeta(1+it) using only real analysis. Unfortunately, for my taste there still is too much complex reasoning in the preceding sections of the paper.

(I'm sure you know that the brevity of Don Zagier's proof of PNT can't be beaten, provided you accept complex analysis.)

You might want to look at the paper "Le théorème des nombres premiers et la transformation de Fourier" by Jean-Benoît Bost, available at http://www.math.polytechnique.fr/xups/xups02-01.pdf It gives a proof of PNT freely using harmonic analysis and some basics of distributions, but as little complex analysis as possible. In particular, section 4 proves a statement that is equivalent to the non-vanishing of zeta(1+it) using only real analysis. Unfortunately, for my taste there still is too much complex reasoning in the preceding sections of the paper.

(I'm sure you know that the brevity of Don Zagier's proof of PNT can't be beaten, provided you accept complex analysis.)