The St. Petersburg School of Number Theory

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HISTOR Y O F MATHEMATIC S • VOLUM E 26

The St Petersburg Schoo l of Number Theor y

B. N. Delone Translate d by Rober t Burn s

AMERICA N MATHEMATICA L SOCIET Y • LONDO N MATHEMATICA L SOCIET Y

https://doi.org/10.1090/hmath/026

Editorial B o a r d

American Mathematica l Societ y Londo n Mathematica l Societ y Joseph W . Daube n Ale x D . D . Crai k Peter Dure n Jerem y J . Gra y Karen Parshall , Chai r Pete r Neuman n Michael I . Rose n Robi n Wilson , Chai r

B. H . H E J I O H E

n E T E P E Y P r C K A J I I I IKOJI A T E O P M H MHCEJ T

M3̂ ;aTejii>CTBO Ana^eMM H Hay K CCC P

MocKBa-JTeHHHrpa^, 194 7

Translated fro m th e Russia n b y Rober t Burn s

2000 Mathematics Subject Classification. Primar y 01A72 , 11-03 ; Secondary 01A55 , 01A60 .

For additiona l informatio n an d update s o n thi s book , visi t www.ams.org/bookpages /hmath-26

Library o f Congres s Cataloging-in-Publicatio n D a t a

Delone, B . N. (Bori s Nikolaevich) , 1890-1980 . [Peterburgskaia shkol a teori i chisel . English ] The St . Petersburg schoo l o f number theor y / B . N . Delone ; translated b y Robert Burns ,

p. cm . — (Histor y o f mathematics, ISS N 0899-242 8 ; v. 26) Includes bibliographica l references . ISBN 0-8218-3457- 6 (acid-fre e paper ) 1. Numbe r theory . 2 . Mathematics—Russi a (Federation)—Sain t Petersburg—History .

3. Mathematicians—Russi a (Federation)—Sain t Petersburg—History . I . Title: Sain t Petersbur g school o f number theory . II . Title. III . Series.

QA241.D413 200 5 512.7—dc22 200504820 5

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Contents

Foreword t o th e Englis h Editio n vii Bibliography xii i Translator's acknowledgment s xi v

Preface x v

Pafnutii L'vovic h CHEBYSHE V 1

Chebyshev's Article s o n th e Prim e Number s 7 On th e Determinatio n o f the Numbe r o f Prime s no t Exceedin g

a Give n Numbe r 7 Commentary 1 4 On th e Prim e Number s 1 6 Commentary 2 2

Aleksandr Nikolaevic h KORKI N 2 5

The Article s o f Korki n an d Zolotare v o n th e Minim a o f Positiv e Quadratic Form s 3 1

On Quaternar y Positiv e Quadrati c Form s 3 1 On Quadrati c Form s 3 4 On Positiv e Quadrati c Form s 3 8 Commentary 4 6

Egor Ivanovic h ZOLOTARE V 6 1

Zolotarev's Memoir s o n th e Theor y o f Idea l Number s 6 7 The Theor y o f Comple x Integer s wit h a n Applicatio n t o th e

Integral Calculu s 6 7 Zolotarev's article s o f 187 8 and 188 5 7 6 Commentary 8 3

Andrei Andreevic h MARKO V 9 1

On Binar y Quadrati c Form s o f Positive Determinan t 9 5 Commentary 10 4

Georgii Fedoseevic h VORONO I 12 9

Voronoi's Dissertation s o n Algebrai c Number s o f the Thir d Degre e 13 5

vi C O N T E N T S

On Integra l Algebrai c Number s Dependin g o n a Roo t o f a n Equatio n of the Thir d Degre e 13 5

On a Generalizatio n o f the Continue d Fractio n Algorith m 13 9

Voronoi's Memoi r o f 1903 : "O n a Problem fro m th e Theor y o f Asymptotic Functions " 16 5

Voronoi's Memoir s o n Quadrati c Form s 16 9 Properties o f Perfec t Positiv e Quadrati c Form s 16 9 Commentary 17 6 Investigations Concernin g Primitiv e Parallelohedr a 17 8 Commentary 19 4

Ivan Matveevic h VINOGRADO V 20 9

Works o f Vinogradov fro m th e Firs t Perio d o f Hi s Mathematica l Activit y 21 3

Waring's Proble m 22 5

The Goldbac h Proble m 24 3

Estimation o f Weyl Sum s an d th e Proble m o f the Fractiona l Part s of a Polynomia l 26 3

A Lis t o f Works i n Numbe r Theor y b y Chebyshev , Korkin , Zolotarev , Markov, Voronoi , an d Vinogrado v 27 3

Foreword t o th e Englis h Editio n

"A series o f Russian mathematicians—Chebyshev , Korkin , Zolotarev , Markov , VoronoT, an d others—hav e worke d i n th e theor y o f numbers . On e ca n becom e ac -quainted wit h th e conten t o f the classica l wor k o f these notabl e mathematician s i n B. N . Delone' s boo k "Th e St . Petersbur g Schoo l o f Numbe r Theory". " S o begin s the Prefac e t o the book "Element s o f Number Theory " b y I . M. Vinogradov. Vino -gradov doe s no t infor m u s tha t h e himsel f i s th e subjec t o f th e longes t an d mos t detailed chapte r o f Delone' s book . Thes e mathematicians , whos e lif e an d wor k i n number theor y for m th e substanc e o f thi s work , ar e indee d a ver y distinguishe d group. Thei r wor k i s of the highes t qualit y an d o f lasting significance.

Delone's boo k wa s published , i n Russian , i n 1947 . Th e presen t volum e i s a translation o f the origina l int o English . Th e tex t i s essentially unaltered , bu t foot -notes ar e adde d t o clarif y th e expositio n an d i n som e case s t o giv e mor e u p t o date references . Fo r each of the six mathematicians considered , a brief biograph y i s provided followe d b y an exposition o f several o f their mos t importan t contribution s to numbe r theory . I n al l cases , th e mathematica l expositio n i s muc h longe r an d more detaile d tha n th e biography . I n spit e o f that , w e ge t a goo d sens e o f thei r professional lives . W e also ge t t o se e how their individua l wor k i s interrelated wit h the wor k o f the others .

It shoul d b e emphasize d tha t thi s boo k i s strictl y abou t numbe r theory . Fo r example, Chebyshe v an d Marko v mad e ver y importan t contribution s t o the theor y of probability , bu t n o mathematica l discussio n o f thei r wor k o n probabilit y i s pre -sented. Anothe r poin t t o mak e t o th e potentia l reade r i s tha t th e mathematica l discussion i s no t superficial . Delon e himsel f wa s a distinguishe d membe r o f th e St. Petersbur g schoo l o f numbe r theory , an d wa s wel l verse d i n th e wor k o f hi s predecessors. Hi s exposition i s at time s taxing, bu t th e effor t expende d i n followin g the argumen t i s amply rewarded . Thi s i s beautifu l material .

One final commen t befor e providin g a n overvie w o f th e mathematica l conten t on this book. I n 1947 , the Cold War was just gettin g under way in earnest. Perhap s because o f this historica l fact , Delone' s treatment i s marred somewha t b y an exces -sive nationalisti c pride . Whil e thi s i s grating o n moder n sensibilities , i t shoul d no t interfere wit h one' s enjoyment whil e reading thi s book. Afte r all , the mathematica l works o f these author s ar e wel l worth bein g prou d of , an d th e Sovie t er a Russian s are no t th e onl y one s wh o hav e bee n guilt y o f thi s fault .

