# Generalized twin prime formulas.

1. Introduction

Golomb used his arithmetic formula [1]

2[LAMBDA](2a - 1)[LAMBDA](2a + 1) - [summation over (d|4[a.sup.2] - [D.sup.2])] [mu](d)[log.sup.2]d, a [greater than or equal to] [a.sub.D], (1.1)

where [LAMBDA](n) is the von Mangoldt function and [mu](n) the Mobius function [2], [3], as coefficients of a power series which naturally converts to a Lambert series. Due to the lack of an Abelian theorem for the latter, no further progress toward a solution of the twin prime problem was possible. However, the formula may be used to construct twin prime Dirichlet series. Such a method is applied here to twin primes p,p' = p + 2D for odd D > 0 first and then even D > 0. Based on the corresponding Golomb identity

2[LAMBDA](2a - D)[LAMBDA](2a + D) = [summation over (d|4[a.sup.2] - [D.sup.2])] [mu](d)[log.sup.2]d, a [greater than or equal to] [a.sub.D], (1.2)

for the generalized twin prime numbers p = 2a - D, p' = 2a + D. The numbers [a.sub.D] > 0 characterize the first twins that are a distance 2D apart. While the running median 2a between p = 2a - D and p' = 2a + D for odd D is an even number independent of D, whereas aD depends on D and is more irregular. For example, [a.sub.D] = 2 for D = 1; [a.sub.D] = 4 for D = 3; [a.sub.D] = 4 for D = 5; [a.sub.D] = 5 for D = 7; [a.sub.D] = 7 for D = 9, etc.

For most even D, such as 2, 4, 8, 10, 14, 16,... the median 3(2a - 1) between the twin primes p = 3 (2a - 1) - D and p' = p = 3 (2a - 1) + D that are a distance 2D apart is again a linear function of the running natural number [alpha]. Golomb's identity for these cases is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

For 6|D the median is more irregular. Therefore, these twin prime numbers and those not of the form (2[alpha] - D, 2[alpha] + D); (3 (2[alpha] - 1) - D, 3 (2[alpha] - 1) + D) will not be considered here.

The strategy will be to decompose the relevant Golomb identity into two factors whose generating Dirichlet series, one acting as the prime number sieve and the other to implement the constraint n = 4[a.sup.2] - [D.sup.2] in [summation over (d|n)] (d) [log.sup.2] d for the twins for odd D and n = 4[(2a - 1).sup.2] - [D.sup.2] for even D, are then used in a product formula for Dirichlet series to construct the twin prime Dirichlet series.

2. Twin Prime Generating Dirichlet Series

Definition 2.1. The generating Dirichlet series of one factor of the term on the rhs of Golomb's identity (1.1) is defined as

Z(s) [equivalent to] [[infinity].summation over (n=2}][1/[n.sup.s]][summation over (d|n)][mu](d)[log.sup.2]d, [Real part](s) [equivalent to] [sigma] > 1. (2.1)

The series converges absolutely for [sigma] > 1. Multiplying termwise the Dirichlet series

[[d.sup.2]/d[s.sup.2]] [1/[zeta](s)] = [[infinity].summation over (n=2)][mu](n)[log.sup.2]n/[n.sup.s] (2.2)

and [zeta](s), which is justified by absolute convergence of both series for [sigma] > 1, gives Eq. (2.1). Carrying out the differentiations yields

Z(s) = [zeta](s) [[d.sup.2]/d[s.sup.2]] [1/[zeta](s)] = 2[([zeta]'(s)/[zeta](s)).sup.2] - [[zeta]"(s)/[zeta](s)] = - [d/ds] [[zeta]'/[zeta]](s) + [([zeta]'(s)/[zeta](s)).sup.2]. (2.3)

Among the poles of the meromorphic function Z(s) are the roots p of the Riemann zeta function in the critical strip 0 < [sigma]< 1, which is clear from Eq. (2.3). The next two lemmas display analytic properties of Z(s) that are needed in Section 6.

Lemma 2.1. A pole expansion of Z(s) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

with [gamma] = 0.57721566... the Euler-Mascheroni constant and p denoting the zeros of [zeta](s) in the critical strip 0 < [sigma] < 1.

