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1 Math-Net.Ru All Russian mathematical portal N. Tchudakoff, On the difference between two neighbouring prime numbers, Rec. Math. [Mat. Sbornik] N.S., 936, Volume (43), Number 6, Use of the all-russian mathematical portal Math-Net.Ru implies that you have read and agreed to these terms of use Download details: IP: November 8, 208, 06:8:58

2 936 МАТЕМАТИЧЕСКИЙ СБОРНИК Т. (43), N. 6 RECUEIL MATHEMATIQUE On the difference between two neighbouring prime numbers N. Tchudakoff (Saratow). Introduction. In this work I apply the results obtained by me in my preceding paper г to some questions of the theory of the distribution of prime numbers. I prove following theorems. Theor em 2 : S A(»)-^> + *(y).. x<in^x-\-y n == / (mod k) where k and I are integers, (ft; I) =, Л (п) is a Mangoldfs number of classes of primes to ft, 0<Y f <l> Theorem II: 3_, 3 log log x y = X A <?i(u>g*)t'' function; <p (k) is the n(x+y; ft; l) n(x; ft; /) = -^~7 f-ofr-^-), v ^ ' ' ' <p(ft)log.* ' Mogx/' адддо тг(лг; ft; /) is the number of primes having the form kn-\-l; ft, /, у have the same value as in the theorem I. Theorem III. Let щ, тт 2,..., тг я,... be a series of primes having the form kn-\-u then:! +. for any e>0. Theorem IV: Every integer thai is large enough may be represented in the form: * Recueil math/, (43): 4, 936, 59Ц 2 It is possible to prove in an analogous way that ]T \i{n) = o{y) 9 J>=--* 4 ц(я) being a Mobius's function.» neel (mod ft)

3 800 N. Tchudakoff while ±n-tfi = 0(n 78 + } {i==i 9 2,..., 8). Theorems I, II and III represent improvements on the analogous theorems due to Hoheisel l and Heilbronn 2. Theorem IV is an improvement on the Chowla's 3 analogous theorem. 2. Notations, k and / are integers, 0 () c i C Q> \, are constants depending only on k and y. Symbol О in the formula Y=0(X) (where X>0) denotes that x^x 0^3, a- \Y\<cX, 3 log log _x x л \_ 3 log log л: 4 ' c x (log xf ' P "^ 4 2^ (log JC) T ' ' Г=л:?, j; = x a, 7j == log л:, а = q(logr)-r, v = * L(s; ]() is the Dirichlet's function corresponding to the character i modulo & 4 ОС 00 (L(s; Z))- = -SX(«)»*(«) «*. Я = for a>l. It is easy to establish following relations (2) gives yl-sl ^U-ffi) : /z=l OO l o ^*c (logx)-t as Г<лт. 3 log log дг.,. Y d-aoд. 2* (tog*)?' = ]/ r v(\ogx) _ 4 в=ь*2 **, (i - gi)( i _ 4 p)^6 loglog *. q(log*) Y log>: Therefore (4) gives: (-)' JI =^Ci-4p)(i-«) > (i ogjc )e, ЛГ 6 log у, ==( 4^) logx = - (logx)rloglogat. (2) (3) (4) (5) (6) G. Hoheisel, Primzahlprobleme in der Analysis ( Sitzungsber. d, Preuss. Akad. d. Wissensch. phys.-math. Kl. a ), (Berlin, 930, S. 580 ff.). 2 H. Heilbronn, Uber den Pfimzahlsatz Herrn Hoheisel ( Mathem. Zeitschrift", Bd. 36, S. 395 ff.). 3 С howl a, The representation of large number as the sum of eight almost equal" cubes ( Quarterly Journ. of Math.% Oxford Series, Vol. 6, N. 22, 935.) 4 We will omit the letterx> if it has a single value in the given research.

