a survey of results on primes in short intervals - … · a survey of results on primes in short...

A Survey of Results onPrimes in Short Intervals

Cem Yalcın Yıldırım

§1. Introduction

Prime numbers have been a source of fascination for mathemati-cians since antiquity. The proof that there are infinitely many primenumbers is attributed to Euclid (fourth century B.C.). The basicmethod of determining all primes less than a given number N isthe sieve of Eratosthenes (third century B.C.). Diophantus (third(?) century A.D.) was occupied with finding rational number solu-tions to equations, extending ancient knowledge from Babylon andIndia on Pythagorean triples. The books of Diophantus lay lostfor ages. It took thousands of years before new aspects of primeswere brought into light until chiefly Fermat and Mersenne (c.1640),influenced by Bachet’s (1621) translation into Latin of the extantbooks of Diophantus, announced various criteria on divisibility byprimes, assertions on primes possessing special forms, and solutionsto Diophantine equations.

A major breakthrough was Euler’s discovery (1737) of the identity

∏p

(1− 1

ps)−1 =

∞∑

n=1

1

ns. (1.1)

1

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Here s = σ+it ∈ C, σ, t ∈ R, and (1.1) is meaningful for σ > 1 whereboth sides are absolutely convergent. This identity of a sum over thenatural numbers and a product over the primes is an analytic way ofexpressing the property of unique factorization of natural numbersinto primes. Euler considered (1.1) and similar identities with nat-ural number values of s. It was Riemann (1859) who initiated thestudy of the quantity in (1.1) as an analytic function of a complexvariable. The two expressions in (1.1) represent the Riemann zeta-function ζ(s) in the half-plane σ > 1. Riemann’s aim was to provethe conjecture of Legendre and Gauss on the number π(x) of primesp ≤ x, that

π(x) ∼ x

log x, (x →∞). (1.2)

This goal was attained in 1896 independently by Hadamard and dela Vallee Poussin who were by then equipped with some essentialknowledge on entire functions.

Riemann showed that ζ(s) can be continued analytically over thewhole complex plane, being meromorphic with a simple pole at s = 1,and satisfies the functional equation

π−s2 Γ(

s

2)ζ(s) = π−

1−s2 Γ(

1− s

2)ζ(1− s). (1.3)

The value of ζ(s) can be calculated at any s with σ > 1 to anydesired accuracy from the expressions in (1.1). Then, using (1.3),ζ(s) can also be calculated for any s with σ < 0. In the rathermysterious strip 0 ≤ σ ≤ 1, one may use

ζ(s) =s

s− 1− s

∫ ∞

1(x)x−s−1 dx (σ > 0), (1.4)

where (x) is the fractional part of x. This is the analytic continuationof ζ(s) to σ > 0, obtained by applying partial summation to theseries in (1.1).

At s = −2,−4,−6 . . ., where Γ(s

2) has poles, ζ(s) vanishes - these

are called the trivial zeros. Upon developing general results on entirefunctions, Hadamard (1893) deduced that ζ(s) has infinitely manynontrivial zeros in 0 ≤ σ ≤ 1. The nontrivial zeros must be situ-ated symmetrically with respect to the real axis, and by (1.3) also

Primes in Short Intervals 3

with respect to the line σ = 12. Applying the argument principle,

von Mangoldt (1895) gave the proof of Riemann’s assertion that thenumber of nontrivial zeros ρ = β + iγ with 0 < γ ≤ T is asymptoti-

callyT

2πlog T , as T →∞. It follows that if the zeros ρ are arranged

in a sequence ρn = βn + iγn with γn+1 ≥ γn, then

γn ∼ 2πn

log n(n →∞). (1.5)

Riemann’s assertion that all of the nontrivial zeros lie on the criti-cal line σ = 1

2is yet unproved. Known as the Riemann Hypothesis

(RH), this has been one of the most profound problems of twentiethcentury mathematics. The Riemann Hypothesis settles the horizon-tal positioning of the zeros of ζ(s). In 1972 Montgomery came upwith the pair correlation conjecture (MC), as to how the nontrivialzeros, assumed to be on the line σ = 1

2, are distributed on this line.

In what follows we narrate the relation between ζ(s) and countingthe number of primes (§2), some unproved strong assertions on thedistribution of primes (§3), primes in arithmetic progressions (§4),the pair correlation conjecture (§5), some details of the connectionsbetween the distribution of primes and the zeta zeros (§6), and wegive the proof of a theorem of Goldston and Yıldırım on primesin arithmetic progressions in short intervals (§7). Finally there are‘Further Notes’ for each section.

§2. The explicit formula

The distribution of primes is closely linked with (the distribu-tion of the nontrivial zeros of) the Riemann zeta-function. Suchconnections are already hinted at by (1.1). Taking the logarithmicderivative of the product in (1.1) gives

ζ ′

ζ(s) = −

∞∑

n=2

Λ(n)

ns(σ > 1), (2.1)

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where Λ(n) is von Mangoldt’s function

Λ(n) =

log p, if n is the power of a prime p,0, otherwise.

(2.2)

Definingψ(x) =

∑

n≤x

Λ(n) =∑

pm≤x

log p , (2.3)

and ψ0(x) = ψ(x)− Λ(x)

2, one has

ψ0(x) =1

2πi

∫ c+i∞

c−i∞[−ζ ′

ζ(s)]

xs

sds (c > 1). (2.4)

Considering the integral from c−iT to c+iT , and moving the line ofintegration all the way to the left in the complex plane one obtains,by the residue theorem, for any x ≥ 2,

ψ0(x) = x− ∑

|γ|<T

xρ

ρ− ζ ′

ζ(0)− 1

2log(1− x−2)

+ O(x log2(xT )

T) + O(log x min(1,

x

T 〈x〉))(2.5)

(here 〈x〉 denotes the distance from x to the nearest prime power -other than x itself if x is a prime power). Equation (2.5) is calledthe explicit formula; it provides an explicit link between a (weighted)count of the primes and a sum over the nontrivial zeros of ζ(s). (Thisform of (2.5) is more useful in applications than the form obtained bytaking the limit T →∞ in (2.5)). The estimate for the sum over ρ in(2.5) depends upon our knowledge about the location of these zeros(this will be dwelt upon in §6). ¿From de la Vallee Poussin’s result

(1899) that ζ(s) 6= 0 for σ > 1− c1

log t, (which could be derived from

a relation between ζ(σ + it) and ζ(σ + 2it) resting on the inequality3 + 4 cos θ + cos 2θ ≥ 0) it follows that

ψ(x) = x + O(x exp[−c2(log x)12 ]) (2.6)

(here ci are appropriate positive constants). This embodies the primenumber theorem in the form (1.2). If one assumes RH, then all theρ’s have real part 1

2, implying

ψ(x) = x + O(x12 log2 x), (2.7)


with a much smaller error term than (2.6). The sharpest possibleestimate for ψ(x) was conjectured by Montgomery [38] by proba-bilistic arguments (upon assuming RH and that the imaginary partsγ > 0 of the nontrivial zeros are linearly independent) to be

limψ(x)− x√

x(log log log x)2= ± 1

2π(x →∞). (2.8)

In the opposite direction we note that Littlewood (1914) proved

ψ(x)− x = Ω±(x12 log log log x) (x →∞) (2.9)

(for the proof see Ingham’s tract [31]).