Proof. The pole expansions [2], [4] of the meromorphic functions [GAMMA]'(s)/[GAMMA]'(s), [zeta](S)/[zeta](S),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

in Eq. (2.3) lead to the corresponding pole expansion of Z(s).

Thus, Z(s) has a simple pole at s = 1 with the residue

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

at s = [rho] with the residue

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.7)

at s = -2n for n = 1, 2,... and double poles at s = [rho], all with coefficients 2, and s = -2n for n = 1, 2,...

Lemma 2.2. The functional equation for Z(s) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

Proof. Differentiating the functional equation [2], [4], [6] of [zeta]'(s)/[zeta](s),

-[zeta]'(1 - s)/[zeta](1 - s) = -log2[pi] - [[pi]/2] tan [s[pi]/2] + [[GAMMA]'(s)/[GAMMA](s)] + [[zeta]'(s)/[zeta](s)] (2.9)

we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.10)

and thus the corresponding functional relation for Z(s).

3. Constraint Generating Dirichlet Series

Definition 3.1. The generating Dirichlet series that implements the constraint in Golomb's identity is defined as

[Q.sub.D](s) = [[infinity].summation over (a>[D/2])] [1/[(4[a.sup.2] - [D.sup.2]).sup.s]], (3.1)

with [D/2] the integer part of D/2. It converges absolutely for [sigma]> 1/2.

Lemma 3.1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.2)

where [D/2] is the integer part of D/2.

Proof. Using a binomial expansion and interchanging the summations, which is justified by absolute convergence for [sigma]> 1 /2, we obtain Eq. (3.2).

Therefore, [Q.sub.D](s) has a simple pole at s = 1/2 with residue 1/4 and at s = 1/2(1 - 2v) with residue [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for v = 1,2,... and is regular elsewhere.

Definition 3.2. The subtracted constraint Dirichlet series is defined as

q(s) = [Q.sub.D](s) - [2.sup.-2s][[zeta](2s) - [[summation over (a[less than or equal to][D/2])][a.sup.-2s] = [summation over (a>[D/2])] {[1/[(4[a.sup.2] - [D.sup.2]).sup.s] - [1/[(4[a.sup.2]).sup.s]). (3.3)

Corollary 3.1. [q.sub.D](s) can be resumed as the contour integral

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

where the contour C runs parallel to the imaginary axis from c - i[infinity] to c + i[infinity] with -[D/2] - 1 < c < -[D/2], and [D/2] the integral part of D/2.

Proof. Adapting a variant of the integral representation of the zeta function due to Kloosterman [4],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)

where the contour runs parallel to the imaginary axis through the abscissa c. The integral representation can also be obtained by folding the contour to the left, running from -[infinity] back to -[infinity] enclosing the point-[D/2]-1 in an anticlockwise sense and applying the residue theorem.

Lemma 3.2. The Dirichlet series [q.sub.D](s) is regular for [sigma] > -1/2, O(1) for [absolute value of t][right arrow][infinity] and, with its general term groupedas [[(2a).sup.2] - [[D.sup.2]].sup.-s] - [(2a.sup.-2s,] converges absolutely for [sigma] > -1/2.

Proof. For a large compared to [absolute value of t], the convergence is the same as for t = 0, i.e. s = [sigma], which is a well known property of Dirichlet series. A binomial expansion of the general term of [q.sub.D]([sigma]),

[(2[alpha).sup.-2[sigma]] [[(1 - [D.sup.2]/[(2[alpha]).sup.2]).sup.-[sigma]] - 1]= O([[alpha].sup.-2[sigma]-2]), (3.6)

shows the absolute convergence and regularity for [sigma]> -1/2.

The convergence of the integral representation of [q.sub.D](s) maybe improved by including more terms of Stirling's asymptotic series [5]. This observation leads to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.7)

Now we deform the contour to enclose the real axis in the left-hand plane along two rays from the origin with opening angle [theta]/[absolute value of t] around [pi] for fixed [Imaginary part](s) = t [not equal to] 0 and -[pi] < [phi] < [pi], i.e. z = -[re.sup.i[phi][absolute value of t]] 0 is used to show that the general term is O(1) for t [right arrow][+ or -][infinity]. The conic area bounded by the rays is closed off by a small circular path around z = -[D/2] - 1. Stirling's series for the logarithmic derivative of the Gamma function applies outside the cone area guaranteeing convergence of the integral in Eq. (3.7). Each term in

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.8)

remains bounded, in absolute value, as we let t [right arrow][+ or -][infinity] for [sigma]> -1/2.