4 On the difference between two neighbouring prime numbers 80 It is not difficult to see that 0og.*)T' It shows that: \ogx = o(v 6 ) (7) for any s^>0. Having taken x 0 sufficiently large we verify that since ~~ Sl >0, ji >-j for x «>. Finally we can take x 0 so large that v = x 2 <dx$=t for x^x 0 Therefore (log x)t log log x > 8 log k. 3, Lemma : ш tte domain: Iog (Щ* > T x (log *V log log x (for * ^ *<>) ( 8 ) (*; x)=0((iog * )T) (D) a>l q(log ^ )-v, / >^4. Proof. It is obvious that it is sufficient to prove our lemma for ^>0. In ray preceding work г I have shown that in the domain o^l 0 0 o &*)~ Y» ^^^o^4, for any у between yr and. Now suppose a 0 =l+^(log/)-t, where с г <^тт (; c Q ).. a^o 0, s = a-\-it, t^t 0, fm< ^ = -^>=o(^)=o«,o g,,>. 2. x (log0~ Y <a<go- Let us describe about the point s 0 = o 0 -\-it which is equal to r = Cl(log*)-» (0=-). It follows from theorem I of my preceding work that:. L(s; y) is regular within the circle K. 2. L (s; i) < <* ^' b (d < y). 3. LAs,) ;Е^=сы-о(^)=о(о^)т). Rec. math.", (43):4. Theorem 2. c 0 does not depend on x- as centre a circle K, the radius of

5 802 N. Tchudakoff Let us put: x) = r y~, x =. L (s 0 ) r It is easy to verify, that f(x) within the circle x ^l satisfies following conditions:. f{x) is regular for * <. 2./(0) l [for L(s 0 )^0]. 3 - \f(x)\<e M where M = c 2 (log<)t-*(i +o(l)), for * «>. and 2 are evident; 3 is proved thus: Let us apply to the function f(x) the well known theorem of the theory of functions i. We obtain: 2Af < ( Ul) 2 ';<*- S e ' -е* where S are zeros of the function f(x) within the circle, taking into consideration their multiplicity. Passing from the function f(x) to the function L(s; y) we obtain by means of some simple calculations:? SQ! <«>-E(.-V p r* (s ^ s 0 ) {p s 0 )" < Wr (r * 5в ) Я (9) where p are zeros of the function L (s) within the circle ] s s Q In particular we obtain: But s 0 \\ ^2M r* /^ r ' 2АГ ^ = 0((log<)T-»).0((log^) = 0(Gog*)T), :r -. (0) () (0), () and (2) give: 2(^p+^)h (fl<*w. Now let us take a point s = a-\-it on the segment (2) (3) l-f-«-ci(log/f)-t- -#, <и<-]- (this segment lying] wholly] within' the circle К for t^t 0, because <J 0 = о (г) as d<y If we take t 0 large enough, then for t^t 0 we have -c 0 (log(* + ))-т< _!L+ i(io g^-t H. Heilbronn, Math. Zeitschrift", Bd. 36, S Zeros of f(x) and L{s; x) correspond to each other as L(so)^0- (4)

6 On the difference between two neighbouring prime numbers 803 because log* By theorem 2 of my preceding work x we have: where p is a zero of L(s) within the circle К; Щ ъ Зр are its real and imaginary parts, This gives: where Then we shall have for the point s o-\-it on the segment (4): 2c x (logt)^ \ _ - Pl >!* e _ p! (i!-i=^4)=»i* e PIA i% (log^t-v * =! Bearing in mind (5) we shall obtain: 2c x c 0 +3q 2 Ac c 0 + W>o. Sc x ^C p ± 2 (5) \s-p >,*-(s~so){ P -s 0 )} Uo-p r* I - s,- p P) (д 0 - s) f/«- (s - p) (p-^)] G-p)(5 0 -p)^(^-(s- So )(p_ So )) r2 ] -So p a < (o 0 o) J* p-j«o Pi But it is easy to verify that i^ (s~p)(p s t ) / (/ ~(S-~8,)(f-St)) (6) since 5ip<a^o 0 and moreover we have for t 0 being large enough: since Therefore (log*)-t = o(r). rz-(s-p)(p-s 0 ) (7) Further it is not difficult to see that 0<(а 0 -Яр) Recueii math/, (43):4, 936, 59. i 0 ~p ^ " :a(-l_. p «*o /*2 (8)

