Prime Numbers

(4)For odd n, investigate the possibility of “close calls” to perfection. For example, show (by machine perhaps) that every odd n with 10 < n < 106 has |σ(n) − 2n| > 5.

(5)Explain why σ(n) is almost always even. In fact, show that the number of n ≤ x with σ(n) odd is √x + x/2 .

(6)Show that for any fixed integer k > 1, the set of integers n with k|σ(n) has asymptotic density 1. (Hint: Use the Dirichlet Theorem 1.1.5.) The case k = 4 is easier than the general case. Use this easier case to show that the set of odd perfect numbers has asymptotic density 0.

(7)Let s(n) = σ(n) − n for natural numbers n, and let s(0) = 0. Thus, n is abundant if and only if s(n) > n. Let s(k)(n) be the function s iterated k times at n. Use the Dirichlet Theorem 1.1.5 to prove the following theorem of H. Lenstra: For each natural number k there is a number n with

It is not known whether there is any number n for which this inequality chain holds true for every k, nor is it known whether there is any number n for which the sequence s(k)(n) is unbounded. The smallest n for which the latter property is in doubt is 276. P. Erd˝os has shown that for each fixed k, the set of n for which n < s(n), yet (1.47) fails, has asymptotic density 0.

1.31. [Vaughan] Prove, with cq (n) being the Ramanujan sum defined in relation (1.37), that n is a perfect number if and only if

1.32.It is known [Copeland and Erd˝os 1946] that the number

0.235711131719 . . . ,

where all the primes written in decimal are simply concatenated in order, is “normal to base 10,” meaning that each finite string of k consecutive digits appears in this expansion with “fair” asymptotic frequency 10−k. Argue a partial result, that each string of k digits appears infinitely often.

In fact, given two finite strings of decimal digits, show there are infinitely many primes that in base 10 begin with the first string and—regardless of what digits may appear in between—end with the second string, provided the last digit of the second string is 1, 3, 7, or 9.

The relative density of primes having a given low-order decimal digit 1, 3, 7, or 9 is 1/4, as evident in relation (1.5). Does the set of all primes having a given high-order decimal digit have a similarly well-defined relative density?

1.33. Here we use the notion of normality of a number to a given base as enunciated in Exercise 1.32, and the notion of equidistribution enunciated in

Exercise 1.35. Now think of the ordered, natural logarithms of the Fermat numbers as a pseudorandom sequence of real numbers. Prove this theorem: If said sequence is equidistributed modulo 1, then the number ln 2 is normal to base 2. Is the converse of this theorem true?

Note that it remains unknown to this day whether ln 2 is normal to any integer base. Unfortunately, the same can be said for any of the fundamental constants of history, such as π, e, and so on. That is, except for instances of artificial digit construction as in Exercise 1.32, normality proofs remain elusive. A standard reference for rigorous descriptions of normality and equidistribution is [Kuipers and Niederreiter 1974]. A discussion of normality properties for specific fundamental constants such as ln 2 is [Bailey and Crandall 2001].

1.34.Using the PNT, or just Chebyshev’s Theorem 1.1.3, prove that the set of rational numbers p/q with p, q prime is dense in the positive reals.

1.35.It is a theorem of Vinogradov that for any irrational number α, the sequence (αpn), where the pn are the primes in natural order, is equidistributed modulo 1. Equidistribution here means that if #(a, b, N )

denotes the number of times any interval [a, b) [0, 1) is struck after N primes are used, then #(a, b, N )/N (b − a) as N → ∞. On the basis of this Vinogradov theorem, prove the following: For irrational α > 1, and the set

S(α) = { kα : k = 1, 2, 3, . . .},

the prime count defined by

πα(x) = #{p ≤ x : p P ∩ S(α)}

behaves as

1 x πα(x) α ln x .

What is the behavior of πα for α rational?

As an extension to this exercise, the Vinogradov equidistribution theorem itself can be established via the exponential sum ideas of Section 1.4.4. One uses the celebrated Weyl theorem on spectral properties of equidistributed sequences [Kuipers and Niederreiter 1974, Theorem 2.1] to bring the problem down to showing that for irrational α and any integer h = 0,

EN (hα) = e2πihαp p≤N

is o(N ). This, in turn, can be done by finding suitable rational approximants to α and providing bounds on the exponential sum, using essentially our book formula (1.39) for well-approximable values of hα, while for other α using (1.41). The treatment in [Ellison and Ellison 1985] is pleasantly accessible on this matter.

As an extension, use exponential sums to study the count

πc(x) = # {n [1, x] : nc P} .

