In its simplest form the prime number theorem states that π(x) x/(log x). For a more accurate version we define the logarithmic sum, ls(x) = 2 m x

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1 THREE PRIMES T Hardy Littlwood circl mtod is usd to prov Viogradov s torm: vry sufficitly larg odd itgr is t sum of tr prims Toy Forbs Nots for LSBU Matmatics Study Group Fbruary Backgroud W sall closly follow Modr Prim Numbr Tory by T Estrma CUP 96 W adopt t covtio tat t variabl p wit or witout a subscript always rags ovr t prims Lt us also gt som fuctio dfiitios out of t way: x πix ; if v is suar-fr µv p v t Möbius fuctio; otrwis φv Eulr s pi fuctio; c v <v gcdv < gcd v Ramauja s sum Obsrv tat c v is just t sum of t v-t powrs of t primitiv -t roots of Lmma i For ay X ad ay itgr v X+ X vxdx if v ad if v ii T Möbius fuctio ad Eulr s pi fuctio ar multiplicativ Ramauja s sum is multiplicativ ovr iii If gcdv t c v c iv If gcdv t c v µ Proof Proprty i is fudamtal to a lot of wat follows Its proof is straigtforward If gcd t c vc v v + v + c + sic rus ovr t primitiv -t roots of Tat taks car of Ramauja s sum T otr two ar lmtary umbr tory Tis provs ii Part iii is obvious I tik For iv w ca assum v by iii If k c p k is t sum of t p k -t roots of mius t sum of t p k -t roots of ; tat is w k ad otrwis Part iv follows by multiplicativity I its simplst form t prim umbr torm stats tat πx x/log x For a mor accurat vrsio w dfi t logaritmic sum lsx log m mx Tis is lik t logaritmic itgral xcpt tat it is a sum ad it is asily s tat t diffrc btw t two is boudd: lsx lix O

2 Torm T9 : Torm 9 i Estrma s book log x πx lsx + O x xp Tis wit lix istad of lsx was provd by d la Vallé Poussi i 898is Lik t lix form it is a vry good approximatio Tat complicatd rror trm is ultimatly suprior to x/log x m for ay fixd positiv m but it is wors ta x δ for ay fixd δ > owvr small T bst rror trm for t prim umbr torm is still t 45-yar-old rsult of H-E Ricrt 967 Tr xists a positiv costat C suc tat πx lix + O x xp C log x 3/5 log log x /5 provd by stablisig tat t Rima zta fuctio ζσ + it as o zros wit σ C log t /3 log log t /3 for som positiv costat C T Holy Grail of t subjct is of cours to xtd t zro-fr rgio wstwards all t way up to t li σ / wit t cosut improvmt i t rror trm of Torm to O x log x Tis is t Rima Hypotsis I t otr dirctio Littlwood sowd tat πx lix Ω ± x / log log x log x W will also d t prim umbr torm for aritmtic progrssios Torm T55 Lt u > Lt log x u ad gcd T t umbr of prims p x p mod is giv by πx; lsx log x φ + O x xp wr t costat implid by t O otatio is idpdt of ad T proof ruirs t svr rstrictio o to guarat uiformity wit rspct to wic is vital for our applicatio Uiformity wit rspct to is trivial A log-stadig problm i tis ara is to xtd t rag of Writ Ex max πx gcd πx; φ T Torms ad stat tat for log x 5 say log x Ex O x xp T Elliott Halbrstam cojctur is tat o avrag o ca rlax t coditio o : for vry θ < ad A > tr xists a costat C 3 > suc tat x θ Ex C 3x log x A Bombiri ad Viogradov sowd tat t Elliott Halbrstam cojctur olds for θ < /