7 804 N. Tchudakoff Therefore it follows from (5), (6), (7) and (8) that since (.4 P / -(*- Hence: S(^ p Г* ( S -5to)(p-Jb)7 S 0 S 0 ) ^-U-p~ #«, /J Therefore y(_l + V (-J L 9=* ) ( ^- Ve-p /*-(s-s 0 )(p-s 0 ) / \ p <±y 3 i(j_+2eil) < l Y'( *з <" U~P ' 2 / 'з ^ ^O +JL^W 0. *o-? P % \ /- 2 / i P~*о \ iza =^) < fi + l) y/j_ i fzzi) (9) As for t 0 being sufficiently large it follows from () (9), (3), (9) and (20) give Lemma 2: 2Mr (r *-* 0 I)*' г (*) s s, 0 I S5S5S <>» -^Г< = 0((log*)*). = 0((log<)T). = 0-((log*)T) (20) ш the domain a^l c (logt)~'(, t^t x^t 0. Proof. Let us suppose again 00 = + *! (log*)"*.. а^а л L(s) X(n)v(n) Л* /г= ; S i = - <*o) = (гч) = ((log ' )Y) ' 2. ^(log/)-y<a^a 0, s==a + tf, log L(s) = 5HogI(s) = UlogL(v + 5? б* \^(s)ds flog Z.(5 0 ) I + J r'(*) *. (2) i Integration is carried on over the segment connecting the points s and «о-

8 On the difference between two neighbouring prime numbers 805 But Therefore \L(s 0 ) = L {so) = v^ х(я)м*) /7 = ' «-= Sf?l<S^-twoo ;S-i=W- «=l log I (* e ) < log С (J,) = log ii^0_) = Y i 0 g log t + О (), (22) j (s) do=(o 0 o)max L ' i \ (23> But Lemma gives: (23), (24) and (25) give: a 0 o<2 Cl (logo-^. max.- (5) O((log?)i), (24> (25) rfa=0((logq-t).0«logt)t) = 0(l), (26) (2), (22) and (26) give: log Z.(s) < T loglog^4-0(l). Therefore log it(5)l = lo S I L ( 5 > I < T lo g l0 S l + W» Lemma 3. For 3 < < < T, 3 -j s^ о «^ 2 following inequalities hold true: I» = X X(e)log«.«-*4-o(logr.^-*). For the proof see Heilbronn, Math. Zeitschrift", 36, Ljemma 4. For Г^З, ~^о<, where J Л (5)?dt=0(T4 d- e > log* Г) о l^resat len^j?' For the proof see Heilbronn, Math. Zeitschrift"» 36, 45. Lemma 5: («V -rw=2; t s- ) +w «= (27)

9 S06 where N. fchudakoff x^2x 0, t t^v^t^t, s = Q ~\~it i g(n) = g(m *), \g(n)\^z(\og(kt))d,(n) td±(n) the number of the representations of n in the form п^=и г и 2 и в и 4 щ being integers (7=, 2, 3, 4)] т \/i(s)\dt = 0(v*(logx)-*) (28) (T, v, a x having values introduced in 2), Proof. For the sake of shortness we will introduce denotations: kt kt _yxi^) y^ = V х(/г) **(*) kt (29) To begin with we have: /z = l -r.">- ==' Further, lemma 3 and (3) give: By lemma 2 we have: Therefore by (7) we have: L'p+Lp' I < Г lpl- 2,, pl=o(^) = o((iog*r*^). _ I^JI Jog T " log x * pz=o(\l(s)\) for x юо and consequently = (bi) ) =0((log ' )T) (see lemma 2). (3), lemmas and 3, (32), (33) give:! Г (30) (3) (32) (33) О ((log*)t) О ( -y-j О ((logx)r) + But (by (3)). Therefore + О (logt^p) 0 ((log*)т)» О ((log x)* (^p). Г IT _ JL i- -t; 2 2 (logx) i. _ l = О ((log x) 2 г» 5") = О ((log лт)т). (34) (35)