Heuristically, one might expect the asymptotic behavior

1 x πc(x) c ln x .

Show first, on the basis of the PNT, that for c ≤ 1 this asymptotic relation indeed holds. Use exponential sum techniques to establish this asymptotic behavior for some c > 1; for example, there is the Piatetski-Shapiro theorem [Graham and Kolesnik 1991] that the asymptotic relation holds for any c with 1 < c < 12/11.

1.36. The study of primes can lead to considerations of truly astoundingly large numbers, such as the Skewes numbers

10101034 , eeee7.705 ,

the second of these being a proven upper bound for the least x with π(x) > li0(x), where li0(x) is defined as 0x dt/ ln t. (The first Skewes number is an earlier, celebrated bound that Skewes established conditionally on the Riemann hypothesis.) For x > 1 one takes the “principal value” for the singularity of the integrand at t = 1, namely,

The function li0(x) is li (x) + c, where c ≈ 1.0451637801. Before Skewes came up with his bounds, J. Littlewood had shown that π(x) − li0(x) (as well as π(x) − li (x)) not only changes sign, but does so infinitely often.

An amusing first foray into the “Skewes world” is to express the second Skewes number above in decimal-exponential notation (in other words, replace the e’s with 10’s appropriately, as has been done already for the first Skewes number). Incidentally, a newer reference on the problem is [Kaczorowski 1984], while a modern estimate for the least x with π(x) > li0(x) is

One interesting speculative exercise is to estimate roughly how many more years it will take researchers actually to find and prove an explicit case of π(x) > li0(x). It is intriguing to guess how far calculations of π(x) itself can be pushed in, say, 30 years. We discuss prime-counting algorithms in Section 3.7, although the state of the art is today π 1021 or somewhat higher than this (with new results emerging often).

Another speculative direction: Try to imagine numerical or even physical scenarios in which such huge numbers naturally arise. One reference for

this recreation is [Crandall 1997a]. In that reference, what might be called preposterous physical scenarios—such as the annual probability of finding oneself accidentally quantum-tunneled bodily (and alive, all parts intact!) to planet Mars—are still not much smaller than A−A, where A is the Avogadro number (a mole, or about 6·1023). It is di cult to describe a statistical scenario relevant to the primes that begs of yet higher exponentiation as manifest in the Skewes number.

Incidentally, for various technical reasons, the logarithmic-integral function li0, on many modern numerical/symbolic systems, is best calculated in terms of Ei(ln x), where we refer to the standard exponential-integral function

z

Ei(z) = t−1et dt,

−∞

with principal value assumed for the singularity at t = 0. In addition, care must be taken to observe that some authors use the notation li for what we are calling li0, rather than the integral from 2 in our defining equation (1.3) for li . Calling our book’s function li , and the latter li0, we can summarize this computational advice as follows:

li (x) = li0(x) − li0(2) = Ei(ln x) − Ei(ln 2) ≈ Ei(ln x) − 1.0451637801.

1.37. In [Schoenfeld 1976] it is shown that on the Riemann hypothesis we have the strict bound (for x ≥ 2657)

where li0(x) is defined in Exercise 1.36. Show via computations that none of the data in Table 1.1 violates the Riemann hypothesis!

By direct computation and the fact that li (x) < li0(x) < li (x) + 1.05, prove the assertion in the text that assuming the Riemann hypothesis,

√

It follows from the discussion in connection to (1.25) that (1.48) is equivalent to the Riemann hypothesis. Note too that (1.48) is an elementary assertion, which to understand one needs to know only what a prime is, the natural logarithm, and integrals. Thus, (1.48) may be considered as a formulation of the Riemann hypothesis that could be presented in, say, a calculus course.

1.38. With ψ(x) defined as in (1.22), it was shown in [Schoenfeld 1976] that the Riemann hypothesis implies that

By direct computation show that on assumption of the Riemann hypothesis,

|ψ(x) − x| < √x ln2 x for x ≥ 3.

Then using Exercise 1.37 give a proof that the Riemann hypothesis is equivalent to the elementary assertion

where L(n) is the natural logarithm of the least common multiple of 1, 2, . . . , n. If (1.48) is to be the “calculus-course” version of the Riemann hypothesis, perhaps (1.49) might be referred to as the “precalculus-course” version, in that all that is used in the formulation here is the concept of least common multiple and the natural logarithm.

1.39. Using the conjectured form of the PNT in (1.25), prove that there is a prime between every pair of su ciently large cubes. Use (1.48) and any relevant computation to establish that (again, on the Riemann hypothesis) there is a prime between every two positive cubes. It was shown unconditionally by Ingham in 1937 that there is a prime between every pair of su ciently large cubes, and it was shown, again unconditionally, by Cheng in 1999, that this is true for cubes greater than ee15 .