After the prime number theorem it is natural to ask for whichfunctions Φ(x), as x →∞,

π(x + Φ(x))− π(x) ∼ Φ(x)

log x? (2.10)

Here one would try to find Φ(x) as slowly increasing as possible.

Heath-Brown [28] proved that one can take Φ(x) = x712−ε(x) (ε(x) →

0, as x →∞), and assuming RH Φ(x) = x12+ε is allowed. Of course

Φ(x) cannot be too small, and we know due to Rankin [46] that

there exist intervals around x of length > clog x log2 x log4 x

(log3 x)2(logk is

the k-fold iterated logarithm) which don’t contain a prime. More-over Maier [35] showed that (2.10) is false even for Φ(x) as large as(log x)λ with any λ > 1, contrary to what was expected from theheuristic probabilistic arguments of Cramer [7]. On the other handSelberg [51] showed assuming RH that, (2.10) holds for almost all x

ifΦ(x)

(log x)2→∞ as x →∞. Here what is meant by ‘almost all x’ is

that, while X → ∞ the measure of the set of x ∈ [0, X] for which(2.10) doesn’t hold is o(X). Without assuming RH, this almost-all

result is known to hold with Φ(x) = x16+ε (Huxley [30]).

6 Yıldırım

§3. Some unproved conjectures on thedistribution of primes

In this section we briefly relate some of the deepest conjectures onthe distribution of primes. One of the oldest of all is the Goldbachconjecture (1742), that every even number > 2 is the sum of twoprime numbers. The furthest that has been proved in this directionis the remarkable theorem of Chen [3], that every sufficiently largeeven number can be expressed as the sum of a prime and a num-ber which has at most two prime factors - counted with multiplicity.It should also be noted that Vinogradov, using his method of esti-mating exponential sums, proved that every sufficiently large oddnumber can be expressed as a sum of three primes. The methodsdeveloped for attacking the Goldbach conjecture can also be usedfor other problems of an additive nature. But still we do not knowwhether or not there are infinitely many twin primes (e.g. p andp + 2 both prime). The more general situation was asserted as theprime r-tuple conjecture by Hardy and Littlewood [25]. The r-tupleconjecture is an asymptotic formula for the number πd(N) of posi-tive integers n ≤ N for which n + d1, . . . , n + dr are all prime (hered1, . . . , dr are distinct integers and d = (d1, . . . , dr)). The formula is

πd(N) ∼ ℘dN

logr N(N →∞) (3.1)

when ℘d 6= 0, where

℘d =∏p

pr−1(p− νd(p))

(p− 1)r,

and νd(p) is the number of distinct residue classes modulo p occu-pied by d1, . . . , dr. With r = 1, this reduces to the prime numbertheorem. For r ≥ 2 the conjecture remains unproved for any d. As-suming that for each r, (3.1) holds uniformly for 1 ≤ d1, . . . , dr ≤ h,Gallagher [14] showed that if Pk(h,N) is the number of integersn ≤ N for which the interval (n, n + h] contains exactly k primes,

then Pk(λ log N, N) ∼ N e−λλk

k!as N →∞, i.e. the distribution tends

to the Poisson distribution with parameter λ.


A heuristic way, depending on the prime number theorem and thecounting of appropriate residue classes to certain moduli, of derivingthe r-tuple conjecture (in the special case r = 2, d1 = 0, d2 = 2) canbe found in the book of Hardy and Wright [26, §22.20]). Hardy andLittlewood developed the circle method for attacking such additivearithmetical problems, which when written in the form of summa-tions can be re-expressed as integrals over the circle |z| = % < 1 witha power series of radius of convergence 1 in the integrand. The maincontribution comes from those z’s with arguments close to fractionswith small denominators while % → 1. The arithmetical informationis then extracted from the singularities of the power series on theunit circle.

For an upper bound on the difference between consecutive primesCramer [7] conjectured on probabilistic grounds that

lim supn

pn+1 − pn

(log pn)2= 1, (3.2)

where pn is the n-th prime. The known estimates for this limit, evenunder the unproved assumptions RH and MC, fall dismally short ofCramer’s guess:

pn+1 − pn ¿ p0.535n (unconditional) [1]

pn+1 − pn ¿ p12n log pn (on RH) [6]

pn+1 − pn = o((pn log pn)12 ) (on RH + MC) [19]

(3.3)

(in (6.1) and (6.15) below, further conditional estimates for the dif-ference between consecutive primes are given).

§4. Primes in arithmetic progressions

Dirichlet (1837) proved that if a and q are two coprime naturalnumbers, then there are infinitely many primes of the form kq + a.Davenport begins his book [8] with the remark that this work ofDirichlet may be regarded as the origin of analytic number theory.

8 Yıldırım

The proof involved the so-called Dirichlet’s L-functions, defined by

L(s, χ) =∞∑

n=1

χ(n)

ns, (4.1)

in σ > 1 where the series is absolutely convergent. Here χ is a Dirich-let’s character to the modulus q, a function of an integer variable nwhich is multiplicative and periodic with period q. It follows that if(n, q) = 1, then χ(n) is a root of unity. For (n, q) > 1, it is conve-nient to define χ(n) = 0. The character χ0 which assumes the value1 at all n coprime to q is called the principal character. It could bethat for values of n coprime to q, the least period of χ(n) is a properdivisor of q, in which case χ is called an imprimitive character, andotherwise primitive. There are φ(q) characters in all to the modu-lus q, which form an abelian group (defining χ1χ2(n) = χ1(n)χ2(n))isomorphic to the group of relatively prime residue classes to themodulus q. The characters satisfy

∑

n ( mod q)

χ(n) =

φ(q), if χ = χ0,0, otherwise,

or equivalently

∑

χ ( mod q)

χ(n) =

φ(q), if n ≡ 1 (mod q),0, otherwise.

Thus by using Dirichlet’s characters we can select from integers ina given set those that are in a particular residue class modulo q asin (4.3) below. It also follows that for nonprincipal χ the series in(4.1) is conditionally convergent in the strip 0 < σ ≤ 1. Dirichlet’sproof hinges on the fact that L(1, χ) 6= 0 for nonprincipal χ. Thetheory of Dirichlet’s L-functions parallels that of ζ(s) for the mostpart, and the Generalized Riemann Hypothesis (GRH) states thatall zeros of Dirichlet’s L-functions lie on the line σ = 1

2.