The boo k begins with a consideration o f P. L. Chebyshev an d hi s contribution s to numbe r theory . Chebyshe v wa s a first rat e mathematicia n wh o contribute d important result s i n a variet y o f fields. Hi s mai n wor k i n numbe r theor y concern s the distribution o f prime numbers. Fo r a positive real number x, Chebyshe v denote s by 4>(x) th e number o f positive prime number s which ar e less than o r equa l to x. I n modern notation hi s function i s called TT(X), bu t w e will follow Chebyshev . Legendr e

vii

viii FOREWOR D T O TH E ENGLIS H EDITIO N

proposed th e approximatio n

^ ~ ln(ar)-1.08366 '

This agree s wel l wit h table s o f prim e numbers , bu t i n hi s first memoi r o n th e subject, Chebyshe v show s tha t i t canno t b e true . H e show s that , i n fact , th e bes t approximation (i n a certain precis e sense ) t o (j)(x) is

r dt x Hx)=L w)~w'

As usual , f(t) ~ g(t) mean s tha t f(t)/g(t) tend s t o 1 as t tend s t o infinity . I n th e proofs, Chebyshe v use s propertie s o f th e zet a function , ((s), a s define d b y Euler . Namely, h e use s th e produc t formul a an d th e fac t tha t (s — 1)C( 5) — > 1 a s s —> 1 from above .

In hi s secon d memoi r o n th e subjec t o f prim e numbers , Chebyshe v goe s muc h further an d comes within striking distance of the prime number theorem as proposed by Gauss , namely ,

A,( \ f dt

In thi s work , Chebyshe v introduce s th e auxilliar y function s

0(a:) = ^ l n ( p ) an d ip(x) = ^ ln(p) , p<x p m<a:

which hav e playe d a ke y rol e i n th e literatur e o n th e distributio n o f prime s eve r since. Perhap s th e mai n resul t o f thi s pape r i s

.92129 — ^ - < <t>(x) < 1.10555 X

ln(x) ln(x) '

A consequence o f this i s a proof o f Bertrand's hypothesis , whic h state s tha t fo r all n > 1 , there i s a prim e numbe r betwee n n an d 2n.

Chebyshev's wor k o n prim e number s wa s a n absolutel y majo r ste p forwar d i n our knowledge of how prime numbers ar e distributed. I n spite of this, i t took almos t fifty mor e year s befor e th e firs t proof s o f the prim e numbe r theore m wer e given b y J. Hadamar d and , independently , b y Ch.-J . d e l a Vall e Poussin .

We no w pas s o n t o th e wor k o f late r member s o f th e St . Petersbur g circle . There i s a problem i n the theory o f quadratic form s tha t provide s a unifying threa d in th e wor k o f Korkin , Zolotarev , Markov , an d Voronoi , s o i t i s a goo d ide a t o begin b y explainin g th e problem . W e wil l wor k i n Euclidea n spac e IR n, whic h w e can thin k o f a s being give n b y colum n vector s o f real number s wit h th e usua l inne r product (x,y) = ^ x ^ - A quadrati c for m i s a homogeneou s polynomia l o f degre e 2, usuall y writte n Q(x) = J2J2

We requir e th e coefficient s t o b e rea l numbers an d th e symmetr y conditio n a ^ = a^ . Th e quantit y D = det(a^ ) i s called th e discriminan t o f Q. I f Q(x) > 0 fo r al l ir , wit h Q(x) = 0 i f an d onl y i f x — 0, w e sa y tha t Q i s positiv e definite . Le t G — Zn c W 1 denot e th e intege r lattice. Se t /x(Q ) equa l t o th e minimu m valu e o f Q(~i) a s 7 run s throug h th e nonzero element s o f G. C . Hermit e wa s abl e t o sho w tha t th e quantit y /JL(Q)/ \[T) was bounde d b y som e constan t dependin g onl y o n th e dimensio n n. Th e smalles t value o f thi s constan t i s sometimes calle d Hermite' s constan t fo r dimensio n n , an d the problem i s to determin e it s value for eac h n > 1 . Th e answe r fo r n = 2 is 2/y/3. For n = 3 , th e answe r i s \ /2 . O n th e basi s o f thi s (an d perhap s more) , Hermit e

F O R E W O R D T O T H E ENGLIS H EDITIO N i x

conjectured tha t th e answe r i n genera l i s 2/ y/n + 1 . Thi s wa s disprove d b y Korki n and Zolotare v i n a shor t pape r (1872 ) wher e the y sho w tha t fo r n = 4 the answe r is \f\. Delon e give s a fairly complet e overvie w o f thi s pape r an d it s successors .

The problem referred t o in the las t paragrap h ca n be reformulated a s a problem about th e smalles t vecto r i n a lattice . Thi s problem , i n turn , i s closel y relate d t o the proble m o f optimal spher e packing in W1. I n orde r t o orien t th e reade r t o othe r parts o f Delone' s exposition , i t i s worthwhile t o briefl y explai n thes e connections . Let {v\,V2, • • • ,vn} b e a basi s o f M.n. A lattice , L , i n R n i s the fre e abelia n grou p generated b y a basis , i.e .

L = 7Lv\ + 2/̂ 2 + • • • + Zv n.

Among al l the nonzer o lattic e vector s A , it i s easy t o se e that ther e wil l be on e (not necessaril y unique) , A 0, suc h tha t ||A 0|| = A/(A 0 , A0) i s least . I f A 0 = J2 nivii then (A 0,A0) = Y^YK vi->vj)nin3-> s o tha t ll^oll 2 * s the ^ eas^ v a m e o f th e quadrati c form Q(x) — ^^2{vi,Vj)xiXj evaluate d a t nonzer o element s o f Z n .

Conversely, i f a positiv e definit e quadrati c for m Q(x) = J2*l2 aijxixj 1S given , set A = (ciij). W e ca n fin d a nonsingula r matri x B suc h tha t B t{aij)B — /, th e identity matrix . Le t C = B~ x. The n A = C lC'. Le t Vi be th e ith colum n vecto r o f C. Fro m C lC = I i t follow s immediatel y tha t (vi,Vj) = a^ . Le t L b e th e lattic e generated b y th e vector s {i^ , t>2, • • •, vn}. Then , th e value s o f Q(x) a t element s o f Zn coincide s wit h th e value s ||A|| 2 a s A varie s throug h L . W e hav e show n tha t th e problem o f finding th e smalles t valu e o f Q{x) a t intege r point s coincide s wit h th e problem o f finding vector s o f leas t lengt h i n a lattice .

The lattic e spher e packin g proble m ha s t o d o wit h placin g nonoverlappin g spheres o f fixed radiu s a t eac h lattic e poin t an d determinin g th e proportio n o f space take n u p by the spheres . Th e ide a i s to find the bes t possibl e lattic e packing . For a fixed lattice , th e maxima l radiu s o f the sphere s t o b e use d i s clearly equa l t o half th e lengt h o f the smalles t nonzer o elemen t o f the lattice . Peopl e als o conside r sphere packing s o f spac e fo r whic h th e center s o f th e sphere s d o no t necessaril y fill ou t a lattice . Thi s proble m ma y see m special , bu t i t ha s wid e application s throughout mathematics . I t i s interestin g t o not e tha t i t i s mentione d b y Hilber t in hi s famous lis t o f problem s (1899) . I t i s discusse d towar d th e en d o f proble m 18. Th e mos t comprehensiv e moder n sourc e o n spher e packin g i s th e 198 8 boo k by J . H . Conwa y an d N . J . A . Sloan e [1] . A shorter , an d perhap s mor e readable , treatment i s give n i n th e boo k o f C . Zon g [11] . Bot h book s refe r t o th e wor k o f Korkin, Zolotarev , an d VoronoT .

In anothe r direction , i t i s interestin g t o not e tha t th e wor k o f Korki n an d Zolotarev ha s foun d application s t o lattic e reductio n algorithm s whic h ar e o f prac-tical importanc e i n cryptography . A famou s 199 0 pape r [7 ] o f J . C . Lagarias , H. W . Lenstra , an d C . P . Schnor r i s th e genesi s o f th e LLL-BK Z algorithm . Th e last tw o letter s i n thi s acrony m stan d fo r Korki n an d Zolotarev , a tribut e t o ho w their fundamenta l wor k continue s t o find ne w applications .