1.40.Show that p≤n−2 1/ ln(n − p) n/ ln2 n, where the sum is over

primes.

1.41.Using the known theorem that there is a positive number c such that

the number of even numbers up to x that cannot be represented as a sum of two primes is O(x1−c), show that there are infinitely many triples of primes in arithmetic progression. (For a di erent approach to the problem, see Exercise 1.42.)

1.42.It is known via the theory of exponential sums that

≤

where R2(2n) is, as in the text, the number of representations p + q = 2n with p, q prime, and where R2(2n) is given by (1.10); see [Prachar 1978]. Further, we know from the Brun sieve method that

Show, too, that R2(2n) enjoys the same big-O relation. Use these estimates to prove that the set of numbers 2p with p prime and with 2p not representable as a sum of two distinct primes has relative asymptotic density zero in the set of primes; that is, the number of these exceptional primes p ≤ x is o(π(x)). In addition, let

A3(x) = # (p, q, r) P3 : 0 < q − p = r − q; q ≤ x ,

so that A3(x) is the number of 3-term arithmetic progressions p < q < r of primes with q ≤ x. Prove that for x ≥ 2,

where C2 is the twin-prime constant defined in (1.6).

In a computational vein, develop an e cient algorithm to compute A3(x) exactly for given values of x, and verify that A3(3000) = 15482 (i.e., there are 15482 triples of distinct primes in arithmetic progression with the middle prime not exceeding 3000), that A3(104) = 109700, and that A3(106) = 297925965. (The last value here was computed by R. Thompson.) There are at least two ways to proceed with such calculations: Use some variant of an Eratosthenes sieve, or employ Fourier transform methods (as intimated in Exercise 1.67). The above asymptotic formula for A3 is about 16% too low at 106. If x2/ ln3 x is replaced with

the changed formula is within 0.4% of the exact count at 106. Explain why the double integral should give a better estimation.

1.43.In [Saouter 1998], calculations are described to show how the validity of the binary Goldbach conjecture for even numbers up through 4· 1011 can be used to verify the validity of the ternary Goldbach conjecture for odd numbers greater than 7 and less then 1020. We now know that the binary Goldbach conjecture is true for even numbers up to 4 · 1014. Describe a calculation that could be followed to extend Saouter’s bound for the ternary Goldbach conjecture to, say, 1023.

Incidentally, research on the Goldbach conjecture can conceivably bring special rewards. In connection with the novel Uncle Petros and Goldbach’s Conjecture by A. Doxiadis, the publisher announced in 2000 a $1,000,000 prize for a proof of the (binary) Goldbach conjecture, but the prize expired unclaimed in 2002.

1.44.Here we prove (or at least finish the proof for) the result of

Shnirel’man—as discussed in Section 1.2.3—that the set S = {p + q : p, q P} has “positive lower density” (the terminology to be clarified below). As in the text, denote by R2(n) the number of representations n = p + q with p, q prime. Then:

(1) Argue from the Chebyshev Theorem 1.1.3 that

for some positive constant A1 and all su ciently large values of x.

Conclude that for distinct primes q, r the Mersenne numbers Mq , Mr are coprime.

1.49.From W. Keller’s lower bound on a factor p of F24, namely,

p > 6 · 1019,

estimate the a priori probability from relation (1.13) that F24 is prime (we now know it is not prime, but let us work in ignorance of that computational result here). Using what can be said about prime factors of arbitrary Fermat numbers, estimate the probability that there are no more Fermat primes beyond F4 (that is, use the special form of possible factors and also the known character of some of the low-lying Fermat numbers).

1.50.Prove Theorem 1.2.1, assuming the Brun bound (1.8).

1.51.For the odd number n = 3 · 5 · · · 101 (consecutive odd-prime product) what is the approximate number of representations of n as a sum of three primes, on the basis of Vinogradov’s estimate for R3(n)? (See Exercise 1.68.)

1.52.Show by direct computation that 108 is not the sum of two base- 2 pseudoprimes (see Section 3.4 for definitions). You might show in passing, however, that if p denotes a prime and P2 denotes an odd base-2 pseudoprime, then

108 = p + P2 or P2 + p

in exactly 120 ways (this is a good check on any such programming e ort). By the way, one fine representation is

108 = 99999439 + 561,

where 561 is well known as the smallest Carmichael number (see Section 3.4.2). Is 108 the sum of two pseudoprimes to some base other than 2? What is the statistical expectation of how many “pseudoreps” of various kinds p+Pb should exist for a given n?