The main question is for which ranges of the relevant variablesare the primes evenly distributed with respect to the permissiblecongruence classes modulo q. To what extent and in which sense thisdistribution is even has been an active area of research. Analogous


to (2.3), for (a, q) = 1 let

ψ(x; q, a) =∑

n≤xn≡a ( mod q)

Λ(n). (4.2)

Writing ψ(x, χ) =∑

n≤x

χ(n)Λ(n), we have

ψ(x; q, a) =1

φ(q)

∑

χ ( mod q)

χ(a)ψ(x, χ). (4.3)

Proceeding as in the proof of the prime number theorem one aimsfor a result of the type

ψ(x; q, a) =x

φ(q)(1 + o(1)). (4.4)

Unconditionally the Siegel-Walfisz theorem says that (4.4) holds uni-formly for q < (log x)N with any fixed N > 0, while assuming GRHyields

ψ(x; q, a) =x

φ(q)+ O(x

12 log2 x) (q ≤ x). (4.5)

Just as for (2.6) and (2.7), these results depend on the knowledge ofthe zero-free region to the left of σ = 1 for Dirichlet’s L-functions.The Siegel-Walfisz restriction on the range of q is quite severe. Onthe other hand (4.5) implies (4.4) for q almost up to x

12 . It is natural

to wonder whether the error term in (4.5) need really be so large.With regard to this we make the following observation. Littlewood’sresult (2.9) was preceeded by the weaker ψ(x)− x = Ω±(x

12 ) due to

E. Schmidt (1903). A proof of this is in Ingham’s tract [31, Thm.33], with the constants implied in Ω± being ± 1

| 12+iγ1| (1

2+ iγ1 is the

zero of ζ(s) with the least positive γ1; γ1 ≈ 14.13 [52, §15.2]). If weadapt this proof for ψ(x, χ), using (4.3) we have

ψ(x; q, a)− x

φ(q)= Ω±(

x12

φ(q)), (4.6)

assuming GRH and some extra hypotheses. In these Ω-results as-suming RH or GRH is not a burden, because zeros off the critical

10 Yıldırım

line, if they ever exist, would cause greater oscillations of the er-ror term in the prime number theorem. The extra hypotheses areL(1

2, χ) 6= 0 for all χ ( mod q), and the critical zero of

∏

χ ( mod q)

L(s, χ)

with the least positive ordinate is a zero of just one of the functionsL(s, χ). If this zero 1

2+ iγ1,q has multiplicity m1 then (4.6) holds

with the constants ± m1

| 12+iγ1,q | (the latter condition may be somewhat

relaxed or modified, and the constants would accordingly be modi-fied). The best that can be hoped for was conjectured by Friedlanderand Granville [12] as

ψ(x; q, a) ¿εx

φ(q)(q <

x

(log x)2+ε), (4.7)

ψ(x; q, a) =x

φ(q)+ O((

x

q)

12 xε) (q ≤ x). (4.8)

When various averages over q and a are taken results that holdin greater ranges of the parameters can be obtained. The Bombieri-Vinogradov theorem says that given any constant A > 0, we have

∑

q≤Q

maxy≤x

maxa

(a,q)=1

|ψ(y; q, a)− y

φ(q)| ¿ x

(log x)A(4.9)

with Q =x

12

(log x)Bwhere B = B(A), thereby saving an arbitrary

power of log x from the trivial estimate (see e.g. [36, Chapter 15]).The meaning of the Bombieri-Vinogradov theorem is that the as-ymptotic formula for ψ(x; q, a) usually holds for q roughly as large

as x12 , the same extent that can be handled by GRH for an individ-

ual ψ(y; q, a), compared with q restricted to powers of log x in theSiegel-Walfisz theorem. The Elliott-Halberstam conjecture is that(4.7) should hold beyond x

12 up to Q = x1−ε. It has been proved by

Friedlander et al. [13] that it cannot hold up tox

(log x)C. The im-

portant ingredients in the proof of the Bombieri-Vinogradov theoreminclude a so-called large-sieve inequality

∑

q≤Q

q

φ(q)

∑*

χ

|M+N∑

M+1

anχ(n)|2 ¿ (N + Q2)M+N∑

M+1

|an|2 (4.10)


(here∑

*χ

denotes a sum over all primitive characters χ(mod q)),

and the Polya-Vinogradov inequality for a nonprincipal character tothe modulus q,

M+N∑

M+1

χ(n) ¿ q12 log q. (4.11)

The Barban-Davenport-Halberstam theorem, the proof of whichalso depends upon the large-sieve inequality (4.10), reads in its as-ymptotic form (proved by Montgomery)

∑

q≤Q

q∑a=1

(a,q)=1

|ψ(x; q, a)− x

φ(q)|2 ∼ Qx log x (

x

(log x)A≤ Q ≤ x),

(4.12)where A > 0 is any fixed number ([36, Thm. 17.2]). Upon GRH

this holds for x12 log2 x ≤ Q ≤ x ([11], [21]). Here the range of q is

much longer but a mean-square over the residue classes is consideredinstead of the maximum in Bombieri-Vinogradov theorem.

§5. Pair correlation and simple zeros

In 1972 Montgomery [37], manipulating the explicit formula, wasled to define the function F (α, T ) as

F (α, T ) = (T

2πlog T )−1

∑

0<γ,γ′≤T

T iα(γ−γ′)w(γ − γ′) (5.1)

(12

+ iγ and 12

+ iγ′ run through the zeros of ζ(s)), where w(u) is asuitable weight function; in [37] it was w(u) = 4

4+u2 but other weightfunctions can also be used (cf. (5.8) below and Hejhal’s w(u) =e−au2

in [29]). By using the large sieve result (a quantitative form ofParseval’s identity for Dirichlet series)

∫ T

0|∑

n

ann−it|2 dt =∑n

|an|2(T + O(n)) , (5.2)

Montgomery showed that upon RH

F (α, T ) = (1 + o(1))T−2α log T + α + o(1) (0 ≤ α ≤ 1) (5.3)

12 Yıldırım

as T →∞. For larger α in bounded intervals Montgomery, drawingupon the prime r-tuple conjecture with r = 2, conjectured that

F (α, T ) = 1 + o(1) (1 ≤ α ≤ A). (5.4)

(In (5.3) and (5.4) the estimates are uniform in the respective do-mains of α).

Convolving F (α, T ) in (5.1) with a kernel r(α) gives

∑

0<γ,γ′≤T

r((γ − γ′)log T

2π)w(γ − γ′) = (

T

2πlog T )

∫ ∞

−∞F (α, T )r(α) dα ,

(5.5)where r and r are Fourier transforms of each other,

r(α) =∫ ∞

−∞r(u)e−2πiαu du .

Since on RH, F (α, T ) can be calculated for |α| ≤ 1 as in (5.3), onecan use (5.5) with r(α) supported in [−1, 1] to see the implications of

RH. By taking r(u) = (sin παu

παu)2 Montgomery derived that at least

23

of the zeros of ζ(s) are simple. The pair correlation conjecture(5.4) implies that almost all zeros are simple. In this connection wemention that Mertens hypothesis in its weaker form

∫ X

1(M(x)

x)2 dx = O(log X), (5.6)

where

M(x) =∑

n≤x

µ(n) (5.7)

(µ(n) is the Mobius function) implies that all zeros of ζ(s) whichare on the critical line are simple (see [52, §14.29]). Recall that

the Riemann Hypothesis is equivalent to |M(x)| = O(x12+ε) ([52,

§14.25]). The Mertens conjecture, in the form |M(x)| < x12 , was

disproved by Odlyzko and te Riele [42].