Delone spend s a lo t o f time o n anothe r aspec t o f Zolotarev' s work . I n tw o pa -pers, one published i n 1874 , the other in 1880, Zolotarev establishes a general theor y of algebrai c integer s an d prim e decomposition . Th e wor k i s somewha t late r tha n Dedekind's famou s Supplemen t X I t o Dirichlet' s boo k "Vorlesunge n fiber Zahlen -theorie", i n which he establishes a general theory o f algebraic integer s base d o n th e new notio n o f ideals a s certain subset s o f the rin g o f integers . Zolotarev' s idea s ar e more i n keepin g wit h Rummer' s treatmen t o f the rin g o f cyclotomic integer s usin g

x F O R E W O R D T O T H E ENGLIS H EDITIO N

the notio n o f ideal numbers . I n hi s first pape r o n thi s subject , h e assume s tha t th e ring o f integer s i s of the for m Z[£] . I f f(x) £ Z[x ] i s the irreducibl e polynomia l fo r £, the prim e decompositio n o f a rational prim e p i s related t o the decompositio n o f the polynomia l f(x) G Z/pZ [x\. I n general , the ring of integers in a number field i s not o f the for m Z[£] . Zolotare v overcome s thi s difficult y i n his (posthumous ) pape r of 1880 . I t i s a sad fac t tha t i n 1878 , at th e ag e of 31 , Zolotare v wa s fatally injure d in a train acciden t an d die d shortl y thereafter .

Up t o now , w e hav e bee n concerne d wit h positiv e definit e quadrati c forms . Markov discusse d i n detai l th e theor y o f indefinit e binar y quadrati c forms . Thes e behave quite differently, bu t Marko v was able to give a rather complete theory which connects the theory of indefinite binar y quadrati c forms in a nontrivial way with th e theory of continued fractions . A s a by-product o f his investigation he is able to give a procedure for finding al l integer solutions to the diophantine equation x 2 + y2 + z2 = Sxyz. Today , thi s i s called Markov' s equatio n i n hi s honor . Delon e give s a carefu l discussion o f Markov's work on binary quadrati c forms . I n addition h e reformulate s much o f i t i n geometri c terms . Th e reade r wil l find thi s reformulatio n t o b e a valuable ai d t o understandin g Markov' s analysis .

Although i t is not directly related to the work described in the present book , i t is worth giving the statement o f the Oppenhei m conjecture , whic h concerns indefinit e quadratic form s Q(x) i n thre e o r mor e variables . Thi s state s tha t unles s Q(x) i s proportional t o a rational form, i.e . on e with rational coefficients , th e values of Q(x) on th e intege r lattic e Z n ar e dens e i n R . Th e Russia n mathematicia n G . Marguli s (now at Yal e University) prove d thi s conjecture i n 1986 . Althoug h th e statement o f the conjectur e i s certainly i n the spiri t o f the wor k o f the classica l mathematician s covered i n th e presen t book , th e method s o f proo f ar e ver y different .

Voronoi mad e importan t contribution s t o man y aspect s o f numbe r theory . While stil l i n graduat e schoo l h e publishe d a pape r o n Bernoull i number s whic h simultaneously foun d ne w result s abou t thei r arithmeti c propertie s an d gav e a ne w method o f deriving some of their know n properties . Thes e ar e based o n some beau -tiful congruence s whic h toda y bea r Voronoi' s name . A complete treatmen t i s given in the text o f Uspensky and Heasle t [8] . Thi s approach i s used in the book of Ireland and Rose n [6] , which als o shows how these congruences ar e related t o the construc -tion o f p-adic L-function s whic h were first define d an d explore d b y H . W. Leopold t in 1975 .

In another direction , Voronoi made an interesting contribution t o analytic num -ber theory . Le t r{n) denot e th e numbe r o f positive divisor s o f a positiv e intege r n. Dirichlet wa s abl e t o sho w tha t

Y^ r(n) = n ln(n) + (2 7 - l)r a + 0(y/n). n,x

Here, 7 denotes Euler's constant. I t has been a big problem over the years to im-prove the erro r ter m i n this formula . Vorono i gave the first significant improvemen t by showin g tha t th e erro r ter m wa s (9(^/nln(n)) . On e o f th e first achievement s of I . M . Vinogradov , who m w e wil l conside r later , wa s t o significantl y generaliz e Voronoi's work . I t i s conjecture d tha t th e erro r ter m i s C^n 1/4-1-6), whic h woul d follow fro m th e Rieman n hypothesis . O n th e othe r hand , i t i s known tha t 0{n 1^) won't do .

A furthe r importan t contributio n o f Voronoi wa s hi s thoroug h investigatio n o f cubic number fields. I n all cases he was able to construct a n explicit integra l basis of

F O R E W O R D T O T H E ENGLIS H EDITIO N x i

the ring of integers of such a field and to discuss in detail the prime decomposition of rational primes . Perhap s hi s main resul t i s to give an efficien t algorith m fo r finding a se t o f fundamenta l units . Here , h e take s a s hi s mode l th e continue d fractio n algorithm fo r finding th e fundamenta l uni t i n a real quadratic numbe r field. Other s have give n algorithm s fo r finding independen t units , bu t Voronoi' s i s bot h mor e efficient an d has the advantage of always leading to a set o f fundamental units . Thi s work i s covere d a t lengt h i n th e presen t book . A mor e elaborat e presentatio n i s given in the book "Th e theory of irrationalities o f the third degree " b y B. N. Delone and D . K . Faddee v [3] .

The las t topi c concernin g Voronoi' s contribution s t o numbe r theor y whic h i s covered b y Delon e i s his work o n quadrati c form s an d lattices . Vorono T introduce d important ne w concepts , e.g. , perfec t quadrati c form s an d Vorono T cells of a lattice . We will not g o into the notio n o f a perfect quadrati c for m i n thi s introduction , bu t will briefl y describ e wha t i s mean t b y a VoronoT cell o f a lattice . Le t L C W 1 b e a lattice. Fo r each A G L le t V (A) b e the se t o f points x i n R n suc h tha t x i s closer t o A than t o an y othe r lattic e poin t i n L. I t i s no t har d t o see tha t V(A ) i s a conve x polytope. I t i s calle d a Voronoi cell o f th e lattice . Th e Vorono T cell s ar e disjoin t and th e unio n o f thei r closure s fills al l o f W 1. Thi s decompositio n i s discusse d i n detail i n th e boo k b y Conwa y an d Sloan e [1] . Anothe r goo d plac e t o rea d abou t VoronoT cell s i s th e charmin g boo k b y Conwa y entitle d "Th e sensua l (quadratic ) form" [2] .

Finally, w e com e t o I . M . Vinogradov . Befor e surveyin g th e chapte r o n Vino -gradov, i t shoul d b e pointe d ou t tha t th e mathematica l expositio n wa s writte n b y B. A . Venkov . I n th e Preface , Delon e refer s t o Venko v a s hi s coautho r an d ex -tends hi s gratitud e fo r writin g th e expositio n o f th e wor k o f Vinogrado v an d tw o parts o f the wor k o f VoronoT (o n sum s o f divisors an d o n perfec t quadrati c forms) . Nevertheless, Venkov' s nam e doe s no t appea r o n th e titl e page .

The first contributio n o f Vinogradov whic h i s discussed i s his generalizatio n o f VoronoT's work o n th e diviso r problem . Th e su m

N

n = l

which i s to b e estimated , i s easil y see n t o b e th e numbe r o f integra l lattic e point s in th e first quadran t unde r th e curv e xy = N wit h x-coordinat e betwee n 1 an d N. Wha t Vinogrado v set s ou t t o d o i s t o estimat e th e numbe r o f integra l lattic e points i n suc h a regio n wher e th e to p curv e y — N/x i s replace d b y a muc h mor e general functio n y = f(x). Wit h som e restriction s o n / (x) , Vinogrado v i s abl e t o show tha t th e numbe r o f lattice point s i s equal t o th e are a o f the regio n plu s a well controlled erro r term . Th e reade r shoul d consul t th e tex t fo r details . I n th e cas e of Dirichlet's diviso r problem , Vinogrado v get s an erro r ter m o f order tfN\ln(N)] 2, which i s only slightl y wors e tha n VoronoT's . O n th e othe r hand , hi s resul t i s muc h more widely applicable .