3 Tr prims Viogradov provd i 937 tat vry sufficitly larg odd itgr ca b rprstd i t form p + p + p 3 Prviously i 93 Hardy ad Littlwood ad sow tat tis is tru if tr xists a umbr δ < 3/4 suc tat o of Diriclt s L-fuctios as zros i t alf-pla Rz > δ Mor rctly i 989 C & Wag sowd tat t tr prims rprstatio olds ucoditioally for odd > 43 Lt r dot t umbr of solutios of p + p + p 3 ; tat is r p + p + p 3 prim p p p 3 Rptitios ar allowd ad ordr is rlvat; so r 6 bcaus Lt ρ log m log m log m 3 m + m + m 3 m m m 3 Ultimatly w wat t followig Torm 3 Lt T S µ φ 3 c r Sρ + O log 4 T proof will occupy t xt tr sctios of ts Nots For ow w obsrv tat t tig big summd i Torm 3 µc /φ 3 is multiplicativ as a fuctio of s Lmma Morovr µ µp ad µp k for k So S as t simpl product form: S p c p p 3 If is v t c / S ad Torm 3 dos t say aytig itrstig O t otr ad w is odd w av c c p p for odd p ad c S p> c p p 3 p> p m m Furtrmor w ca stimat ρ For 6 t umbr of trms i t sum for ρ is 4 m ad ac trm is at last /log 3 Trfor m ρ > log 3 4 m m 3 > 3log 3 3

4 for sufficitly larg Tus w av our dsird rsult: r > 3log + O 3 log 4 T computatio of r For v lt fx v pv px T ad fx v 3 p v p + p + p 3 x p v p 3 v x + r fx 3 xdx x Curiously tis formula actually works at last for small Puttig it ito Matmatica givs tis tabl r But our mai task is to fid a o-trivial gral lowr boud for r by provig Torm 3 W sall stimat t itgral i Hcfort w will assum tacitly tat is sufficitly larg For covic w fix log 5 x wic w wat to b small ad it will b providd is sufficitly larg It apps tat fx 3 is small ulss x is ar a ratioal umbr wit a small suar-fr domiator Ev for t tiy valu 6 w ca clarly s t spiks at / /3 /3 /6 ad 5/6 as wll as lssr paks at j/ for j 9 j 5; but tr ar o at /4 3/4 /8 3/8 5/8 ad 7/8 It turs out tat t alf-widt of t six most promit spiks is about ad if w itgrat fx 6 3 x ovr just t itrvals a ± a / /3 /3 /6 5/6 w obtai 766 compard wit t tru valu r

5 W split t itrval [x x + ] ito major arcs ad mior arcs T major arcs will cosist of all tos umbrs tat ar witi x of a ratioal umbr wit a domiator ot xcdig log 5 T mior arcs cosist of vrytig ls i [x x + ]; as w sall s latr Torms 6 ad 7 ts umbrs ar always witi x of a ratioal umbr wit a domiator i t rag log 5 /log 5 W attmpt to fid a rasoably accurat stimat of t itgral for r ovr t major arcs wr w oft xpct fx to b larg O t mior arcs w ar cott to fid a o-trivial uppr boud for fx For a typical major arc w writ J /+x / x fx 3 xdx W ca assum is so larg tat t major arcs do ot ovrlap Hc r J + fx 3 xdx log 5 < gcd mior arcs T mior arcs W stimat t itgral ovr t mior arcs wit t xt fw torms bgiig wit a uality wic says tat addig somtig small to x opfully wo t cag fx v too muc Torm 4 5 : Estrma 5 T fuctio fx v satisfis t idtity fx + y v vyfx v πiy uyfx udu Proof Obsrv tat Hc fx + y v pv vy wy πiy uydu w pxpy pv px vy πiy vy px πiy uy pxdu pv pu vyfx v πiy uyfx udu Torm 5 T56 Lt ad v T log 5 < f v gcd log 5 log 3 p uydu Proof Torm 5 is t vital mior arcs stimat tat maks t circl mtod work for t tr prims problm Tis is wr Viogradov succdd aftr H & L faild T proof is lmtary but complicatd ad is omittd S Estrma pp