Goldston [17] showed assuming RH, that the following asymptoticestimates as T →∞ are equivalent:


(i)∫ a+δ

aF (α, T ) dα ∼ δ (fixed α ≥ 1, δ > 0)

(ii)∑

0<γ,γ′≤T

0<γ−γ′≤ 2πβlog T

1 ∼ (T

2πlog T )

∫ β

01− (

sin πu

πu)2 du (fixed β > 0)

(iii)∫ T κ

1(ψ(u +

u

T)− ψ(u)− u

T)2u−2 du ∼ (κ− 1

2)log2 T

T,

for fixed κ ≥ 1 (if 0 < κ ≤ 1, then this integral is ∼ κ2

2

log2 T

Tassuming RH).

Some of these results have been extended to Dirichlet’s L-functionsand primes in arithmetic progressions by Ozluk [43] and Yıldırım[55]. By estimating a q-analogue of F (α, T ),

FQ(α) =∑

q≤Q

1

φ(q)

∑

χ ( mod q)

∑

γ,γ′Qiα(γ−γ′)(

sin γ

γ)2(

sin γ′

γ′)2, (5.8)

(in the innermost summation, assuming GRH, 12

+ iγ and 12

+ iγ′

run through the zeros of L(s, χ)) in 0 ≤ α ≤ 2 − ε, Ozluk showedthat at least 11

12of the zeros of all Dirichlet’s L-functions are simple.

It was because an ensemble of L-functions were considered togetherthat the barrier α ≤ 1 could be overcome.

It was shown by Goldston and Montgomery [20] that upon RH,assuming

F (T, x) ∼ 1

2πT log T

for xB1(log x)−3 ≤ T ≤ xB2(log x)3, (0 < B1 ≤ B2 ≤ 1) implies

∫ x

1ψ(y(1 + δ))− ψ(y)− yδ2 dy ∼ 1

2δx2 log

1

δ, (5.9)

uniformly for x−B2 ≤ δ ≤ x−B1 . (There is also a converse implica-tion. Another such equivalence was mentioned above). Upon RH,for 0 < δ ≤ 1 this integral is ¿ δx2(log 2

δ)2 (see [50]; also in Eq.

(7.9) below we mention a lower bound of the correct order of mag-nitude). Yıldırım defined a function which correlates the zeros of

14 Yıldırım

all pairs of Dirichlet’s L-functions to the same modulus. Under aconjecture for this function, analogous to MC, the author got theasymptotic result corresponding to (5.9) for the second moment forprimes in an individual arithmetic progression.

The appearance in (ii) of 1− ( sin πuπu

)2 as the pair correlation func-tion of the zeros of ζ(s) has opened up new avenues of progress. Theeigenvalues of a random complex Hermitian matrix of large ordertaken from the Gaussian Unitary Ensemble (GUE) have the samepair correlation function. So one might expect that there exists alinear operator whose eigenvalues characterize the zeros of ζ(s). Re-cently higher order correlations of zeros of ζ(s) have been understudy. Hejhal [29] calculated the triple correlation function, similarto Montgomery’s work. Rudnick and Sarnak [48] [49] defined then-level correlation sums for the Riemann zeta-function and moregeneral L-functions. They showed that the n-level correlations arein accordance with the predictions by the GUE model. Farmer [9]has given some consequences of the ‘GUE Hypothesis’ that the dis-tribution of gaps between the zeros of ζ(s) is like the distribution ofgaps between the eigenvalues of large random Hermitian matrices.The numerical results of Odlyzko [40], [41] constitute great evidencefor the truth of RH and the GUE model. Also the heuristic andnon-rigourous methods of Bogomolny and Keating (in a series of ar-ticles, the last being [2]) assuming the Hardy-Littlewood conjecturewith r = 2, indicate that the n-level correlations of zeta zeros arein agreement with the results for GUE beyond the ranges that werepossible in the works of Hejhal, and Rudnick – Sarnak who assumedmerely RH. The equivalence of (iii) with the pair correlation conjec-ture (cf. also (5.9)) reflects that the second moment for primes isdetermined by the pair correlation of the zeros of ζ(s). The highermoments analogues of (5.9) (which can be expected to be calculablefrom the general r-tuple conjecture (3.1), see Gallagher [14]), andtheir connections with the distribution of zeta zeros seems not tohave been worked out yet.


§6. Sums over zeta zeros and theerror term in the prime number theorem

The pair correlation conjecture (5.4), assumed in varying degreesof strength depending on the problem, implies (see Heath-Brown[27])

lim infn→∞

pn+1 − pn

log pn

= 0, (6.1)

ψ(x) = x + o(x12 log2 x) (6.2)

(cf. Eq.s (2.6)-(2.9) and (3.3)). Such implications are natural, as(5.4) (or weaker forms of it) imply nontrivial estimates on sums like∑ρ

xρ

ρ, the quantity which appears in the explicit formula (2.5). To

describe this briefly we take the pair correlation function in the form

F (T, x) =∑

0<γ,γ′≤T

xi(γ−γ′) 4

4 + (γ − γ′)2, (6.3)

and let ∑(T, v, u) =

∑

0<γ≤T

e(γ(v + u)). (6.4)

Then ∫ ∞

−∞2πe−4π|v||∑(T, v, log x)|2 dv = F (T, x), (6.5)

and from here it follows that∑

0<γ≤T

xiγ ¿ T12max

t≤TF (t, x) 1

2 . (6.6)

So, roughly speaking, the assumption that the size of F (T, x) (for

appropriate ranges of T and x) is O(T log T ) will save a log12 T from

the trivial estimate ∑

0<γ≤T

xiγ ¿ T log T. (6.7)

Since ψ0(x) is discontinuous at the prime powers, so is∑ρ

xρ

ρ(=

limT→∞

∑

|γ|<T

xρ

ρ) by the explicit formula (2.5); the series is boundedly

16 Yıldırım

convergent in fixed intervals 1 < a ≤ x ≤ b. The sum∑

0<γ≤T

xρ is

also discontinuous at the prime powers. By considering∫ ζ ′(s)

ζ(s)xs ds

taken around a rectangular contour Landau [33] proved that

∑

0<γ≤T

xρ = − T

2πΛ(x) + O(log T ) (6.8)

for every fixed x > 1, as T → ∞. Gonek [23] proved a uniform (inboth x and T ) version of (6.8), that for x, T > 1,

∑

0<γ≤T

xρ = − T

2πΛ(x) + O(x log 2x log log 3x)

+ O(log x min(T,x

〈x〉)) + O(log 2T min(T,1

log x)).

