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1 Annals of Mathematics An Elementary Proof of Dirichlet's Theorem About Primes in an Arithmetic Progression Author(s): Atle Selberg Source: Annals of Mathematics, Second Series, Vol. 50, No. 2 (Apr., 1949), pp Published by: Annals of Mathematics Stable URL: Accessed: :26 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics

2 ANNALS OF MATHMATICS Vol. 50, No. 2, April, 1949 AN ELEMENTARY PROOF OF DIRICHLET'S THEOREM ABOUT PRIMES IN AN ARITHMETIC PROGRESSION By ATLE SELBERG (Received August 18, 1948) 1. A classical theorem by Dirichlet asserts that every arithmetic progressio ky + 1, where the positive integers k and 1 are relatively prime, represents infinitely many primes as y runs over the positive integers. The object of this paper is to give a new and more elementary proof of this theorem. More elementary in the respect that we do not use the complex characters mod k, and also in that we consider only finite sums. More precisely the theorem that is proved in this paper is the following: For every positive integer k, there exist positive numbers Ck and xo depending only on k, such that, when (k, 1) = 1 we have 1o 1 ( 1. 1 ) E log P > Ck log x, f or x > Xo. pe I(k) The value we will obtain for the Ck could easily be improved, but this is of little interest. 2. Notations. We write in the following, supposing the k fixed, (2.1) Si(x) = log p QI(x) = Sl(x) P:!z p log x' PHIT(k) Further li(d) denotes the M6bius function, and (2.2) Xd = Xd,z = t(d) log. We make use of the formulas (2.3) log P - log x + 0(1), and, lr(x) denoting the number of primes < x (2.4) Tr(x) = 0 o x) which are both well known results, w Finally x(n) will always denote a real, 3. In this paragraph we shall prove 1 Here and in the following, p always denote 297

3 298 ATLE SELBERG We start with the expression on = On,: - z Xci din where n is a positive integer and the summation is extended over all divisors d of n. We shall prove that log2 x, for n = 1, (3.1) OR = J log p logx2/p, for n = pa, a > 1, 2log p log q, forn =paq, a > 1, >1, 0, for all other n. This follows immediately when we remark, that it is clearly enough to prove it for quadrat-frei numbers. Writing n = PlP2 *. pi where the primes are different from each other, one has OVlV2*PVitZ = 0P1P2*Vi-1.z - 0PVP2*..Pi. -,/Pi l from which the result follows at once by induction. Hence, for (1, k) = 1 2 O2 = log p lcg -+ lgplog q 2 n5p5 paz; pj pa<z n l k a_ I (A;) VaQ0 (k)i +*O(logz)= x j log p + logp log q + O (log p log ) P:5 X q< p< p (3.2' 1)p (k pqz \I(k) + 0 logx j logp + 0 ( F log p log q + 0(log2 x) j auz J paq$~ < P<a 2 pab c- =2/ \ a22 / a~2 / = i lcg2 p + E log p log q + O(x), p < pq <a p - I (A;) pq --I (k) where the 0-terms are easily estimated by means of (2.4) and (2.3). On the other hand we have on@ = E Xd E 1 = - E -d + O( E I X ) X E I'd + o(x), npx d<s din k d5z d d~x k d<z d n Il(k) (d.k)-l n S x (d.c)=l (d.k)=1 n - I (k) since E I Xd E log? = O(x). d~x d z d 2 We may omit the factor 2 befor