While thi s earl y wor k clearl y show s promise , th e nex t tw o accomplishment s show tha t th e earl y promis e wa s more than fulfille d b y th e matur e mathematician . Vinogradov mad e signa l contribution s t o tw o o f th e mos t venerabl e problem s o f number theory , Waring' s proble m an d th e Goldbac h conjecture .

xii F O R E W O R D T O T H E ENGLIS H EDITIO N

In 177 0 E . Warin g conjecture d tha t fo r ever y n > 2 there i s a numbe r r suc h that ever y positiv e intege r i s the su m o f a t mos t r nt h powers , i.e .

N = x\ + x"l + • • • + xnr

is solvabl e i n positiv e integer s fo r ever y T V > 0 . Th e prototyp e fo r suc h a resul t i s Lagrange's theore m tha t ever y positiv e intege r i s the su m o f four squares .

In 1909 , Hilbert prove d Waring' s conjectur e i n ful l generality . Thi s was a grea t step forward , bu t ther e remaine d th e questio n o f finding a goo d boun d o n r a s a functio n o f n. Hilbert' s proo f produce d value s whic h wer e muc h to o large . I n the earl y 1920s , Hardy an d Littlewood , usin g thei r newl y invente d "circl e method " produced th e first reasonabl e bounds . Le t u s define G(n) t o be the smalles t intege r such tha t ever y N wit h a t mos t finitely man y exception s i s th e su m o f G(n) nth powers. B y allowin g fo r a finite numbe r o f exceptions , th e theor y i s mad e muc h easier.

It i s not har d t o se e that n + 1 is the lowe r boun d fo r G(n) (se e Chapte r 1 8 of the boo k "Introductio n t o numbe r theory " b y H . K . Hu a [4]) . Th e uppe r boun d obtained b y Hard y an d Littlewoo d i s

G(n) < ( n - 2 ) 2 n ~ 1 + 5 .

In 1937 , Vinogradov wa s abl e t o prov e tha t fo r n > 16

G(n) < 6n(ln(n) + l) .

This i s a remarkabl e result . On e goe s fro m a boun d whic h i s exponentia l i n n t o one tha t i s almos t linea r i n n. Vinogrado v wa s no t content , an d late r (1959 ) h e showed tha t fo r larg e n on e ha s

G(n) < n(21n(n) + 41nln(n ) + 21nlnln(n ) + 13) .

This uppe r boun d wa s superseded , thirt y year s later , b y a resul t o f T . D . Woole y which show s tha t

G(n) < n(ln(n) + lnln(n ) + 0(1)) .

In 1995 , Wooley furthe r improve d th e 0(1 ) term . Thi s seem s t o b e th e bes t resul t at present . I n les s precis e bu t simple r terms , w e no w kno w tha t fo r al l larg e n , n + 1 < G(n) < n1 + e fo r an y e > 0 .

The Goldbac h conjectur e state s tha t ever y eve n numbe r greate r tha n fou r i s the su m o f tw o od d prime s an d tha t ever y od d numbe r greate r tha n seve n i s a sum o f thre e od d primes . Thi s famou s conjectur e wa s first state d i n a lette r fro m C. Goldbac h t o L . Eule r i n 1742 . Onc e again , th e first bi g ste p forwar d wa s du e to Hard y an d Littlewood . The y show red tha t almos t al l eve n number s ar e th e su m of tw o od d primes . Thi s mean s tha t f(m)/m — > 0 a s m — > oo, wher e f(m) i s the numbe r o f positiv e eve n number s < m whic h canno t b e writte n a s th e su m o f two od d primes . The y als o showe d tha t ever y sufficientl y bi g od d numbe r i s th e sum o f three od d prime s provide d tha t th e generalize d Rieman n hypothesi s i s true . Unfortunately, t o thi s day , neithe r th e Rieman n hypothesi s no r th e generalize d Riemann hypothesi s ha s bee n proved .

This wa s th e situatio n i n 193 7 when Vinogrado v proved , unconditionally , tha t every sufficientl y larg e od d numbe r i s th e su m o f thre e od d primes . Ye t anothe r utterlv remarkabl e result !

BIBLIOGRAPHY xii i

It i s wort h notin g tha t neithe r Vinogrado v no r anyon e els e ha s bee n abl e t o resolve th e Goldbac h conjectur e fo r eve n numbers . Thi s remain s inaccessibl e a t th e present time .

Vinogradov's succes s wa s du e i n large par t t o hi s ingeniou s method s fo r evaluat -ing arithmeti c sums , i n particular , trigonometri c sums . Th e las t par t o f th e chapte r on hi s wor k i s concerne d wit h hi s estimate s o f Wey l sum s (name d afte r Herman n Weyl an d introduce d i n Weyl' s 191 6 pape r entitle d "Ube r Gleichverteilun g vo n Zahlen mod . Eins") . Th e sum s i n questio n loo k lik e

TV

n=l

where f(x) i s a polynomia l wit h rea l coefficient s an d positiv e degree . The reade r wh o woul d lik e t o se e Vinogradov' s metho d exposite d i n ful l gener -

ality shoul d consul t hi s boo k "Th e metho d o f trigonometri c sum s i n th e theor y o f numbers" [10] . Anothe r excellen t resourc e i s th e recen t boo k o n analyti c numbe r theory b y H . Iwanie c an d E . Kowalsk i [5] .

This bring s u s t o th e conclusio n o f thi s Foreword . O f course , th e St . Petersbur g school o f numbe r theor y di d no t en d i n 194 7 whe n th e origina l o f th e presen t boo k was published . A s Delon e point s ou t i n hi s Prefac e (somewha t immodestly) , "Th e next mos t importan t wor k o f th e St . Petersbur g number-theoreti c school , i n th e chronological orde r establishe d above , belong s t o th e autho r o f th e presen t book , and t o Venkov , Kuz'min , Tartakovskh , an d Linnik. " Whil e thes e ar e al l estimabl e mathematicians, i t i s fai r t o sa y tha t i f th e volum e wer e t o continu e beyon d 1947 , it woul d hav e t o includ e Venko v an d Linni k an d perhap s th e others . I n an y case , by composin g th e presen t volume , Delon e di d a grea t servic e t o posterity . Th e ne w translation shoul d hav e a wid e readershi p amon g Englis h speakin g mathematician s with enoug h backgroun d t o enjo y it .

Michael Rose n Brown Universit y

Bibl iography

[1] J . H . Conwa y an d N . J . A . Sloane , Sphere packings, lattices, and groups, Springer-Verlag , New York-Berlin-Heidelberg , 1988 .

[2] J . H . Conway , The sensual (quadratic) form, Caru s Math . Monograp h No . 26 , MAA , Wash -ington, DC , 1997 .

[3] B . N . Delon e an d D . K . Faddeev , The theory of irrationalities of the third degree, Amer . Math. Soc , Providence , RI , 1964 .

[4] L . K . Hua , Introduction to number theory, Springer-Verlag , Berlin-Ne w York , 1982 . [5] H . Iwanie c an d E . Kowalski , Analytic number theory, Amer . Math . Soc , Colloquiu m Publ. ,

Vol. 53 , Providence , RI , 2004 . [6] K . Irelan d an d M . Rosen , A classical introduction to modern number theory, Grad . Text s i n

Mathematics, Col . 84 , Springer-Verlag , Ne w York-Berlin-Heidelberg , 1990 . [7] J . C . Lagarias , H . W . Lenstr a Jr. , an d C . P . Schnorr , Korkin-Zolotarev bases and successive

minima of a lattice and its reciprocal lattice, Combinatoric a 1 0 (1990) , 333-348 . [8] J . V . Uspensk y an d M . A . Heaslet , Elementary number theory, McGra w Hill , 1939 . [9] I . M . Vinogradov , Elements of number theory, Dove r Publ. , Ne w York , 1954 . (reprint , 2004 )

[10] , The method of trigonometric sums in the theory of numbers, Interscience , London -New York .