(6.9)(It is possible to calculate the sum also for 0 < x < 1 from (6.9) byusing the symmetry of the zeros of ζ(s) with respect to the criticalline). In (6.9) if one assumes RH, then

∑

0<γ≤T

xiγ ¿ (Tx−12 + x

12 ) log x log log x. (6.10)

Comparison with (6.7) shows that (6.10) is nontrivial for 2 ≤ x ≤T 2−ε. If one assumes further that xiγ’s behave like independent ran-dom variables (cf. (2.8)), then one expects that for almost all x > 1

∑

0<γ≤T

xiγ ¿ T12+ε. (6.11)

By Dirichlet’s theorem on Diophantine approximation there existarbitrarily large x with

∑

0<γ≤T

xiγ À T log T, (6.12)

so (6.11) doesn’t hold for all x > 1. The observation (6.11) alongwith the heuristics of using n

log nin place of γn (see (1.5)) in

∑xiγ

led Gonek to conjecture that∑

0<γ≤T

xiγ ¿ (Tx−12+ε + T

12 xε) (x, T ≥ 2). (6.13)


This would imply that for 1 ≤ h ≤ x

ψ(x + h)− ψ(x) = h + O(h12 xε), (6.14)

which in turn impliespn+1 − pn ¿ pε (6.15)

(cf. (3.3) and (6.2)).

Averages of the error term in the prime number theorem have alsobeen of interest not just for their own sake but because quantitiesinvolving them crop up in many problems (e.g. Eq.s (7.15) and(7.24) below). For brevity call

R(x) = ψ(x)− x . (6.16)

The results on the order of magnitude of R(x) or various averages ofit are in correspondence with the estimates for sums over zeta zerosas clearly seen from the explicit formula. Upon RH one has (2.7),and one can only hope to have improvements in the logarithmic part(cf. (2.8), (2.9), (6.2)). By (2.9) it is known that, as x → ∞, R(x)changes sign infinitely many times. Cramer showed that on RH

∫ X

1

(R(u))2

udu = O(X), (6.17)

∫ X

1(R(u)

u)2 du ∼ C log X . (6.18)

Gallagher’s article [15] contains compact proofs of such results. ByCauchy-Schwarz inequality (6.17) implies

∫ X

1|R(u)| du = O(X

32 ).

Pintz [44] has shown that for all sufficiently large X

X32

400≤

∫ X

1|R(u)| du ≤ X

32 , (6.19)

where the lower bound is unconditional and the upper bound de-pends essentially on RH. Jurkat [32], by developing concepts on

almost-periodic functions proved upon RH that, with d(x) =1

log log x

1

d(x)

∫ x+x d(x)

x

R(u)

u12

du

u= Ω±(log log log x) (6.20)

18 Yıldırım

(with the implied constants ±12), and that this cannot be improved

upon much for he also showed that this quantity is O((log log log x)2).Eq. (6.20) implies (2.9), Littlewood’s result without averages. From(6.19) and (2.7) we see that as x → ∞, |R(x)| spends most of its

time roughly around the value x12 (instead of much smaller values),

and (6.20) reveals the existence of quite long intervals throughoutwhich |R(x)| is almost as large as possible.

§7. Some recent results on the second momentsfor primes

In this section, as an example of recent work in our topic, wepresent a theorem of Goldston and Yıldırım. This is Thm. 3 of [22],where the details of the proof have not been included. The resultsgiven by Eq.s (7.2), (7.6) and (7.9) below are also proved in [22]. Wedefine for

x ≥ 2, 1 ≤ q ≤ x, 1 ≤ h ≤ x,

I(x, h, q) =∑

*

a(q)

∫ 2x

x

(ψ(y + h; q, a)− ψ(y; q, a)− h

φ(q)

)2

dy,

(7.1)

where∑

*

a(q)

is the sum over a reduced set of residues modulo q.

If h ≤ q, the interval (y, y + h] contains at most one integer whichbelongs to the congruence class a (mod q), so the situation is rathertrivial and one has unconditionally

I(x, h, q) ∼ hx log x (h ≤ q). (7.2)

So henceforth we will take 1 ≤ q ≤ h ≤ x.

It was shown by Prachar [45] that, assuming GRH

I(x, h, q) ¿ hx log2 qx. (7.3)

It is possible to evaluate the asymptotic value of I(x, h, q), asx →∞, assuming RH and a strong form of the twin prime conjecture


(the case r = 2 of the Hardy-Littlewood conjecture mentioned in §3above). Let

N1 = N1(k) = max(0,−k), N2 = N2(x, k) = min(x, x− k),

E(x, k) =∑

N1(k)<n≤N2(x,k)

Λ(n)Λ(n + k)−S(k)(x− |k|), (7.4)

where

S(k) =

2C∏

p|kp>2

(p− 1

p− 2

), if k is even, k 6= 0;

0, if k is odd,

with

C =∏

p>2

(1− 1

(p− 1)2

).

Assuming the twin prime conjecture in the form that for 0 < |k| ≤ x,and some given ε ∈ (0, 1

2)

E(x, k) ¿ x12+ε, (7.5)

it follows for 1 ≤ hq≤ x

12−ε and h ≤ x that

I(x, h, q) ∼ hx logxq

h. (7.6)

Upon GRH only, it can be shown that (7.6) holds for almost all q

with h34 log5 x ≤ q ≤ h (see [22]). Moreover by Goldston’s method

([18]) of using the auxiliary arithmetic function

λR(n) =∑

r≤R

µ2(r)

φ(r)

∑

d|rd|n

dµ(d), (7.7)

which in relevant cases mimics the behaviour of Λ(n), starting fromthe inequality

∑

a(q)

∫ 2x

x

∣∣∣∣∑

y<n≤y+hn≡a(q)

(Λ(n)− λR(n))∣∣∣∣2

dy ≥ 0 (7.8)

20 Yıldırım

a lower-bound of the correct order of magnitude for I(x, h, q) canbe obtained. The last inequality enables one to replace the trouble-some sums involving Λ(n)Λ(n+k)’s by sums of λR(n)Λ(n+k)’s andλR(n)λR(n + k)’s, which can be evaluated with no need for a con-jecture like (7.5). The result is that for any ε > 0, and 0 ≤ α ≤ 1

3,

where we write hq

= xα,

I(x, h, q) ≥ (1

2− 3

2α− ε)hx log x. (7.9)

Let us remember that Lavrik [34] showed that for B > 0 andy ≥ y0(B),

y2∑

k=1

(E(y, k))2 ¿ y2(log y)−B. (7.10)

This was used by Montgomery [36] in proving the asymptotic versionof the Barban-Davenport-Halberstam theorem (4.10), in the light ofwhich we expect that assuming only GRH one can get an asymptoticresult for sums of I(x, h, q) over certain ranges of q. This is indeedthe case, and here we will prove the following theorem.

Theorem. Assume the Generalized Riemann Hypothesis. Thenwe have, for h

12 log6 x ≤ Q ≤ h ≤ x, as x →∞

∑

q≤Q

I(x, h, q) ∼ Qhx log(xQ

h).

In fact we shall obtain the more detailed formula (7.48). Notethat the range of validity of this theorem is even greater than thatof (7.6) when h is very close to x. In the proof we will use methodsand results of Friedlander and Goldston [11].


Proof. Expanding the integrand of (7.1) we have

I(x, h, q) =∑

*

a(q)

∫ 2x

x(ψ(y + h; q, a)− ψ(y; q, a))2 dy

− 2h

φ(q)

∑*

a(q)

∫ 2x

x(ψ(y + h; q, a)− ψ(y; q, a)) dy +

h2x

φ(q)

= S1 − 2h

φ(q)S2 +

h2x

φ(q), say.

