Chapter 2 Lambert Summability and the Prime Number Theorem

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1 Chapter 2 Lambert Summability and the Prime Number Theorem 2. Introduction The prime number theorem (PNT) was stated as conjecture by German mathematician Carl Friedrich Gauss ( ) in the year 792 and proved independently for the first time by Jacques Hadamard and Charles Jean de la Vallée-Poussin in the same year 896. The first elementary proof of this theorem (witho using integral calculus) was given by Atle Selberg of Syracuse University in October 948. Another elementary proof of this theorem was given by Erdös in 949. The PNT describes the asymptotic distribion of the prime numbers. The PNT gives a general description of how the primes are distribed among the positive integers. Informally speaing, the PNT states that if a random integer is selected in the range of zero to some large integer N, the probability that the selected integer is prime is abo = ln.n /, where ln.n / is the natural logarithm of N. For eample, among the positive integers up to and including N D 3, abo one in seven numbers is prime, whereas up to and including N D, abo one in 23 numbers is prime (where ln.3/ D 6: and ln./ D 23:25859). In other words, the average gap between consecive prime numbers among the first N integers is roughly ln.n /: Here we give the proof of this theorem by the application of Lambert summability and Wiener s Tauberian theorem. The Lambert summability is due to German mathematician Johann Heinrich Lambert ( ) (see Hardy [4, p. 372]; Peyerimhoff [8, p. 82]; Saifi [86]). 2.2 Definitions and Notations (i) Möbius Function. The classical Möbius function.n/ is an important multiplicative function in number theory and combinatorics.this formula is due to German mathematician August Ferdinand Möbius (79 868) who M. Mursaleen, Applied Summability Methods, SpringerBriefs in Mathematics, DOI.7/ , M. Mursaleen 24 3

2 4 2 Lambert Summability and the Prime Number Theorem introduced it in 832..n/ is defined for all positive integers n and has its values in f ; ; g depending on the factorization of n into prime factors. It is defined as follows (see Peyerimhoff [8, p. 85]): 8 < ; n is a square-free positive integer with an even number of prime factors;.n/ D ;nis a square-free positive integer with an odd number of prime factors; : ; n is not square-free; that is, 8 < ; n D ;.n/ D. / : ;ndp p 2 p ;p i prime, p i p j ; ; otherwise: (2.2.) Thus (a).2/ D, since 2 D 2; (b)./ D, since D 2 5; (c).4/ D, since 4 D 2 2. We conclude that.p/ D, ifp is a prime number. (ii) The Function./. The prime-counting function./ is defined as the P number of primes not greater than, for any real number, that is,./ D p< (Peyerimhoff [8, p. 87]). For eample,./ D 4 because there are four prime numbers (2, 3, 5, and 7) less than or equal to. Similarly,./ D ;.2/ D ;.3/ D ;.4/ D 2;./ D 68;. 6 / D 78498; and. 9 / D (Hardy [43, p. 9]). (iii) The von Mangoldt Function ƒ n. The function ƒ n is defined as follows (Peryerimhoff [8, p. 84]): log p;ndp for some prime p and ; ƒ n D ; otherwise: (iv) Lambert Summability. Aseries P nd a n is said to be Lambert summable (or summable L) tos,if lim!. / a D s: (2.2.2) D In this case, we write P a n D s.l/: Note that if a series is convergent to s, then it is Lambert summable to s: This series is convergent for jj <, which is true if and only if a n D O.. C "/ n /, for every ">(see [6, 52, 99]).