[11] C . Zong , Sphere packings, Springer-Verlag , Berlin-Heidelberg-Ne w York , 1999 .

xiv F O R E W O R D T O T H E ENGLIS H EDITIO N

Translator's acknowledgment s

I thank Steve n Chen, Nina loukhoveli, Yurh Kazmerchuk, Keit h Matthews, an d once agai n Ab e Shenitzer , fo r hel p an d encouragemen t whil e thi s translatio n wa s being carrie d ou t a t Yor k University , Toronto , an d Th e Universit y o f Queensland , Brisbane.

200 4 R. G . Burn s

Preface

The wor k o f Russia n mathematician s i n th e theor y o f number s constitute s a glorious contributio n t o Russia n science . Althoug h mentio n shoul d perhap s b e made i n thi s connectio n o f th e brillian t wor k o f th e grea t Leonhar d Euler , sinc e he wa s a membe r o f th e Russia n Academ y o f Science s fo r a considerabl e portio n of hi s life 1, i n it s prope r sens e th e Russia n schoo l o f numbe r theor y begin s wit h Chebyshev. Th e St . Petersbur g schoo l wa s illumine d b y name s suc h a s tha t o f Chebyshev himself , Korkin , Zolotarev , Marko v an d Voronoi 2; an d a t th e presen t time3 th e Sovie t schoo l ca n boas t o f severa l excellen t numbe r theorists , le d b y Academician Vinogradov .

The ai m o f th e presen t boo k i s to acquain t th e love r o f mathematic s wit h th e most importan t work s o f the above-name d si x preeminen t member s o f th e St . Pe -tersburg schoo l o f number theory . Fo r eac h o f these si x a shor t biograph y i s given , followed b y a n expositio n o f tw o o r thre e o f th e mos t significan t o f hi s number -theoretical contributions . Eac h such contribution i s first expounde d i n the author' s original terminolog y an d notation , i.e. , i n the for m o f a summary , a s i t were , facili -tating th e readin g o f the origina l work itself , an d thi s i s then followe d b y a more o r less broa d commentar y o n it . Certai n o f th e work s i n question , fo r instanc e thos e of Chebyshev o n primes, hav e proved amenabl e t o a relatively complet e exposition , while others , mor e wide-rangin g i n nature , ar e deal t wit h muc h mor e briefly—fo r instance Zolotarev' s dissertatio n o n integra l comple x numbers. 4

The nex t mos t importan t wor k o f the St . Petersbur g number-theoreti c school , in th e chronologica l orde r establishe d above , belong s t o th e autho r o f th e presen t book, an d t o Venkov , Kuz'min , Tartakovsk h an d Linnik . Outstandin g contribu -tions t o numbe r theor y hav e als o bee n mad e b y Chebotarev , an d b y th e Mosco w mathematicians Khinchin , Shnirel'ma n an d Gel'fond . Howeve r consideration s o f space hav e mad e i t impossibl e t o includ e exposition s o f the wor k o f these authors , let alon e th e grea t man y othe r lesse r Russia n contribution s t o numbe r theory .

I wis h t o expres s m y deepes t gratitud e t o my coauthor , Professo r Bori s Alek -seevich Venkov , fo r writin g th e expositio n o f the wor k o f I . M . Vinogrado v an d o f two o f Voronoi' s works—o n th e su m o f th e number s o f divisor s an d o n complet e forms. I n additio n I than k bot h hi m an d Correspondin g Membe r o f th e Academ y P. O . Kuz'mi n fo r valuabl e advic e concernin g th e compilatio n o f materia l fo r thi s book.

1947 B . Delon e

11725-1741 an d 1766-1783 . Trans. 2These hav e stresse d syllable s a s follows : Chebyshov , Korkin , Zolotaryov , Markov , Voronoi .

Trans. 3In 1947 . Trans. 4Algebraic integers . Trans.

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A Lis t o f Work s i n Numbe r Theor y b y Chebyshev , Korkin, Zolotarev , Markov , Voronoi , an d

Vinogradov

P. L . Chebyshe v

[1] The theory of congruences, book ; Russia n editions : first , St . Petersburg , 1849 ; second , St. Petersburg , 1879 ; third , St . Petersburg , 1901 ; Translations : Theorie der Congruenzen, Berlin, 1888 ; Theoria delle congruenze, Roma , 1895 . Als o reprinte d i n th e Collecte d Work s of P . L . Chebyshev , publishe d b y th e Academ y o f Science s o f th e USS R i n 194 4 (Russian) .

[2] On the determination of the number of primes not exceeding a given number, first publishe d in 184 9 a s a thir d appendi x t o th e boo k The theory of congruences (Russian) , an d als o a s a separate articl e unde r th e titl e Sur la fonction qui determine la totalite des nombres premiers inferieures a une limite donnee, Memoire s de s savant s etranger s d e l'Acad . Imp . Sci . d e St. -Petersbourg, VI , 1848 . Publishe d als o unde r th e sam e titl e i n th e Journa l d e math , pure s e t appl., I serie , XVIII , 1852 .

[3] Memoire sur les nombres premiers, Memoire s d e l'Acad . Imp . Sci . d e St.-Petersbourg , VII , 1850, an d unde r th e sam e titl e i n th e Journa l d e math , pure s e t appl. , I serie , XVII , 1852 .

[4] Sur les formes quadratiques, Journa l d e math , pure s e t appliquees , I serie , XVI , 1851 . [5] Note sur differentes series, Journa l d e math , pure s e t appl. , I serie , XVI , 1851 . [6] On an arithmetical question, A n appendi x t o Note s o f th e Imp . Akad . o f Sciences , X , No . 4 ,

1868. (Russian ) [7] Lettre a M. Fuss sur un nouveau theoreme relatif aux nombres premiers contenus dans les

formes 4 n + l and 4n+3 , Bull , de la classe phys.-math. d e l'Akad. Imp . Sci . de St.-Petersbourg , XI, 1853 .

[8] Sur la generalisation de la formule de M. Catalan et sur une formule arithmetique qui en resulte, C . R . d e l'Assoc . franc, , pou r l'avancemen t de s sciences , 1876 .

[9] Sur une transformation des series numeriques, Nouv . corr . math , redig e pa r M . Catalan , IV , 1879.

[10] Proof of a theorem of Chebyshev (Russian) , not e writte n u p b y A . A . Marko v fro m paper s of Chebyshe v remainin g afte r hi s death , C . R . d e l'Acad . Sci . Paris , CXX , 1895 .

(Except fo r th e last , al l o f thes e work s wer e reproduce d i n bot h Russia n an d Frenc h i n th e first editio n o f Chebyshev' s collecte d work s (Akad . Nauk , 1899-1907) , an d i n Russia n i n vol . 1 of a ne w edition , A N SSSR , 1944). )

A. N . Korki n

[1] Sur les formes quadratiques positive quaternaires (join t wit h E . I . Zolotarev) , Math . Ann. , Bd. 5 , 1872 .

[2] Sur les formes quadratiques (join t wit h E . I . Zolotarev) , Math . Ann. , Bd . 6 , 1873 . [3] Sur les formes quadratiques positives (join t wit h E . I . Zolotarev) , Math . Ann. , Bd . 11 , 1877. [4] Sur I'impossibilite de resoudre Vequation X n + Y n + Z n = 0 e n fonctions entieres, C . R . d e

l'Acad. de s Sci. , Paris , 1880 . A pape r read y fo r publicatio n wit h th e followin g titl e wa s foun d amon g Korkin' s remains ; i t was subsequentl y incorporate d i n par t i n D . A . Grave' s cours e i n numbe r theory :

273

274 A LIST O F WORK S I N NUMBE R THEOR Y

[5] Sur la distribution des nombres entiers suivant le module premier, et les congruences binomes, avec une table des racines primitives et des caracteres qui s'y rapportent pour les nombres premiers inferieurs a 4.000.

E. I . Zolotare v

[1] On an indeterminate equation of the third degree, dissertatio n fo r th e degre e o f Maste r o f Mathematics, St . Petersburg , 1869 . (Russian )

[2] Sur les formes quadratiques positives quaternaires (join t wit h A . Korkin) , Math . Ann. , Bd . 5, 1872 .

[3] Nouvelle demonstration de la loi de reciprocity de Legendre, Nouv . Ann . d e Math. , 2- e serie ,

t. 11 , 1872 .