(7.11)Here

S2 =∑

(n,q)=1

Λ(n)f(n, x, h), (7.12)

where

f(n, x, h) =∫

[x,2x]∩[n−h,n)1 dy =

n− x, if x ≤ n < x + h ,h, if x + h ≤ n ≤ 2x,2x− n + h, if 2x < n ≤ 2x + h,0, otherwise.

Since

∑

x≤n≤2x+h(n,q)>1

Λ(n) =∑

p|q

∑ν

x≤pν≤2x+h

log p ¿ ∑

p|qlog p ¿ log q, (7.13)

we can lift the condition (n, q) = 1, so that

S2 =∑

x<n≤2x+h

Λ(n)f(n, x, h) + O(h log q).

The sums involving f(n, x, h) will be evaluated by the following par-tial summation formula. Let C(x) =

∑

n≤x

cn. Then

∑

x<n≤2x+h

cnf(n, x, h) =∫ 2x+h

2xC(u) du−

∫ x+h

xC(u) du (7.14)

(cv = 0 if v is not an integer). Taking C(x) = ψ(x) in (7.14) andrecalling (6.16) we obtain

S2 = hx + ∫ 2x+h

2x−

∫ x+h

xR(u) du. (7.15)

22 Yıldırım

¿From (7.11) we have

S1 =∑

x<n≤2x+h(n,q)=1

Λ2(n)f(n, x, h)

+2∑

0<k≤hk≡0(q)

∑

x<n≤2n+h−k(n(n+k),q)=1

Λ(n)Λ(n + k)f(n, x, h− k).(7.16)

A calculation similar to (7.13) shows that we may drop the conditions(n, q) = 1 and (n(n + k), q) = 1 in the above sums with an error¿ h2

qlog2 x. Thus we have

S1 = S3 + 2S4 + O(h2

qlog2 x), (7.17)

whereS3 =

∑

x<n≤2x+h

Λ2(n)f(n, x, h), (7.18)

and

S4 =∑

0<j≤h/q

∑

x<n≤2x+h−jq

Λ(n)Λ(n + jq)f(n, x, h− jq). (7.19)

To evaluate S3 call

P (x) =∑

n≤x

Λ2(n)− x log x + x, (7.20)

and apply (7.14) with cn = Λ2(n) to get

S3 =(2x + h)2

2log(2x + h)− (2x)2

2log 2x

−(x + h)2

2log(x + h) +

x2

2log x− 3xh

2

+∫ 2x+h

2x−

∫ x+h

xP (u) du.

(7.21)

Taking C(u) =∑

0<n≤u

Λ(n)Λ(n + jq) in (7.14) we have

S4 =∫ 2x+h

2x

∑

0<j≤u−2xq

∑

n≤u−jq

Λ(n)Λ(n + jq) du

−∫ x+h

x

∑

0<j≤u−xq

∑

n≤u−jq

Λ(n)Λ(n + jq) du.(7.22)


Note that by (7.4) the difference of the two integrals in (7.22) is

x∑

0<j≤h/q

(h−jq)S(jq)+∫ 2x+h

2x

∑

0<j≤u−2xq

−∫ x+h

x

∑

0<j≤u−xq

E(u, jq) du

(7.23)On combining (7.11), (7.15), (7.17), (7.21), and effecting the cancel-lations that occur in (7.21), we obtain

I(x, h, q) = hx log x + x2 log((1+ h

2x)2

(1+hx)12) + hx(log(

4(1+ h2x

)2

(1+hx)

)− 32)

+h2

2log(

2x + h

x + h)− h2x

φ(q)+ 2S4 + O(

h2 log2 x

q)

+∫ 2x+h

2x−

∫ x+h

x(P (u)− 2h

φ(q)R(u)) du

(7.24)If we use (7.22), (7.23) and (7.5) in (7.24), we obtain (7.6). Also(7.2) follows from (7.24) on using the prime number theorem (2.6)

to estimate the integrals in (7.24) (whenh

q≤ 1, S4 is void, and

instead of the O-term of (7.24) there is O(h log2 x). Note also thatq

φ(q)¿ log log q [26, §18.4] to deal with the term

h2x

φ(q)in (7.24)).

Now note that by (7.3)

∑

q≤Q

I(x, h, q) =∑

Q0≤q≤Q

I(x, h, q) + O(Q0hx log2 x), (7.25)

so we must calculate∑

Q0<q≤Q

S4. Letting

Su(α) =∑

n≤u

Λ(n)e(nα), (7.26)

Wv,u(α) =∑

Q0<q≤v

∑

0<j≤uq

e(−jqα), (7.27)

we can express

∑

n≤u−jq

Λ(n)Λ(n + jq) =∫ 1

0|Su(α)|2e(−jqα) dα, (7.28)

24 Yıldırım

and upon taking v = min(Q, u),

∑

Q0<q≤Q

S4 =∫ h

Q0

∫ 1

0(|Su+2x(α)|2 − |Su+x(α)|2)Wv,u(α) dα du

+O(h2 log3 x).(7.29)

We shall need an upper bound on the size of Wv,u(α). Changing theorder of summation in (7.27) gives

Wv,u(α) =∑

0<j≤ uQ0

∑

Q0<q≤uj

e(−jqα)− ∑

0<j≤uv

∑

v<q≤uj

e(−jqα). (7.30)

We now employ the estimate of Vinogradov and Vaughan (see [8,Chapter 25])

∑

0<j≤ uY

∑

Y <q≤uj

e(−jqα) ¿ (u

r+

u

Y+ r) log(

2ru

Y), (7.31)

which rests upon the assumption

α =b

r+ β, |β| ≤ 1

r2, (b, r) = 1, (7.32)

to obtain for v ≥ Q0

Wv,u(α) ¿ (u

r+

u

Q0

+ r) log(2ru

Q0

). (7.33)

Letting

Ju(β) =µ(r)

φ(r)Iu(β), Iu(β) =

∑

n≤u

e(nβ), (7.34)

we will consider separately each term on the right-hand side of

∫ 1

0|S(α)|2W (α) dα =

∫ 1

0|J |2W (α) dα +

∫ 1

0|S − J |2W (α) dα

+2∫ 1

0Re(S − J)JW (α) dα

(7.35)where the subscripts (which are either u + 2x or u + x for S, andv, u for W ) have been suppressed. The range of integration is de-composed into Farey arcs of order R; MR(r, b) = ( b+b′

r+r′ ,b+b′′r+r′′ ], where


1 ≤ r ≤ R, (b, r) = 1, and b′r′ < b

r< b′′

r′′ are consecutive frac-tions in the Farey sequence. We call θR(r, b) the translated interval( −1

r(r+r′) ,1

r(r+r′′) ]. Note that ( −12rR

, 12rR

] ⊂ θR(r, b) ⊂ (−1rR

, 1rR

], and

∫ 1

0=

∑

r≤R

∑*

b(r)

∫

θR(r,b)

(we will abbreviate θR(r, b) as θ). To estimate∫ 1

0|S − J |2W (α) dα,

recall Lemma 7.1 of [11] which says upon GRH

∑*

b(r)

∫ δ

−δ|S − J |2 dβ ¿ δrx(log rx)4, (δ ≥ 1

x). (7.36)

Taking δ = 1rR

(so that we must have R ≤ x12 ), (7.33) and (7.36)

give∫ 1

0|S − J |2W (α) dα ¿ xu

Rlog6 x +

xu

Q0

log5 x + xR log5 x. (7.37)

Next we have∫ 1

0Re(S − J)JW (α) dα

¿ ∑

r≤R

(u

r+

u

Q0

+ r) log(2ru

Q0

)µ2(r)

φ(r)

∑*

b(r)

∫

θ|I||S − J |.