10 On the difference between two neighbouring prime numbers 807 But lemma gives: Consequently ^(5) = 0((log/)T) 0((log^)T) :0((l0gX)T). (36) Further it follows from (30): Hence we obtain: Now if we put: 2r y^ ^Z-r2-и- 2*т2*т\ 2*T) ^H Y (37) g(n) = 2 2 X(e)logax(o)- i(»). 2 X(B)io?B]((ti)x(w) i(»)][(?) i(f), Ш7 = яз q^kt then in view of (29) by means of some simple calculations we shall obtain: ZMTZ^T 2ит2^т\2-*т) 2.Л м = Also it is not difficult to verify that: Finally let us put: (34), (36), (37), (38) and (40) give us Therefore: \R(s)\ 4-/ \g(n)\^s(log(kt))d i (n). /7 = -rw-^+^'f -г<'>- : + т "т = 0^(logjf)«(log*) 2 r j + 0 (\А\Цо ё х)). (38) (39) (40) J R{s) <#= 0((logje) 2 P-Uog Г) + Q((log*) T JI Л \*dt). (4) But in virtue of lemma 4, (3) and T<^x we have: \\A\^t==0(T^'^log 4 T) = 0(v 2 (logx)-4og i x) = 0(v^{\ogx)^). (42) Therefore (7), (4) and (42) give: т ^ ^\R(s)\dt^O(Vv(iogx) 2 )^0(v^(logx)-^) = 0(v^(\ogx)^); (43) (39), (40) and (43) prove our lemma.

11 808 N. Tchudakoff Lemma 6: t\^it j" <*±^=*^(,)Л = о(у). a t d=/t Proof. In virtue of symmetry it is sufficient to prove the theorem for-\-t. Integralion will be carried on over the segment connecting the points а г -j- IT, '7}~f- IT: Ч + /Г J Ч-YiT since by lemma But Therefore: But it is easy to see that v ^ г (s) ds < ) jj l i-= О ((log л:)т) (44) ±(s)\^0((logtf) = 0((logxW. 4 ail<2, v ^ =Q(^-P)., n + it f ( * +y)s ~ XS (syds\ = 0(xi-Hogx). J S L. j (45) *3^= J fi-<-*bgx = e- m *,ogu " x.]dgx-+0 i (46) (45) and (46) prove our lemma. Lemma 7: for z^x, x >oo, 5 = a t -(-#. Proof. Lemma gives: s t + tf t by (7)). Lemma 8: 0(+/r /or z^x, x юо, s = G l -\-it. Proof. Lemma 5 gives: f z s -^^-(s)ds = o(l) oi -}- fo d'-fit ^-{-/v (*T>* r '. <2«иаг' «Кт)"*<«2^(тГ" я. и lofif

