Proposition 2.1. There are an infinite number of primes of the form p = 4n 1. Proof. Suppose there are only a finite number of such primes, say

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1 Chater 2 Euclid s Theorem Theorem 2.. There are a ifiity of rimes. This is sometimes called Euclid s Secod Theorem, what we have called Euclid s Lemma beig kow as Euclid s First Theorem. Proof. Suose to the cotrary there are oly a fiite umber of rimes, say Cosider the umber, 2,..., r. N = 2 r +. The N is ot divisible by i for i =,..., r, sice N has remaider whe divided by each of these rimes. Take ay rime factor q of N. We kow from the Fudametal Theorem that there is such a rime.) The q differs from all of the rimes,..., r, sice it divides N. Hece our assumtio that the umber of rimes is fiite is uteable. 2. Variats o Euclid s roof Proositio 2.. There are a ifiite umber of rimes of the form = 4. Proof. Suose there are oly a fiite umber of such rimes, say Cosider the umber, 2,..., r. N = 4 2 r. Sice N is odd, it is a roduct of odd rime factors. Ay odd umber is of the form 4 + or 4. If all the rime factors of N were of the form 4 + their roduct N would be of this form. Sice it is ot, we coclude that N has a rime factor of the form 4. This must differ from,..., r, sice oe of these rimes divides N. Hece we have a further rime of the form 4, cotradictig our origial assumtio. Rather surisigly, erhas, we caot show i the same way that there are a ifiity of rimes of the form 4 +, although that is true. 2

2 2.2 The zeta fuctio Havig established that there are a ifiity of rimes, the questio arises: How are these rimes distributed? Riema s zeta fuctio is the major tool i this study. Defiitio 2.. Riema s zeta fuctio ζs) is defied by whe this series coverges. ζs) = + 2 s + 3 s + 4 s +, Although Riema s ame is give to this fuctio, it was i fact itroduced by Euler. However, Euler oly cosidered the fuctio for real s. Riema s cotributio was to cosider the fuctio for comlex s, i a revolutioary aer O the umber of rimes less tha a give value, ublished i 859, usig the theory of comlex fuctios laid dow by Cauchy some 20 years before. Note that the terms i the series ca be defied, for real ad comlex s, by s = e s l. We see from this that ad so x+iy)) = e x l e iy l, s = Rs), sice e iθ = for all real θ. A simle but useful tool allows us to determie whe the series coverges. Lemma 2.. If fx) is a mootoe fuctio the f) coverges fx)dx coverges. The lower limits o each side so ot matter; it is sufficiet that fx) is defied for x X. Oe might thik it should be secified that fx) is cotiuous. But i fact ay mootoe fuctio fx) is ecessarily Riema itegrable ad so Lebesgue itegrable). This follows from the fact that fx) has oly a eumerable set of discotiuities, so the artitios i Riema sums ca be chose with ed-oits avoidig these oits. Proof. We may assume relacig fx) by fx) if ecessary) that fx) is decreasig. We may also assume that fx) 0 as x ; for we kow that fx) teds to a limit l ossibly ), ad if l 0 the it is easy to see that both sum ad itegral diverge. If x + the f) fx) f + ). 2 2

3 Hece Thus f) fm)+fm+)+ +f ) from which the result follows. + m fx)dx f + ). fx)dx fm+)+fm+2)+ +f), Proositio 2.2. The series for ζs) coverges for Rs) >. Proof. For real s > this follows from the revious lemma, sice x s dx = s x s ). Ad it follows from this that s is absolutely coverget if Rs) >, sice s = Rs). 2.3 Euler s Product Formula If a, a 2,... is a ifiite sequece of real of comlex umbers, we say that the ifiite roduct a a 2 coverges to l 0 if the artial roducts A = a a 2 a coverge to l. If A 0 the we say that the roduct diverges to 0.) If the a are real ad ositive we ca covert a ifiite roduct to a ifiite series by takig logarithms: a coverges l a coverges. Because of this logarithmic coectio we usually take the roduct i the form + a ). This allows us to ass to comlex a rovided a ) <, sice i that case l + a ) = a 2 a2 + 3 a3 4 a4 +. Lemma 2.2. Suose a 2 is absolutely coverget. The + a ) coverges a coverges. I articular the roduct is coverget if the series is absolutely coverget. Proof. Sice 2 a2 3 a3 + 4 a4 2 a a a 4 + a 2 + a 3 + a 4 + ) 2 = a 2 2 a 2 a 2, 2 3

4 if a /2. It follows that l N + a ) N a N a 2 rovided a /2 for [, N], from which the result follows. Theorem 2.2. For Rs) >, ζs) = s ), where the ifiite roduct exteds over all rime umbers. Proof. The formula ca be writte + 2 s + 3 s + 4 s + = + 2 s + 2 2s + ) + 3 s + 3 2s + ) + 5 s + 5 2s + ). If = 2 e 2 3 e3 5 e5 the s = 2 e 2s 3 e3s 5 e 5s ; ad we see that s o the left is matched by 2 e 2s from the first factor o the right, 3 e 3s from the secod factor, ad so o. Theorem 2.3. The series where rus over the rimes) diverges. Proof. Takig s = i the above formula, the series diverges. So the roduct ) also diverges. It follows that the iverse ) = 0, ie the artial roduct P = ) 0 as. We say that the ifiite roduct diverges to

5 Takig logarithms, it follows that log ) =. Recall that log x) = x + x 2 /2 x 3 /3 +. If x is small, say x < /2, we ca combie the secod ad later terms: Thus where a coverges, sice x 2 /2 x 3 /3 + x 2 /2 + x + x 2 + ) x 2 = 2 x) x 2. = log ) + a. a 2, ad / 2 coverges with / 2. We coclude that / is the sum of a diverget series ad a coverget series, ad therefore diverges. Note that coverges for r >, sice r r coverges by comariso with the itegral /x r ). 2.4 Dirichlet s Theorem Theorem 2.4. There are a ifiity of rimes i ay arithmetic sequece with d > 0 ad gcda, d) =. a + d = 0,, 2,... ) 2 5