(7.38)

Using I ¿ min(x, ||β||−1), and the Cauchy-Schwarz inequality weget from (7.36)

∑*

b(r)

∫ 1x

− 1x

|I||S − J | ¿ ∑*

b(r)

x12 (

∫ 1x

− 1x

|S − J |2) 12 ¿ rx

12 log2 x.

On the rest of θR(r, b), calling uj = (2jrR)−1, we similarly see that

∑*

b(r)

∫ uj

uj2

|I||S − J | ¿ u−1j

∑*

b(r)

∫ uj

−uj

|S − J | ¿ rx12 log2 x.

There are O(log x) such u′js. So (7.38) gives

∫ 1

0Re(S − J)JW (α) dα ¿ ux

12 log5 x+

ux12 R

Q0

log4 x+R2x12 log4 x.

(7.39)

26 Yıldırım

Next∫ 1

0|Ju+x|2Wv,u(α) dα =

∑

r≤R

µ2(r)

φ2(r)

∑*

b(r)

e(−jqb

r)∫

θ

∑

n,m≤u+x

∑

Q0<q≤v

∑

0<j≤uq

e(n−m− jqβ) dβ.

Thinking of∫

θas

∫ 1

0−

∫

[0,1]\θ, we will first consider upon changing

the order of summations

∑

Q0<q≤v

∑

0<j≤uq

∑

r≤R

µ2(r)

φ2(r)

∑*

b(r)

e(−jqb

r)∫ 1

0

∑

n,m≤u+x

e((n−m− jq)β) dβ.

(7.40)With

cr(k) =∑

*

b(r)

e(kb

r) =

∑

d|(r,k)

dµ(r

d) =

µ( r(r,k)

)φ(r)

φ( r(r,k)

),

for Ramanujan’s sum (see [8, Chapter 20]), one has

∞∑

r=1

µ2(r)

φ2(r)cr(k) = S(k).

We define

SR(k) =∑

r≤R

µ2(r)

φ2(r)cr(k), SR = S−SR,

and (7.40) becomes

∑

Q0<q≤v

∑

0<j≤uq

∑

r≤R

µ2(r)

φ2(r)cr(jq)

∑

n,m≤u+xn−m=jq

1

=∑

Q0<q≤v

∑

0<j≤uq

SR(jq)∑

m≤u+x−jq

1

=∑

Q0<q≤v

∑

0<j≤uq

(S(jq)− SR(jq))(u + x− jq) + O(|SR(jq)|).

Recall from (7.29) that we need∫ 1

0(|Ju+2x|2−|Ju+x|2)Wv,u(α) dα, so

we should calculate

x∑

Q0<q≤v

∑

0<j≤uq

(S(jq)− SR(jq)) + O(u log2 x), (7.41)


where the last error term is deduced from SR(jq) ¿ log R. Fromthe SR term we have

x∑

Q0<q≤v

∑

0<j≤uq

∑

r>R

µ2(r)

φ2(r)cr(jq) ¿ x

∑q

∑

j

∑

r>R

µ2(r)

φ2(r)

∑

d|(r,jq)d

¿ x∑q

∑

j

∑

d|jq

dµ2(d)

φ2(d)

∑

r1> Rd

µ2(r1)

φ2(r1)¿ x

R

∑q

∑

j

∑

d|jq

d2µ2(d)

φ2(d)

¿ x

R

∑

d≤u

d2µ2(d)

φ2(d)

∑

m≤ud|m

τ(m) ¿ x

R

∑

d≤u

d2µ2(d)τ(d)

φ2(d)

∑

n≤ud

τ(n)

¿ xu

Rlog x

∑

d≤u

d2µ2(d)τ(d)

φ2(d)¿ xu

Rlog3 x.

(7.42)Now we estimate

∑

r≤R

∑*

b(r)

∫

[0,1]\θ|Ju+x|2W (α) dα

≤ ∑

r≤R

∑*

b(r)

µ2(r)

φ2(r)

∫

[0,1]\θ|I(β)|2|W (

b

r+ β)| dβ

(7.43)

by taking a finer Farey decomposition of order T with T > R. ThenθT (r, b) ⊂ θR(r, b), so that

∫

[0,1]\θR(r,b)|I(β)|2 |W (

b

r+ β)| dβ

=∫

[0,1]\MR(r,b)|I(α− b

r)|2|W (α)| dα

≤∫

[0,1]\MT (r,b)|I(α− b

r)|2|W (α)| dα

=∑

t≤T

∑*

c(t)ct6= b

r

∫

θT (t,c)|I(

c

t− b

r+ β)|2|W (

c

t+ β)| dβ

On θT (t, c), we have |β| ≤ 1tT≤ 1

2rtif T ≥ 2R, so that

|I(c

t− b

r+ β)| ¿ ||c

t− b

r+ β||−1 ¿ 2||cr − bt

rt||−1.

28 Yıldırım

Hence by (7.33)

∑

t≤T

∑*

c(t)ct6= b

r

∫

θT (t,c)|I(

c

t− b

r+ β)|2|W (

c

t+ β)| dβ

¿ ∑

t≤T

(u

t+

u

Q0

+ t) log x∑

*

c(t)ct6= b

r

1

tT

(rt)2

|cr − bt|2 ,

where from each pair of residue classes b (mod r) and c (mod t) theb and c which minimize |cr− bt| is chosen. For given r, t the numberof representations of m as m = cr − bt is (r, t) if (r, t)|m, and 0otherwise. So

∑*

b(r)

∑*

c(t)ct6= b

r

1

|cr − bt|2 ≤2ζ(2)

(r, t)¿ 1.

Using these in (7.43) we get

∑

r≤R

∑*

b(r)

∫

[0,1]\θ|Ju+x|2W (α) dα ¿ log x

T

∑

r≤R

r2µ2(r)

φ2(r)

∑

t≤T

u +uT

Q0

+ t2

¿ Ru log x +uR2

Q0

log x + R3 log x,

(7.44)if we take T = 2R. Combining Eq.s (7.35), (7.37), (7.39-44) in

(7.29), and recalling that R ≤ x12 we have

∑

Q0<q≤Q

S4 = x∫ h

Q0

∑

Q0<q≤v

∑

0<j≤uq

S(jq) du

+O(xh2

Rlog6 x) + O(xhR log5 x) + O(

xh2

Q0

log5 x).