12 On the difference between two neighbouring prime numbers 809 But it follows from (5), (6) and (39), that z z \ log log log n\ & n x ь (ktf if x^x Qf \g(n)\^^d^{n)\og(kx). Consequently: \I l \^o(log(kx){\ogx)-h\og\ogx)-^\ogxr^^p). But it is well known that г z Therefore Further n 7 = l / i = 0((logJc) -«+ *) = o(l). / 2 = J z s ~*R(s)ds \^x *- l \R(s)\dt==x^\\R(s)\dt. But by (28) we have: I / 2 = О С*-*.*;* (logjc)-i) о (). Thus the lemma is completely proved. Theorem : A(«) = 4 + *(y) nezl (mod A) where A(n) is the MangoldVs function for the values of y, k, I (see 2). Proof. For the future let us put: It is easy to verify that f^-^. <r Klog n for /г^2. 2 -i i^=0( 7 i ri ), l<a<2. V^ V f(k) ^* i(l) L ( *' X)=-ii^' 00 fa(fl) for /z = /(mod /г), 0 for n "ф. (mod &). Consequently we apply the Landau's 2 theorem to the function f(s) for z^t^s, <7)<2: 2itr 7-7Г l^w^s Heilbronn, «Math. Zeitschrift», Bd. 36, S E. Landau, Ober einige Summen, die von den Nullstellen der Riemannschen Zetafunktion abhangen («Acta Mathematica», 35,(92), ).

13 80 N. Tchudakoff Let T and i\ have values introduced in 2. We will give z two values: z = x, z = x-\-y. Then we obtain: ш J 5 as ~ л Ъ a «-r u { en-i)t J' 2itf fir Г tumlds- У a i-c)f^^±\ Subtracting the second equality from the first one by terms, we obtain: since x-f-^<2x, But (2*x)v = 0(x). лн^(гао(л) for In this way we obtain: 3 xl-^logs^ _ (i 0 g*)y k>glog* г =jci~ a -Hog 3 x = ^ 2r * log 3 JC = o(l). ^Y {X+y?- XS f(s)ds= 2 A W + o(y). (47) In the left hand side of (47) let us substitute for the integration over the rectilinear segment (TJ IT; ч + rt) by the integration over a broken contour (L) consisting of segments: L i (4 it > <*i Wh L 2 ( a i it '> a i u ih L 3 (i x it t \ c 4 UJ, L 4 (c 4 tf x,c A -f- ^), [c i is chosen so that in the domain limited by the contour (L) and by the straight line a=l there should not be zeros L(s)]. It is easily seen that within a domain limited by the contour L and the segment -(*] IT; Ч + /7) /(5) has no other singularities except 5 =. At this point the function f(s) has a pole of the first order with a residue equal to where (p(&) is the number of classes relative prime to k. Thus ч+/г

14 On the difference between two neighbouring prime numbers 8 The integral over the contour (L) is divided into 7 integrals accordingly to the segments L t,...,l 7. It is not difficult to see that j\=0(l) (/ = 3, 4, 5), (it) j (x+y)s x* f(s)ds- : ^ j<±±f= *(- («x)) *c/=i,2,-6,7). 'f(k) (49> (49) shows that it is sufficient to estimate integrals of the form: - ^x+y l S - XS ±(s;r)ds (/=,2,6,7). (h) Further in virtue of the symmetry principle it is sufficient to consider the integrals: 5 and J* Let us study the estimation of each of the above integrals.. Estimation of ^ : (A) ot +it j = _ J" (x + y^-xsv^. i)ds J j ^ Г -.p (5. л л л = But c t -f/r far 2^дг; л: >oo by lemmas 7 and 8. It follows from it: J =o(y). (50> 2. Estimation of \ : (A) -n+jt vi + it ( 6 ) by lemma 6. (47), (48), (49), (50) and (5) give: ^ 4-\-iv T(^;X)^ = o(^) 2 А^=ж).+^) л ЕЕ/(mod &) (5) (52>