3 2.3 Lemmas 5 If we write D e y.y > /; s.t/ D P t a.a D /; g.t/ D te t e t ; then P a is summable L to s if and only if (note that y ) Z lim y! y The method L is regular. te y t Z ds.t/ D lim e y t y! D lim y! y Z t s.t/dg y g t y s.t/dt D s: 2.3 Lemmas We need the following lemmas for the proof of the PNT which is stated and proved in the net section. In some cases, Tauberian condition(s) will be used to prove the required claim. The general character of a Tauberian theorem is as follows. The ordinary questions on summability consider two related sequences (or other functions) and as whether it will be true that one sequence possesses a limit whenever the other possesses a limit, the limits being the same; a Tauberian theorem appears, on the other hand, only if this is untrue, and then asserts that the one sequence possesses a limit provided the other sequence both possesses a limit and satisfies some additional condition restricting its rate of increase. The interest of a Tauberian theorem lies particularly in the character of this additional condition, which taes different forms in different cases. Lemma 2.3. (Hardy [4, p. 296]; Peyerimhoff [8, p. 8]). If g.t/; h.t/ 2 L.; /, and if Z then s.t/ D O./ (s.t/ real and measurable) and lim! Z g.t/t i dt. <</; (2.3.) t g s.t/dt D implies lim! Z t h s.t/dt D : Lemma (Peyerimhoff [8, p. 84]). If n D p p. i D ;2;:::; p i prime/; then P d=n ƒ d D log n: Proof. Since d runs through divisors of n and we have to consider only d D p ;p 2;:::;p ;:::;p ; therefore P d=n ƒ d D log p C 2 log p 2 C C log p D log n: This completes the proof of Lemma

4 6 2 Lambert Summability and the Prime Number Theorem Lemma (Peyerimhoff [8, p. 84]).. / n nd where is a Riemann s Zeta function. Proof. We have n s D. 2 s /.s/.s > /; (2.3.2). / n n s nd D 2 s C 3 s 4 s C D C 2 C 3 2 s s 2 C s 4 C s 6 C s D.s/ 2 2 s C 2 s C 3 s C D.s/ 2 s.s/ D. 2 s /.s/: This completes the proof of Lemma Lemma (Hardy [4, p. 246]). If s>, then.s/ D Y p p s p s : (2.3.3) Lemma (Hardy [43, p. 253]). Proof. From 2.3.3, wehave.s/ D.s/ nd ƒ n n s (2.3.4) log.s/ D p log p s p s : Differentiating with respect to s and observing that d p s log D log p ds p s p s ;

5 2.3 Lemmas 7 we obtain.s/.s/ D p log p p s : (2.3.5) The differentiation is legitimate because the derived series is uniformly convergent for s C ı>;ı>: We can write (2.3.5) in the form.s/.s/ D p log p md p ms and the double series PP p ms log p is absolely convergent when s>. Hence it may be written as p ms log p D ƒ n n s : p;m This completes the proof of Lemma Lemma (Peyerimhoff [8, p. 84]). s n! s.l/, asn!and a n D O n imply s n! s, asn!. Proof. We wish to show that a n D O.=n/ is a Tauberian condition. In order to apply Wiener s theory we must show that (2.3.) holds. B for "> i.e., Z Z t ic" g.t/dt D.i C "/ D.i C "/ Z D nd t ic" g.t/dt Z D.i C "/. C " C i/ t ic" e.c/t dt D. C / C"Ci t i g.t/dt D. C i/ lim "!.i C "/. C " C i/: This has a simple pole at and is on the line Rez D : A stronger theorem is true, namely, L Abel, i.e., every Lambert summable series is also Abel summable (see [42]), which implies this theorem. For the sae of completeness we give a proof that. C i/ for real. The formula (2.3.4) implies. C i/ : This completes the proof of Lemma

6 8 2 Lambert Summability and the Prime Number Theorem Lemma (Peyerimhoff [8, p. 86]). nd.n/ n D Proof. This follows from O-Tauberian theorem for Lambert summability, if D O.L/.B P nd.n/ n.n/ n. / D. /.n/ n nd D. / nd D n.c/.n/ D. /: md n=m A consequence is (by partial summation)./ D O.n/ (2.3.6) n which follows with the notation m.t/ D./ from./ D n t Z n tdm.t/ D nm.n/ Z n m.t/dt: This completes the proof of Lemma Lemma (Hardy [43, p. 346]). Suppose that c ;c 2 ;:::; is a sequence of numbers such that C.t/ D nt c n and that f.t/is any function of t. Then c n f.n/ D C.n/ff.n/ f.nc /gcc./f.œ /: (2.3.7) n n If, in addition, c j D for j<n and f.t/has a continuous derivative for t n, then Z c n f.n/ D C./f./ C.t/f.t/dt: (2.3.8) n n