4 DIRICHLET 'S THEOREM ABOUT PRIMES 299 Comparing this with (3.2) we get E log2p? + E log p log p =X X + +0(x). p5z pqcz k d<z d p_- Z(k) p qi- I (k) (d.k)=l By summing over all residues 1 mod. k and observing that for (1, k) > 1 the left-hand side of the above formula is O(x) by (2.4), we get if wp(k) denotes Eulers function, E log2 p + E log p log q P?Z ' pq?z < (3p3) pq-i (k) From this we deduce by partial summation -1 ) E log2 p + E logplogq} + O(x). sp(k) p<p pq<-z E log2 p + E log p log q 1 p5 X, p pq:z Pq V(k) (3r4) p t-_l (k) pq oi (k) _ pg pq J + 0(log x) =l -lg log2 x + 0(log X), the last form being obtained by summing EpZ5 log2 p/p a Eq-<Xp log q/q by means of (2.3). or In the same way we deduce from (3.3) that log' +ogpq= log3x + l0(log2 )o P:5x P pq:!z pq 3pr(k) p2 I (k) Pq Si (k) (3.5) E +2 E log P 3 (k) log3 X + 0(log2 X), P- I (k) pq- I (k) writing now z log p log2 q _ 0 log p log2 q, pqcz pq pcx P qx/f p pq-i(k) prk q Ip(k) q where p is determined by p- 1(k log p log2 q _ () Xl p p - _ log p pq'z pq _(k) p ~x p P P5x p qk) log q log r + O log p log x) qr!-f X/p ( _ ) lg qrrglp(k) gr \P:5? P = -lq oploqlog r + 1 lo3 x + 0(log2 x), Pq r:5z pqr 3p(k) pqr_ I(k) 3 It is possible to estimate this series elementary, one finds it is 2' A/o(k) log x From the resulting formula one can give an elementary proof of the prime-number theorem.

5 300 ATLE SELBERG since one easily deduces from (2.3) that E log P log2$ = log3 x + O(log2 x). pzs P p Inserting the above result in (3.5) one gets (3.6) E -P = 2 E log p log q log r+ O(log2 x). p<z P pqrv_ pqr P-= I k Pqr l (A;) Again from (3.4) we deduce that log2 P?l1 log2x + O(log x), p p =o(k) p I(k) from which we easily find, by partial summation, Now by (3.7) 3 log p 2 logx + O(loglogx). p<sz p fk)~ '" ' p- (k) z logp logq i < logp logqq +2 logp logq pqz $ pq pli XIq1zII q I3<p.X P - q Pqql(k) pq wk(k) I'3< psq p p + O(log log x 5 logp log p log q p<x P P6x1/3 q<sz13 pq pq- I (k) + 9 log2 X + O(log x log log x by (2.3). Inserting this in (3.4) we get log2p > 1 log2 x logp logzq + O(logZ loglogx) p$ p = 9'p(k) pfxl/3 q<=1i3 pq P~~~~ 1(k) ~~~~~~pqa t(k) or, for x> xo logx log P > 1 log - / logplog q pa P~5 p I l~p (k) (k) pq- log x ~13 I (k) q;szlf3 pq from which log x * Sz(x) > log2 x - Sm(x 18)Sm'(X 113) O(p (k) mi_ I W

6 DIRICHLET 'S THEOREM ABOUT PRIMES 301 where the sum is taken over all pairs of residues mod. k with mm' -(k). Dividing by log2 x we get (3.8) Q1(x) > 1 1 Qm(x"')Qm'(zx11), for x > xo. In a similar way, we get from (3.6) that (3.9) Q1(x) E2 - Qm(xl1 )Qmt#(X13)Qmii(X113) 0 ( ) 27 mm'am"_i=(k) log x (3.7) gives (3.10) Qz(x) :?(2 + 0 (log log x) 4. We now proceed to prove the following LEMMA 1. For every real, nonprincipal character x mod k we have log P > 1 log x for x > xo. X (P)-1 We make use of the fact, which can be deduced from the law of reciprocity for the quadratic residue symbols, that to each x there exist an integer D which is not a square and I D I < k2, such that for all primes p we have x(p) = (Dip) where (Dip) is the ordinary quadratic residue symbol.4 What we shall prove then is (4.1) E log p > 1 log x for x > xo. (D/ p)-1 We consider the product (4.2) P= II' _ u?-dvi? I u. I/z12 where the dash HI' indicates that th seen that for x > xo (4.3) log P > logx. Let us estimate the exponent of the highest power of a prime p which divides P. First suppose that (Dip) = 1. We try to estimate how many solutions the congruence 2u U2-2 DyV2 _ (P), 4 See for instance Dirichlet-Dedekind: Vorlesungen uiber Zahlentheorie, the beginning of?135.