[4] Sur I'equation Y 2 - ( - l ) ^ p Z 2 = AX, ibid . [5] Extrait d'une lettre de M. G. Zolotareff, Nouv . Ann . d e Math. , 2- e serie , t . 12 , 1873 . [6] Sur les formes quadratiques (join t wit h A . Korkin) , Math . Ann . Bd . 6 , 1873 . [7] The theory of integral complex numbers with an application to the integral calculus, Doctora l

Dissertation, St . Petersburg , 1874 . (Russian ) [8] Sur les nombres complexes, Bull , d e l'Acad . Sci . d e St.-Petersbourg , 3- e serie , 24 , 1878 . [9] Sur les formes quadratiques positives (join t wit h A . Korkin) , Math . Ann. , Bd . 11 , 1877.

[10] Sur la theorie des nombres complexes, Journa l d e math , pure s e t appliquees , (3) , t . 6 , 1885 .

A. A . Marko v

[1] On indefinite ternary quadratic forms, Izv . AN , vol . XIV , 1901 . (Russian ) [2] On indefinite quadratic forms in four variables, Izv . AN , vol . XVI , 1902 . (Russian ) [3] On three indefinite ternary quadratic forms, Izv . AN , vol . XVII , 1902 . (Russian ) [4] G. F. Voronoi. 1868-1908. Obituary , Izv . AN , vol . II , 1908 . (Russian ) [5] Table des formes quadratiques ternaires indefinies ne representant pas zero, pour tous les

determinants positifs D < 50 , Note s o f A . N . i n th e physic s an d mathematic s department , series VIII , vol . XXII , No . 7 , 1909 .

[6] On binary quadratic forms of positive determinant, Master' s Dissertation , St . Petersburg , 1880. (Russian )

[7] Sur les nombres entiers dependants d'une racine cubique d'un nombre entier ordinaire, Mem . de l'Acad . Sci . d e St.-Petersbourg , VI I serie , t . 38 , 1892 .

G. F . Vorono i ( A complet e lis t o f hi s works )

On the Bernoulli numbers, Soobshch . Khar'kov . Mat . Obshch. , 1890 . (Russian ) On integral algebraic numbers depending on a root of an irreducible equation of the third degree, Master' s dissertation , 1893 . (Russian ) On a generalization of the continued fraction algorithm, Doctora l dissertation , Warsaw , 1896 . (Russian) On the number of roots of a congruence of the third degree with respect to a prime modulus, Notes o f th e Xt h conferenc e o f scientifi c professions , 1897 . (Russian ) On the sum of the quadratic residues of the form 4 m + 3 modulo a prime number p, Protokol y St.-Peterburg. Mat . Obshch. , 1899 . (Russian ) A generalization of the concept of the sum of a series, Note s o f th e Xlt h conferenc e o f scientific professions , 1901 . (Russian ) Sur un probleme du calcul des fonctions asymptotiques, Crelle , Bd . 126 , 1903 . Sur une fonction transcendente et ses applications a la sommation de quelques series, Ann . de l'Ecol e Norm . Sup. , t . 20 , II I serie , 1903 . Propriete du discriminant des fonctions entieres, Ver . de s II I internationale n math . Kongr. , 1904. Sur le developpement a I'aide des fonctions cylindriques des sommes doubles ^2f(pm 2 + 2qmn + rn 2), ibid . Sur quelques proprietes des formes quadratiques positives parfaites, Crelle , Bd . 133 , 1907 . Recherches sur les paralleloedres primitifs, Crelle , Bde . 134 , 136 , 1908 , 1909 .

[i]

[2]

[3]

[4]

[5]

[6]

[7] [8]

[9]

[10]

[11] [12]

A LIST O F WORK S I N NUMBE R THEOR Y 275

I. M . Vinogradov 2 7 9

A new method for obtaining asymptotic expressions for arithmetic functions, Izv . Ros . Akad . Nauk, vol . II , No . 16 , 1917 , pp . 1347-1378 . (Russian ) On an asymptotic equivalence in the theory of quadratic forms, Zh . Fiziko-Matematicheskog o Obshch. Perm . Univ. , vol . 1 , 1918 , pp . 18-28 . (Russian ; Frenc h summary ) On the average value for the number of classes of purely fundamental forms of negative determinant, Soobshch . Khar'kov . Mat . Obshch. , vol . 16 , Nos. 1/2 , 1918 , pp. 10-38 . (Russian ) Sur la distribution des residus et des non-residus des puissances, Zh . Fiziko -Matematicheskogo Obshch . Perm . Univ. , issu e 1 , 1918 , pp . 94-98 . (Russian ) On the distribution of quadratic residues and non-residues, Zh . Fiziko-Matematicheskog o Obshch. Perm . Univ. , vol . 2 , 1919 , pp . 1-16 . (Russian ; Frenc h summary ) On asymptotic equivalences in number theory, Izv . Ros . Akad . Nauk , vol . 5 , 1921 , pp . 158 -160. (Supplemen t t o th e proceeding s o f th e XV I meetin g o f th e Phys.-Math . Sectio n o f th e Acad, o f Science s o n Novembe r 23 , 1921. ) (Russian ) Sur un theoreme general de Waring, Matem . Sb. , vol . 31, 1924, pp. 490-507. (French ; Russia n summary) An elementary proof of a general theorem of analytic number theory, Izv . Akad . Nau k SSSR , vol. 19 , Nos . 16/17 , 1925 , pp . 785-796 . (Russian ; Frenc h summary ) An elementary proof of a general proposition of analytic number theory, Izv . Leningrad . Politekhn. Inst. , vol . 29 , 1925 , pp . 3-12 . (Russian ; Frenc h summary ) On the distribution of indices, Dokl . Akad . Nau k SSSR , Apri l 1926 , pp . 73-76 . (Russian ) On a bound for the least non-residue of degree n, Izv . Akad . Nau k SSSR , vol . 20 , Nos . 1/2 , 1926, pp . 47-58 . (Russian ) On the fractional parts of an integer polynomial, Izv . Akad . Nau k SSSR , vol . 20 , No . 9 , 1926 , pp. 585-600 . (Russian ) On the question of the distribution of the fractional parts of the values of a function of one variable, Zh . Leningrad . Fiz.-Mat . Obshch. , vol . 1 , 1926 , pp . 56-65 . (Russian ; Frenc h summary) On a general theorem concerning the distribution of residues and nonresidues of powers, Bull. Amer . Math . Soc , vol . 32 , No . 6 , 1926 , p . 596 . Analytic proof of a theorem on the distribution of the fractional parts of an integer polyno-mial, Izv. Akad . Nau k SSSR , vol . 21 , Nos. 7/8 , 1927 , pp . 567-578 . (Russian ) On the distribution of the fractional parts of the values of a polynomial in two variables, Izv . Leningrad. Politekhn . Inst. , vol . 30 , 1927 , pp . 31-52 . (Russian ; Frenc h summary ) Demonstration elementaire d'un theoreme de Gauss, Zh . Leningrad . Fiz.-Mat . Obshch. , vol. 1 , 1927 , pp . 187-193 . (French ; Russia n summary ) On a general theorem concerning the distribution of the residues and non-residues of powers, Trans. Amer . Math . Soc , vol . 29 , 1927 , pp . 209-217 . On the bound of the least non-residues of n-th powers, Trans . Amer . Math . Soc , vol . 29 , 1927, pp . 218-226 . On Waring's theorem, Izv . Akad . Nau k SSS R (Otd . Fiz.-Mat . Nauk) , Nos . 4/5, 1928 , pp. 393 -400. (Russian ) On the representation of a number by an integer polynomial in several variables, Izv. Akad . Nauk SSS R (Otd . Fiz.-Mat . Nauk) , Nos . 4 /5 , 1928 , pp . 401-414 . (Russian ) On a class of simultaneous Diophantine equations, Izv . Akad . Nau k SSS R (Otd . Fiz.-Mat . Nauk), No . 4 , 1929 , pp . 355-376 . (Russian ) On the least primitive root, Dokl . Akad . Nau k SSSR , No . 1 , 1930 , pp . 7-11 . (Russian ) The elements of higher mathematics, Leningrad. , Izd . KUBUCH , Par t 1 , Analytic geometry, 1932, 24 8 pages . (Russian )

2 7 9 The origina l lis t o f Vinogradov' s publication s include s appointment s a t variou s time s to editoria l position s o n th e followin g journal s an d series : Zhurna l Leningradskog o Fiziko -Matematicheskogo Obshchestva ; Mathematic s i n Monographs . Surve y series ; Matematichecki i Sbornik. Ne w series ; Izv . Akad . Nau k SSSR , OMEN , Ser . Mat. ; Mathematic s i n Monographs . Basic series .