7 2.3 Lemmas 9 Proof. If we write N D Œ, the sum on the left of (2.3.7)is C./f./ CfC.2/ C./gf.2/CCfC.N/ C.N /gf.n/ D C./ff./ f.2/gccc.n /ff.n / f.n/gcc.n/f.n/: Since C.N/ D C./, this proves (2.3.7). To deduce (2.3.8), we observe that C.t/ D C.n/ when n t<nc and so Z nc C.n/Œf.n/ f.nc / D n C.t/ f.t/dt: Also C.t/ D when t<n : This completes the proof of Lemma Lemma (Hardy [43, p. 347]). n where C is Euler s constant. n D log C C C O ; Proof. P c n D and f.t/d =t. WehaveC./ D Œ and (2.3.8) becomes where is independent of and n E D n D Œ Z C Œt t dt 2 D log C C C E; Z C D D Z Z D O t Œt dt t 2 t Œt Œ dt t 2 O./ dt C O t 2 This completes the proof of Lemma

8 2 2 Lambert Summability and the Prime Number Theorem Lemma 2.3. (Peyerimhoff [8, p. 86]). If./ D h C log CC i and./ D n ƒ n ; then./ D O.log. C //: Proof. Möbius formula (2.2.) yields that./ C log C C D d.d/: (2.3.9) d From log n D P d=n ƒ d Lemma 2.3.2, it follows that log n D ƒ d D ƒ d D n n ddn d= : Therefore, we obtain./ D n log n log C C C O C Œ log log C Œ C; i.e.,./ D O.log. C //: (2.3.) This completes the proof of Lemma Lemma 2.3. ([Aer s Theorem] (Peyerimhoff [8, p. 87])). If (a)./ is of bounded variation in every finite interval Œ; T ; (b) P a D O./; (c) a n D O./; (d)./ D O. / for some < <; then a D O./: Proof. Let <ı<. Then a D O. /ı D O ı : ı

9 2.3 Lemmas 2 Assuming that m <ı m, N <NC (m and N integers), we have ı a D N Dm D O./ D O./ Z ı Z =ı C ˇ ˇd ˇˇˇ C O./ t jd.t/jco./: s C s N s m N m This completes the proof of Lemma Lemma (Peyerimhoff [8, p. 87])../ D O./. Proof. It follows from (2.3.6), (2.3.9), (2.3.), and Aer s theorem, that./ D O./: This completes the proof of Lemma Lemma (Peyerimhoff [8, p. 87]). Let #./ D P p log p.pprime/; then (a) #././ D O./; (b)./ D #./ C #. p / CC#. p /, for every > log log 2. Lemma (Peyerimhoff [8, p. 87])../ D #./ C O./ log p : log 2 Proof. It follows from part (b) of Lemma that./ D #./ C O./ log p : log 2 This completes the proof of Lemma Lemma (Peyerimhoff [8, p. 87]). #./ D C O./: (2.3.) Proof. Lemma implies that #./ D C O./:

10 22 2 Lambert Summability and the Prime Number Theorem 2.4 The Prime Number Theorem Theorem The PNT states that./ is asymptotic to =log (see Hardy [4, p. 9]), that is, the limit of the quotient of the two functions./ and =ln approaches, as becomes indefinitely large, which is the same thing as Œ./ log =!, as!(peyerimhoff [8, p. 88]). Proof. By definition and by./ D Z 3=2 D #./ log C log t d#.t/ Z D #./ log C O [note that #./ D O./ : Using (2.3.) we obtain the PNT, i.e., or #.t/ t.log t/ dt 2.log / 2 3=2 lim./ D! log ;./log lim D :! This completes the proof of Theorem 2.4..

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