7 302 ATLE SELBERG has in the given range for u and then we see that if (u, v) is another we have or one of the congruences (uvp)2 - (up) - 0(p), UVO? UOV B 0(p), must be satisfied. The number of solutions in the given range for u and v of these congruences is easily estimated to be less than 4x px/5+ O (O. Thus at most 8x/(pV/D) + 0(x/p) of the numbers u2 - Dv2 contain the prime p as a factor, and in the same manner, we prove that at most 8x of the u2 - Dv contain less than 8 xoq11f= 8 x+(44). On the other hand if (D/p) = -1, we easily see, that since p has to divide both u and v in order to divide u2 - DV2, the product P will contain p to a power less than OQ( ). Finally if (Dlp) = 0, or p/d, we have that P contains p to a power less than 0 (x). These results give logp 5 8 a E + op 0 E ogp (D/ p)l + + 0(x210 x log P) + O ( (x?log p g: zp\ pid io =8 l + O(x) (DI p)=1 S(U., v.) is assumed to be a nontrivial solution, i.e. (u0, p) - (v.

8 DIRICHLET 'S THEOREM ABOUT PRIMES 303 Comparing this with (4.3) we get (D/p)=l E log P+ 0(1) > log x, or which proves our lemma. E og P > log x for x > xo, P<X P (D/p)=l 5. LEMMA 2. Suppose that we have a set of different residues ml, M2, **m mod k, such that they all are relatively prime to k. Further suppose that h > (o(k)), and that to each real character x mod k, we can find an m in the set with x(m) = 1. Let (1, k) = 1, and suppose that there is a m and m', not necessarily different, belonging to the set for which mm' _(k). Then we can find a triple of residues belonging to the set (m, m', mrn)6 such that or Assume that always mim'm" _(k). Min^m^2i3 1(k), (5.1) milmi2 0 lfnim3(k). Since the left-hand side can assume at least h different7 values and the right-hand side h different values, we see that the lemma is true for h > 1(w(k)). Thus we will assume h = 2(w(k)). Then we see from (5.1) that the product mi1mi2 can assume only h different values. Writing ni = mifm1 the same will be the case with the h residues ni = 1, n2,..., nh. From this we see that these residues form a group with respect to multiplication. We then define a real character x(n) which is 1 for n1, n2, * * *, nh and -1 for the other h residues. For this character we would have for i = 1, 2,...,h x(mi) = x(mi). According to the assumption, there is at least one mi with x(mi) = 1, so th all mi, x(mi) = 1. From this we get x(l) x(m)y(m') = 1 so that both 1 and 1 will be found in the set of mi, since (k), this contradicts (5.1) so our lemma is completely proved. 6. We are now able to prove the theorem stated in section 1. More precisely we will show that (6.1) QjX() > ( (k)) for x > X 6 The m and m' do not necessarily mean the same residues as in the preceding congruence. 7When we say different we mean different mod k.

9 304 ATLE SELBERG Let us assume that (6.2) QI(x) <30p(k)' for some large x. From (3.10) we see that since by (2.3), Ad Qn(x ) = 1 +? (1s)X we can find at least I(po(k)) values m for which Qm(XI1/ ) > 20( (k) for x > xo. Further from Lemma 1, for any real non-principal character Z Qn(Xll') > for x > xo, X(n)=l so that there exist at least one Qm(xl/3) > 2(9%p(k)) with x(m) = 1. Finally from (3.8) and (6.2), E Qn(X"3)Qn1(X"3) > 1 n n'i= 1(k)lok) so that there exists at least one pair of residues m, m' with mm' I(k) and Qm(X113)Qm'(X1/3) > or by (3.10) / / QM(xl/ ) > 31<,o(k) > Thus we can find a set of different residues mod k, min, m m, with h > (((k)) so that Qmj(x1/3) > 20Qp(k))2 for i = 1, 2, * *, h, and that further to each real character x there is a m, with X(m,) = 1, and finally there exist residues m, m' belonging to this set, such that mm' = (k). From Lemma 2 we then conclude that there exist residues m, mt', i" in set with mm'min" 1(k). Then (3.9) gives >2 /~i3 f~13)q 13) - 1 \> 1 Q1(x) > 2 Qt(log )Qn(Zx )Q,,(/ - (I ) > (20)4%p(k))' for x > xo, which proves our theorem. THE INSTITUTE FOR ADVANCED STUDY