(7.45)The integral in (7.45) is evaluated by pulling the summations outsidethe integral sign as

∑

Q0<q≤Q

∑

0<j≤hq

(h− jq)S(jq)


which in turn is, by Proposition 3 and Lemma 6.1 of [11],

x2

∑

Q0<q≤Q

h2

φ(q)− h log(

h

q)− h(γ + log 2π − 1 +

∑

p|q

log p

p− 1)

+O(min(Q32 h

12 x log

32 Q,Qhx))

(7.46)Hence by (7.24), (7.25), (7.45) and (7.46) we obtain

∑

q≤Q

I(x, h, q) = Q(2x + h)2

2log(2x + h)− (2x)2

2log 2x

−(x + h)2

2log(x + h) +

x2

2log x− 3xh

2

+Qxh logQ

h−Qxh(γ + log 2π +

∑p

log p

p(p− 1))

+O(min(Q32 h

12 x log

32 Q,Qhx))

+O(xh2

Rlog6 x) + O(xhR log5 x) + O(

xh2

Q0

log5 x)

+O(Qhx12 log3 x) + O(Q0hx log2 x).

(7.47)In writing (7.47), we have used the RH estimate (2.7) for R(u) and

P (u) of (7.24). Choosing Q0 = R =h

12

2, the O-terms in (7.47) can be

gathered in O(min(Q32 h

12 x log

32 Q, Qhx))+O(xh

32 log6 x), and (7.47)

may be recast as

∑

q≤Q

I(x, h, q) = Qhx log(xQ

h) + Qx2 log(

(1 + h2x

)2

(1 + hx)

12

)+

Qhx(log( 2π

(1+ h2x

)2

1+hx

)− 32− γ −∑

p

log p

p(p− 1))+

Qh2

2log(

2x + h

x + h) + O(min(Q

32 h

12 x log

32 Q,Qhx)) + O(xh

32 log6 x).

(7.48)This completes the proof of the theorem. The result of this theoremis expected to be related to the distribution and simplicity of zerosof Dirichlet’s L-functions as was recounted on p. 319.

30 Yıldırım

Further notes

Section 1: The book of Hardy and Wright [26] contains almostall of the classic results of number theory. For an exposition ofthe theory of the Riemann zeta-function, Dirichlet’s L-functions anddistribution of primes we refer the reader to the books of Davenport[8] and Ingham [31]. Titchmarsh’s book (revised by Heath-Brown)[52] is an extensive treatise on the Riemann zeta-function.

¿From the work of Conrey [4] at least 25

of the zeta zeros are knownto lie on σ = 1

2.

Section 2: We note that with more work involving estimates onexponential sums a greater region than de la Vallee Poussin’s hasbeen shown to be free of zeta zeros by Vinogradov, so (2.6) can bewritten with any number less than 3

5replacing 1

2(see [52, Chapter

6]).

There are many problems about the prime counting functionswhich have not been included in this survey, a few of which willbe mentioned briefly here. By (2.9) it is clear that ψ(x)− x changessign infinitely often as x →∞. This is less involved than the prob-lem that gave rise to it, the sign changes of π(x) − li(x), becauseψ(x) is more directly related to ζ(s) than is π(x). Riemann hadasserted that π(x) < li(x) for x > 2, which was proved to be falseby Littlewood’s result

π(x)− li(x) = Ω±(x

12

log xlog log log x) (x →∞).

Skewes succeeded in giving an upper bound (a huge number) asto where the first change of sign occurs. The frequency of thesign changes was considered by Polya, Ingham, Turan, Knapowski,Levinson, Pintz and others. There are also estimates concerningψ(x; q, a1)−ψ(x; q, a2) (or with π instead of ψ) originated by Landau,Ingham, Polya and continued by Turan, Knapowski, Stas, Wierte-lak. In these estimates generally q is taken to be either fixed or verysmall compared to x. For all these we refer the reader to Ingham’stract [31], Turan’s Collected Works Vol. 3 [53] and Pintz’s reviewarticles therein.


Section 3: The prime r-tuple conjecture was put forth by Hardyand Littlewood [25]. Gallagher’s paper [14] contains results relatedto this conjecture. The methods of attacking problems of an additivenature, and the conjectures on the distribution of primes mentionedin this section are treated in the books by Halberstam and Richert[24], Richert [47] and Vaughan [54].

Section 4: The latest studies related to the Barban-Davenport-Halberstam theorem were conducted by Friedlander and Goldston[11], and Goldston and Vaughan [21]. For in-depth comments onthe error terms in the prime number theorems we refer the reader toFriedlander’s survey article [10].

Section 5: With a better choice of r(u) Montgomery showed (uponRH) that at least 0.6725.. of the zeros of ζ(s) are simple. Conrey,Ghosh and Gonek [5], assuming RH and an upper bound for averagesof sixth moments of Dirichlet’s L-functions, improved this to 19

27.

The sixth moment estimate is implied by the Generalized LindelofHypothesis that for any ε > 0

L(s, χ) ¿ε (q(1 + |t|))ε (σ ≥ 1

2).

The Generalized Riemann Hypothesis implies the Generalized Lin-delof Hypothesis. The entirely different approach in [5] rests uponusing appropriate Dirichlet polynomials instead of some of the Dirich-let series involved, thus being able to handle certain higher-momentcalculations.

For the theory of correlation functions and relations of zeta zeros toeigenvalues of a Hermitian operator the reader may consult the bookby Mehta [39]. The links between the Gaussian unitary ensembleof random matrix theory and quantum chaology are recounted byBogomolny and Keating [2].

Section 6: For the convergence properties of∑ xρ

ρwe refer the

reader to Ingham [31, Chapters 4, 5]. It was remarked after Eq. (6.9)that one may consider x ∈ (0, 1) as well. In this case the explicit

32 Yıldırım

formula is′∑

n≤ 1x

Λ(n)

n= log

1

x− γ +

∑ρ

xρ

ρ− x +

1

2log

1 + x

1− x

(∑′

means that when 1x∈ Z, the term corresponding to n = 1

x

is 12

Λ(n)n

; γ is Euler’s constant). The series∑

γ>0

(xρ

ρ+

xρ

ρ) exhibits

a Gibbs phenomenon in the neighbourhood of the points p±m. Atx = 1 this series is absolutely convergent, but no explicit formula isvalid. This series cannot be boundedly convergent on either side ofx = 1 because of the logarithmic terms in (2.5) and the last formula.It is this infinite jump at x = 1 that Jurkat [32] exploited remarkably.

Section 7: It was Selberg [51] who first obtained an upper bound(assuming RH) for the second moment on primes. The integral Sel-berg considered was similar to (5.9), but its integrand was dampedby an extra factor of y−2. The result expressed in (7.48) and other re-lated results will be in a forthcoming paper by Goldston and Yıldırım[22].

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36 Yıldırım

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Cem Yalcın YıldırımMathematics DepartmentBilkent UniversityAnkara 06533Turkeye-mail: [email protected]

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