100 " d|n
2. Modular Arithmetic
dµ(d)
"
1≤q≤n/d
q=
"
dµ(d)
+ 1) 1 " µ(d)n2 " = + nµ(d) , 2 2 d
n n d(d
d|n
d|n
d|n
where the penultimate equality comes from Theorem 1.1 on page 2 acting on the previous second sum. We now invoke Theorem 2.17 on the page before and nφ(n) Theorem 2.14 on page 96 to get that the above equals 21 n2 φ(n) n = 2 . Hence, we have demonstrated the important fact that for n > 1, "
1≤k≤n gcd(k,n)=1
k=
nφ(n) . 2
Biography 2.4 Franz Carl Joseph Mertens (1840–1927) was born on March 20, 1840 in Schroda, Posen, Prussia (now Sroda, Poland). Mertens studied at the University of Berlin with Kronecker and Kummer as his advisors, obtaining his doctorate, on potential theory, in 1865. His first position was at the Jagiellonian University at Cracow, and he worked his way up to ordinary professor by 1870. He also held positions at the Polytechnic in Graz, Austria, and the University of Vienna from which he retired in 1911. Among his students at Vienna were Ernst Fischer (1875–1954), and Eduard Helly (1884–1943). Fischer is best known for the Riesz-Fischer theorem in the theory of Lebesgue integration, and Helly proved the Hahn-Banach theorem in 1912, some fifteen to twenty years before Hahn and Banach provided their versions. Mertens’ areas of interest included not only number theory and potential theory, but also geometric applications to algebra and matrix theory. Other than the conjecture that bears his name, he is known for his elementary proof of Dirichlet’s Theorem — see Biography 1.8 on page 35 and Theorem 1.19 on page 35. He also has his name attached to three number-theoretic results on density of primes, one of which is an asymptotic formula for the fraction of natural numbers not divisible by the primes less than a given x. Although his conjecture was proved to be false, as noted above, it stood for almost a century before it fell. It is unfortunate since a proof of his conjecture would have meant that the Riemann hypothesis is true — see page 72. Merten’s died on March 5, 1927 in Vienna.
Exercises ! ! 2.22. Prove that if d ! n ∈ N, then φ(d) ! φ(n). Use this fact to show that 2|φ(n) for any n > 2. 2.23. Prove that
p−1 " j=1
j p−1 ≡ −1 (mod p)