1.53.Prove: If the binary expansion of a prime p has all of its 1’s lying in an arithmetic progression of positions, then p cannot be a Wieferich prime. Prove the corollary that neither a Mersenne prime nor a Fermat prime can be a Wieferich prime.

1.54.Show that if u−1 denotes a multiplicative inverse modulo p, then for

each odd prime p,

1.55. Use the Wilson–Lagrange Theorem 1.3.6 to prove that for any prime p ≡ 1 (mod 4) the congruence x2 + 1 ≡ 0 (mod p) is solvable.

1.56. Prove the following theorem relevant to Wilson primes: if g is a primitive root of the prime p, then the Wilson quotient is given by

Then, using this result, give an algorithm that determines whether p with primitive root g = 2 is a Wilson prime, but using no multiplications; merely addition, subtraction, and comparison.

1.57. There is a way to connect the notion of twin-prime pairs with the Wilson–Lagrange theorem as follows. Let p be an integer greater than 1. Prove the theorem of Clement that p, p + 2 is a twin-prime pair if and only if

4(p − 1)! ≡ −4 − p (mod p(p + 2)).

1.58. How does one resolve the following “Mertens paradox”? Say x is a large integer and consider the “probability” that x is prime. As we know,

primality can be determined by testing x for prime divisors not exceeding

√

x. But from Theorem 1.4.2, it would seem that when all the primes less

√

than x are probabilistically sieved out, we end up with probability

Arrive again at this same estimate by simply removing the floor functions in (1.46). However, the PNT says that the correct asymptotic probability that x is prime is 1/ ln x. Note that 2e−γ = 1.1229189 . . ., so what is a resolution?

It has been said that the sieve of Eratosthenes is “more e cient than random,” and that is one way to envision the “paradox.” Actually, there has been some interesting work on ways to think of a resolution; for example, in [Furry 1942] there is an analysis of the action of the sieve of Eratosthenes on a prescribed interval [x, x + d], with some surprises uncovered in regard to how many composites are struck out of said interval; see [Bach and Shallit 1996, p. 365] for a historical summary.

1.59. By assuming that relation (1.24) is valid whenever the integral converges, prove that M (x) = O(x1/2+ ) implies the Riemann hypothesis.

1.60. There is a compact way to quantify the relation between the PNT and the behavior of the Riemann zeta function. Using the relation

− ζ (s) = s ∞ ψ(x)x−s−1 dx, ζ(s) 1

show that the assumption

ψ(x) = x + O(xα)

implies that ζ(s) has no zeros in the half-plane Re(s) > α. This shows the connection between the essential error in the PNT estimate and the zeros of ζ.

For the other (harder) direction, assume that ζ has no zeros in the halfplane Re(s) > α. Looking at relation (1.23), prove that

xρ = O(xα ln2 T ),

|ρ|

Im(ρ) ≤ T

which proof is nontrivial and interesting in its own right [Davenport 1980].

Finally, conclude that

ψ(x) = x + O xα+

for any > 0. These arguments reveal why the Riemann conjecture

π(x) = li (x) + O(x1/2 ln x)

is sometimes thought of as “the PNT form of the Riemann hypothesis.”

1.61. Here we show how to evaluate the Riemann zeta function on the critical line, the exercise being to implement the formula and test against some high-precision values given below. We describe here, as compactly as we can, the celebrated Riemann–Siegel formula. This formula looms unwieldy on the face of it, but when one realizes the formula’s power, the complications seem a small price to pay! In fact, the formula is precisely what has been used to verify that the first 1.5 billion zeros (of positive imaginary part) lie exactly on the critical line (and parallel variants have been used to push well beyond this; see the text and Exercise 1.62).

A first step is to define the Hardy function

Z(t) = eiϑ(t)ζ(1/2 + it),

renders Z a real-valued function on the critical line (i.e., for t real). Moreover, the sign changes in Z correspond to the zeros of ζ. Thus if Z(a), Z(b) have opposite sign for reals a < b, there must be at least one zero in the interval (a, b). It is also convenient that

|Z(t)| = |ζ(1/2 + it)|.

Note that one can either work entirely with the real Z, as in numerical studies of the Riemann hypothesis, or backtrack with appropriate evaluations of Γ and so on to get ζ itself on the critical line.

That having been said, the Riemann–Siegel prescription runs like so [Brent 1979]: Assign τ = t/(2π), m = √τ , z = 2(√τ − m) − 1. Then the