276 A LIST O F WORK S I N NUMBE R THEOR Y

[25] Problems of analytic number theory. Summary of the lecture. I n th e book : Lectures on the occasion of the Jubilee session of the Acad, of Sciences of the USSR, devoted to the XV-th anniversary of the October Revolution, Leningrad , Izd . Akad . Nau k SSSR , 1932 , p . 13 . (Russian) On the number of integer points inside a circle, Izv . Akad . Nau k SSSR , No . 3 , 1932 , pp. 313 -336. (Russian ) The elements of higher mathematics, Leningrad , Izd . KUBUCH , Par t 2 , Differential calculus, 1933, 17 6 pages . (Russian ) On the problems of analytic number theory. I n th e book : Proc. of the session devoted to the fifteenth anniversary of the October revolution, November 12-19, 1932, Leningrad , Izd . Akad. Nau k SSSR , 1933 , pp . 27-37 ; se e als o a separat e brochure , 1933 , pp . 1-11 . (Russian ) An application of finite trigonometric sums to the question of the distribution of the fractional parts of an integer polynomial. I n Trudy Fiz.-Mat. Inst. Steklova, vol . 4 , 1933 , pp . 5-8 . (Russian) On a trigonometric sum and its application to number theory, Dokl . Akad . Nau k SSSR , No. 3 , 1938 , pp . 195-204 . (Russian ; Englis h summary ) On certain trigonometric sums and their applications, Dokl . Akad . Nau k SSSR , No . 6 , 1933 , pp. 249-255 . (Russian ; Englis h summary ) New applications of trigonometric sums, Dokl . Akad . Nau k SSSR , vol . 1 , No. 1 , 1934 , pp. 10— 14. (Russian ; Englis h summary ) New asymptotic expressions, Dokl . Akad. Nau k SSSR , vol . 1 , No. 2, 1934 , pp. 49-51 . (Russian ; English summary ) Trigonometric sums depending on a composite modulus, Dokl . Akad . Nau k SSSR , vol . 1 , No. 5 , 1934 , pp . 225-229 . (Russian ; Englis h summary ) New theorems on the distribution of quadratic residues, Dokl . Akad . Nau k SSSR , vol . 1 , No. 6 , 1934 , pp . 289-290 . (Russian ; Frenc h summary ) New theorems on the distribution of primitive roots, Dokl . Akad . Nau k SSSR , vol . 1 , No . 7 , 1934, pp . 366-369 . (Russian ; Englis h summary ) A new solution of Waring's problem, Dokl . Akad . Nau k SSSR , vol . 2 , No . 6 , 1934 , pp . 337 -341. (Russian ; Englis h summary ) On certain new problems of number theory, Dokl . Akad . Nau k SSSR , vol . 3 , No . 1 , 1934 , pp. 1-3 . (Russian ; Englis h version , pp . 4-6. ) Certain theorems of analytic number theory, Dokl . Akad . Nau k SSSR , vol . 4 , No . 4 , 1934 , pp. 185-187 . (Russian ; Englis h summary ) A new estimate for G{n) in Waring's problem, Dokl . Akad . Nau k SSSR , vol . 4 , Nos . 5/6 , 1934, pp . 249-251 . (Russian ; Englis h version , pp . 251-253 ) On an upper bound for G(n) in Waring's problem, Izv . Akad . Nau k SSSR , No . 10, 1934 , pp. 1455-1469 . (Russian ; Englis h summary ) Certain theorems on the distribution of indices and primitive roots. I n Trudy Fiz.-Mat. Inst. Steklova, vol . 5 , 1934 , pp . 87-93 . (Russian ) Sur quelques nouveaux resultats en theorie analytique des nombres, C . R . Acad . Sci . Paris , vol. 199 , No . 3 , 1934 , pp . 171-175 . On approximations by means of rational fractions with denominator an exact power, Dokl . Akad. Nau k SSSR , vol . 2 , No . 1 , 1935 , pp . 1-5 . (Russian ; Frenc h summary ) On certain rational approximations, Dokl . Akad . Nau k SSSR , vol . 3 , No . 1 , 1935 , pp . 3-6 . (Russian) On the fractional parts of polynomials and other functions, Dokl . Akad . Nau k SSSR , vol . 3 , No. 3 , 1935 , pp . 99-100 . (Russian ) New estimates for Weyl sums, Dokl . Akad . Nau k SSSR , vol . 3 , No . 5 , 1935 , pp . 195-198 . (Russian) A new variant of the derivation of Waring's theorem, Trud y Matem . Inst . Steklova , vol . 9 , 1935, pp . 5-15 . (Russian ) The number of integer points in a sphere, Trud y Matem . Inst . Steklova , vol . 9 , 1935 , pp. 17— 38. (Russian ) Une nouvelle variante de la demonstration du theoreme de Waring, C . R . Acad . Sci . Paris , vol. 200 , 1935 , pp . 182-184 . Sur les sommes de M. H. Weyl, C . R . Acad . Sci . Paris , vol . 201 , No. 13 , 1935 , pp . 514-516 . On Waring's problem, Ann . o f Math. , vol . 36 , No . 2 , 1935 , pp . 395-405 .