15 82 N. Tchudakoff In a particular case when ft = we have: where ф(лг) is the Tchebysheffs function. Theorem 2: Mx+y; k; O-i.(x; k; /, = - ^ _ + 0 ( ^ ) where тг (л:; /г; /) & /fte number of primes ^ x having the form kn -(- /. Proof. It is known * that 2 А(л)= X logp + 0(K^) (53) x<n^x-}-y n~il (mod k) x<pis x + v p l(modk) where p is prime number. (52) and (53) give us:. since J/ x = о (у) for x > со. But it is evident that 2 j»te I (mod ) io gp=&)+<>(y) < 54 > logx[u(x-\-y; ft; l) тг (л:; ft; l)]^ ^ logp*^\og(x-{-y)[n(x-\-y; k; l)- n(x; k; /)]. Therefore where 6 <. But p l (mod л) logx[tt(x-{-y; ft; /) тг(л;; k; /)]= log/?-f x<p^x±y p=(lmod k) -\-Mog( x -±y)[n(x+y; k; /) ir(*; k; I)} Therefore: (54) and (55) give: Thus ё( Х -~)=о(%), к(х+у, k; l) = o(x). logx[n(x-]-y; k; l) n(x; ft; /)]= Corollary I. Let x<p^x-\-y pezl (mod&) logx[n(x-]-y; ft; /) тт(лг; ft; 0]=^щ J r-otv). n(x-\-y; ft; /) IT (л; ft; / ) = yw J i ogar + W l> K 2> Я 3» ' ' \ ' be a series of prime numbers of the form kn-\-l. 3 4 Then тг лч тг я =го(я л ") for any s>0. о Р-\- (У)- ( 55 ) 4og л; / E. Landau, «Acta Mathematica», Bd. 35, 27. A particular case is givea there for fc=, but the proof holds true for any k.

16 On the difference between two neighbouring prime numbers 83 Proof. It is evident that it is sufficient to prove it for all n n^x 0. Let us take x 0 so large that a<4--f-s, n(x-\-y\ ft; I) n(x; ft; V)^\ for x^x 0. If we suppose х = тг я, 3/ тг а, all conditions of the theorem 2 for such x and у are satisfied. Therefore: Therefore in the iuterval: *(*n + K> *; ') *(**; ft; /)>! there is at least one prime number ~-Ф=п п. Therefore ^ ^ 7T ^ТГ а^тг 4 n + л n n Hence it is not difficult to obtain: Corollary II. Let a>0, #J0. Then there exists at least one prime number of the form kn-\-l, (ft; t) = \ between (ax-\~bf { * and (a(x-\- ) -\-b) 4! % х;>лг 0, beginning with a sufficiently large x 0. Corollary III. Every integer n^n 0 may be represented in the form: being n = v? x + u\-\-... +«l/i-af = 0(/i -^ + t ) (*==э 2, 3,...,8). The proof coincides with the proof of the Chowla's analogous theorem; it is only necessary to put m occurring in the Chowla's proof equal to 4-j-e. (Поступило в редакцию 27/iV 936.) Математический сборник~ t т. t (43), N. б.

17 84 H. Г. Чудаков О разности двух соседних простых чисел Н. Г. Чудаков (Саратов) (Резюме) В этой работе я даю полное доказательство следующих теорем, краткое сообщение о которых было дано раньше : Теорема I: п = I (mod k) где k и I целые числа, (к; /)~-, А (п) ф нкция Mangoldt'a, <p (k) число классов, взаимно простых с k, ] Z l Z x л ' " Теорем а II: *<*+* k > *>-*& k > ^т^^6.ш) где iz(x;k;l) число простых чисел ^х, имеющих форму kn-\-l; k y I, у имеют такое же значение, как и в теореме /. Теорема III. Пусть "l > ^ 2 ' > Я*., ' ряд простых чиселу имеющих форму kn-\-l; тогда: ** +! *«= ( Tr i. 4 ) при любом 8^>0. Теорема IV. Всякое достаточно большое целое число может быть представлено в форме: л П /»3 f/3, I, = " -+-«з-4- +«*8» причем /I «з=0(л "" 78 + в ) ( =, 2...,8, б>0). 8 «С. R.», t. 202, N. 3 (936 г.).

Math-Net.Ru All Russian mathematical portal

Math-Net.Ru All Russian mathematical portal U. V. Linnik, On the representation of large numbers as sums of seven cubes, Rec. Math. [Mat. Sbornik] N.S., 1943, Volume 12(54), Number 2, 218 224 Use of the