6 Torm 6 T57 Giv ay x ad ay y tr xist ad wit y ad gcd suc tat x < y Proof W may assum < x < Lt m y T [ + x or x + [ + + wr / ad / ar coscutiv fractios i t Fary suc of ordr m; so tat ad + m + If x is i t first itrval w tak / / sic x < + + m + y Similarly w tak / / if x is i t scod itrval Torm 7 5 Suppos x is i a mior arc T fx O log 3 Proof Suppos x is i a mior arc T by Torm 6 wit y /log 5 tr xist coprim ad wit log 5 < < log 5 ad x < log 5 x t first iuality bcaus otrwis x would b i a major arc Trfor by Torm 5 f v log v 3 Puttig z x / ad usig Torm 4 w av fx f + z zf πiz sic z x / < x / < / log + πz 3 + π log 3 log 3 du uzf u du W ar ow rady to stablis t dsird o-trivial uppr boud for t r itgral ovr t mior arcs Torm 8 W av mior arcs fx 3 xdx O log 4 6

7 Proof By Torm 7 mior arcs fx 3 xdx O log 3 fx dx But fx dx p p p p xdx p π ad t rsult follows from t prim umbr torm Torm T major arcs Lt gx v mv mx log m for v for v < So gx v is lik fx v but istad of summig ovr prims w sum ovr itgrs m wit wigt /log m to accout for t approximat dsity of t prims at m Torm 9 4 T fuctio gx v satisfis t idtity gx + y v vygx v πiy uygx udu Proof Similar to Torm 4 Torm T58 Suppos T log 5 gcd ad y x f + y µ φ gy log 69 log 5 Proof Suppos v W av f v pv p p p < But pv p p <l gcdl l pv p l mod <l gcdl l πv; l ad from t prim umbr torm for aritmtic progrssios Torm lsv πv; l φ < gcdl log 7

8 Furtrmor by Lmma µ c <l gcdl l Hc obsrvig tat g v lsv f v µ g v φ < pv p <l gcdl log < p µ φ lsv l πv; l lsv φ log 85 Hc by Torm 4 ad Torm 9 f + y µ gy φ f y µ g πiy φ f µ g φ + πx f v + πx 4 < < log 85 log 7 log 69 vy f v µ φ g v dv µ φ g v dv Obsrv tat for o-suar-fr µ ad fx is small w x is ar / Torm T59 Suppos < log 5 gcd ad y x T 3 f + y µ 3 3 gy 3 φ 3 log 69 Proof Tis follows from Torm togtr wit t trivial stimats fx ad gx Substitutig x / + y i t xprssio for J givs Puttig J x 3 f x + y ydy K x x gy 3 ydy w av by Torm ad rcallig tat x log 5 / µ J φ K 6 3 log 54 8

9 providd tat gcd ad log 5 Just as r ca b xprssd as a itgral ivolvig fx ρ as a similar formula usig gx : ρ / / gy 3 ydy Now for w > ad < y / w my w + y y y m By Abl s lmma summatio by parts gy log + my m k k m my provd by rvrsig t ordr of summatio Hc gy y log + + y log k logk + k si πy y logk + log k y for < y / ad trfor rcallig tat K is lik ρ but itgratig ovr t sortr itrval [ x x ] ρ K x / dy y 3 + / x dy y 3 / x dy y 3 < x log 3 From tis ad w gt a good stimat for a sigl major arc: µ J ρ φ 6 3 log + 54 φ 3 log 3 agai providd tat gcd ad log 5 Summig ovr t major arcs ad usig t dfiitio of Ramauja s sum µ J ρ φ c 3 log 5 < gcd 6 log 4 + log 3 log 5 < φ log 5 7 log 4 3 sic φ is boudd rcall tat φ > 3 9 ad at last w av t stimat ovr t major arcs tat w wat Combiig 3 wit t mior arcs stimat Torm 8 givs µ r ρ φ c O 3 log 4 log 5 ad it is asily sow tat t sam stimat olds w w tak t sum to ifiity Rcallig t dfiitio of S from Torm 3 w trfor av r ρs O log 4 ad t proof of Torm 3 is complt 9