A LIST O F WORK S I N NUMBE R THEOR Y 277

[53] On approximation to zero with the help of numbers of a certain general form, Matem . Sb. , vol. 42 , No . 2 , 1935 , pp . 149-156 . On Weyl's sums, Matem . Sb. , vol . 42 , No . 5 , 1935 , pp . 521-530 . An asymptotic formula for the number of representations in Waring's problem, Matem . Sb. , vol. 42 , No . 5 , 1935 , pp . 531-534 . Fundamentals of number theory, Moscow-Leningrad , ONTI , 1936 . (Russian ) A new improvement of the estimates of trigonometrical sums, Dokl . Akad . Nau k SSSR , vol . 1 , No. 5 , 1936 , p . 195 . (Russian ; Englis h version , pp . 199-200 ) New results concerning the distribution of the fractional parts of a polynomial, Dokl . Akad . Nauk SSSR , vol . 2 , No . 9 , 1936 , pp . 355-357 . (Russian ; Englis h version , pp . 361-364 ) Approximation with the help of certain fractions, Ann . o f Math., vol . 37, No. 1 , 1936 , pp. 101 -106. On fractional parts of certain functions, Ann . o f Math. , vol . 37 , No . 2 , 1936 , pp . 448-455 . Sur les nouveaux resultats de la theorie analytique des nombres, C . R . Acad . Sci . Paris , vol. 202 , 1936 , pp . 179-180 . Sur quelques inegalites nouvelles de la theorie des nombres, C . R . Acad . Sci . Paris , vol . 202 , 1936, pp . 1361-1362 . On the number of fractional parts of a polynomial lying in a given interval, Matem . Sb. , vol. 1 (43) , No . 1 , 1936 , pp . 3-8 . A new method of resolving certain general questions of the theory of numbers, Matem . Sb. , vol. 1 (43) , No . 1 , 1936 , pp . 9-20 . Approximation by means of fractional parts of a polynomial, Matem . Sb. , vol . 1 (43) , No . 1 , 1936, pp . 21-27 . On an asymptotic formula in Waring's problem, Matem . Sb. , vol. 1 (43), No. 2, 1936 , pp. 169 -174. A new method of estimation of trigonometrical sums, Matem . Sb. , vol . 1 (43) , No . 2 , 1936 , pp. 175-178 . Supplement to the paper "On the number of fractional parts of a polynomial lying in a given interval", Matem . Sb. , vol . 1 (43) , No . 3 , 1936 , pp . 405-407 . A new method in analytic number theory, Trud y Matem . Inst . Steklova , vol . 10 , 1937 . (Rus -sian) The distribution of the fractional parts of the values of a polynomial under the condition that the argument ranges over the primes in an arithmetic progression, Izv . Akad . Nau k SSSR, , No. 4 , 1937 , pp . 505-514 . (Russian ; Englis h summary ) Represention of an odd number as a sum of three primes, Dokl . Akad . Nau k SSSR , vol . 15 , Nos. 6/7 , 1937 , pp . 291-294 . (Russian ) Some new problems of the theory of prime numbers, Dokl . Akad . Nau k SSSR , vol . 16 , No . 3 , 1937, pp . 139-141 . (Russian ; Englis h version , pp . 131-132 ) New estimations of trigonometrical sums containing primes, Dokl . Akad . Nau k SSSR , vol . 2 (44), No . 4 , 1937 , pp . 165-166 . A new estimate of a sum containing primes, Matem . Sb. , vol. 2 (44), No. 5, 1937 , pp. 783-792 . (Russian; Englis h summary ) Some theorems concerning the theory of primes, Matem . Sb. , vol . 2 (44) , No . 2 , pp . 179-195 . (English; Russia n summary ) Some general theorems in the theory of primes, Trud y Tbilis . Mat . Inst. , vol . 3 , 1937 , pp. 35 -67. (Russian ) Einige allgemeine Primzahlsatze, Trud y Tbilis . Mat . Inst. , vol . 3 , 1937 , pp . 35-67 . Fundamentals of number theory, Secon d revise d edition , Moscow-Leningrad , ONTI , 1938 . (Russian) A new estimate of a certain trigonometric sum containing primes, Izv . Akad . Nau k SSSR , No. 1 , 1938 , pp . 3-14 . (Russian ; Englis h summary ) Improvement in the estimate of a certain trigonometric sum containing primes, Izv . Akad . Nauk SSSR , Ser . Mat. , No . 1 , 1938 , pp . 15-24 . (Russian ; Englis h summary ) Some new estimates in analytic number theory, Dokl . Akad . Nau k SSSR , vol . 19 , No. 5 , 1938 , pp. 339-340 . (Russian ) The distribution of quadratic residues and non-residues of the form p + k with respect to a prime modulus, Matem . Sb. , vol . 3 (45) , No . 2 , 1938 , pp . 311-319 . (Russian ; Englis h summary)

278 A LIS T O F WORK S I N NUMBE R THEOR Y

[83] Elementary estimates of a certain trigonometric sum involving primes, Izv . Akad . Nau k SSSR, Ser . Mat. , 1939 , pp . 111-122 . (Russian ; Englis h summary )

[84] Estimates of certain of the simplest trigonometric sums involving primes, Izv . Akad . Nau k SSSR, Ser . Mat. , 1939 , pp . 371-398 . (Russian ; Englis h summary )

[85] New perfection of a method for estimating trigonometric sums involving primes, Dokl . Akad . Nauk SSSR , vol . 22 , No . 2 , 1939 . (Russian )

[86] The simplest trigonometric sums involving primes, Dokl . Akad . Nau k SSSR , vol . 23 , 1939 . (Russian)

[87] A certain general property of the distribution of primes, Matem . Sb. , vol . 7 (49) , 1940 , pp. 365-372 . (Russian ; Englis h summary )

[88] The distribution relative to a given modulus of the primes in an arithmetic progression, Izv . Akad. Nau k SSSR , Ser . Mat. , vol . 4 , 1940 , pp . 27-36 . (Russian ; Englis h summary )

[89] Two theorems relating to the theory of the distribution of prime numbers, Dokl . Akad . Nau k SSSR, vol . 30 , No . 4 , 1941 , pp. 287-288 .

[90] Improvement of some theorems of the theory of primes, Dokl . Akad . Nau k SSSR , vol . 37 , 1942, pp . 115-117 .

[91] General theorems on estimations of trigonometric sums, Dokl . Akad . Nau k SSSR , vol . 43 , No. 2 , 1944 , pp . 47-48 .

[92] The analytic theory of numbers, Izv. Akad . Nau k SSSR , Ser . Mat. , vol . 9 , No . 3 ; The works of P. L. Chebyshev in the theory of numbers (join t wit h B . N . Delone) . I n th e collection : Scientific heritage of P. L. Chebyshev, vol . I , Izd . Akad . Nau k SSSR , 1945 . (Russian )

[93] A general distribution law for the fractional parts of the values of a polynomial when the variable ranges over the primes, Dokl . Akad . Nau k SSSR , vol . 1 , No . 7 , 1946 , pp . 491-492 .

Titles i n Thi s Serie s

26 B . N . Delone , Th e St . Petersbur g schoo l o f numbe r theory , 200 5

25 J . M . Plotkin , Editor , Hausdorf f o n ordere d sets , 200 5

24 Han s Niel s Jahnke , Editor , A histor y o f analysis , 200 3

23 Kare n Hunge r Parshal l an d Adrai n C . Rice , Editors , Mathematic s unbound : Th e

evolution o f a n internationa l mathematica l researc h community , 1800-1945 , 200 2

22 Bruc e C . Bernd t an d Rober t A . Rankin , Editors , Ramanujan : Essay s an d surveys ,

2001

21 Arman d Borel , Essay s i n th e histor y o f Li e group s an d algebrai c groups , 200 1

20 Kolmogoro v i n perspective , 200 0

19 Herman n Grassmann , Extensio n theory , 200 0 18 Jo e Albree , Davi d C . Arney , an d V . Frederic k Rickey , A statio n favorabl e t o th e

pursuits o f science : Primar y material s i n th e histor y o f mathematic s a t th e Unite d State s Military Academy , 200 0

17 Jacque s Hadamar d (Jerem y J . Gra y an d A b e Shenitzer , Editors) , Non-Euclidea n geometry i n th e theor y o f automorphi c functions , 199 9

16 P . G . L . Dirichle t (wit h Supplement s b y R . Dedekind) , Lecture s o n numbe r theory , 1999

15 Charle s W . Curtis , Pioneer s o f representatio n theory : Probenius , Burnside , Schur , an d Brauer, 199 9

14 Vladimi r Maz'y a an d Tatyan a Shaposhnikova , Jacque s Hadamard , a universa l mathematician, 199 8

13 Lar s Garding , Mathematic s an d mathematicians : Mathematic s i n Swede n befor e 1950 , 1998

12 Walte r Rudin , Th e wa y I remembe r it , 199 7

11 Jun e Barrow-Green , Poincar e an d th e thre e bod y problem , 199 7

10 Joh n Stillwell , Source s o f hyperboli c geometry , 199 6

9 Bruc e C . Bernd t an d Rober t A . Rankin , Ramanujan : Letter s an d commentary , 199 5

8 Kare n Hunge r Parshal l an d Davi d E . Rowe , Th e emergenc e o f th e America n

mathematical researc h community , 1876-1900 : J . J . Sylvester , Feli x Klein , an d E . H . Moore ,

1994

7 Hen k J . M . Bos , Lecture s i n th e histor y o f mathematics , 199 3

6 Smilk a Zdravkovsk a an d Pete r L . Duren , Editors , Golde n year s o f Mosco w

mathematics, 199 3

5 Georg e W . Mackey , Th e scop e an d histor y o f commutativ e an d noncommutativ e

harmonic analysis , 199 2

4 Charle s W . McArthur , Operation s analysi s i n th e U.S . Arm y Eight h Ai r Forc e i n Worl d

War II , 199 0

3 Pete r L . Dure n e t al. , Editors , A centur y o f mathematic s i n America , par t III , 198 9

2 Pete r L . Dure n e t al. , Editors , A centur y o f mathematic s i n America , par t II , 198 9

1 Pete r L . Dure n e t al. , Editors , A centur y o f mathematic s i n America , par t I , 198 8