4. Twin Prime Dirichlet Series

The following product formula for absolutely converging Dirichlet series is one of our main tools; it is obtained from summing a well-known formula from which mean values of Dirichlet series are usually derived. We mention it for ease of reference and because details of its proof are needed in Sect. 6.

Lemma 4.1. (Product formula.) Let the Dirichlet series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.1)

be single-valued, regular and absolutely convergent for [sigma] > [[sigma].sub.a] and [sigma] > [[sigma].sub.b], respectively. Then the product series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.2)

is regular and converges absolutely for [Real part](w) [equivalent to] u > [sigma] + [[sigma].sub.b], [sigma] > [[sigma].sub.a] + 1. The limit of the integral exists and is a regular function of the variable w [equivalent to] u + iv.

Proof. Substituting the Dirichlet series and using [2], [4]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.3)

we obtain the first term on the rhs of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.4)

which converges absolutely because u [greater than or equal to] [[sigma].sub.a] + [[sigma].sub.b] + [epsilon].

To deal with [summation over (n[not equal to]m)] we split the summation m[not equal to] n into the ranges n [less than or equal to] m/2, m/2 < n < 2m, n [greater than or equal to] 2m. Since [absolute value of log n/m] [greater than or equal to] 1/log 2 for n [less than or equal to] m/2 we obtain the estimate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.5)

provided u - [sigma] > [[sigma].sub.b] and [sigma] > [[sigma].sub.a]. For n [greater than or equal to] 2m, we get the same estimate. For m/2 < n < 2m, we split the range m/2 < n < 2m into m/2 < n < m and m < n < 2m with n [not equal to] m and use

1/log(1 - [1/n]) = n + O(1), [1/log(1 - [2/n]) = [n/2] + O(1), ..., n[n-1.summation over (j=1)] [1/j] = O(n log n) (4.6)

to obtain the estimate

[summation over ([m/2]<n<m)] [1/[absolute value of log [n/m]] = O(m log m), (4.7)

and the same estimate for the range m < n < 2m. Putting all this together, we find for [sigma] > [[sigma].sub.a] + 1, u - [sigma] > [[sigma].sub.b] that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.8)

Using termwise differentiation with respect to the variable w in the integrals in conjunction with the estimate log n = O([n.sub.[epsilon]]) shows the absolute convergence of the termwise differentiated product series and the regularity of the product series.

As a first step toward constructing the twin prime Dirichlet series we apply the product formula to Z(s)[q.sub.D](s - w).

Theorem 4.1. For odd D>0, [Real part](w) = u>[sigma] + 3/2, [sigma] > 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.10)

where the limit of the integral on the rhs is a regular function for u > 5/2 and the asymptotic series A(w) converges for u > 1 /2.

Proof. Since Z(s) converges absolutely for [sigma] > 1, the product [Z(s)2.sup.-2w-2s][[zeta](2w - 2s) [summation over ([alpha][less than or equal to][D/2]) [a.sup.-2w+2s]] converges absolutely for u - [sigma] > 1/2, i.e. u > 3/2. Lemma 4.1 implies that the integral in Eq. (4.9) is O(T) and the limit T [right arrow] [infinity] exists representing a regular function for [sigma] > 5/2. The integral over [[Z(s)2.sub.-2(w-s)][[zeta](2(w - s)) - [summation over ([alpha][less than or equal to][D/2]) in Eq. (4.9) is O(T). We apply the product formula and obtain the first term on the rhs of Eq. (4.9) for u> 5/2.

5. The prime pair generating and asymptotic Dirichlet series

In order to analyze the generating Dirichlet series Z(s) introduced in Section 2 and find the Dirichlet series on the rhs ofEqs. (7.6), (4.10) in Theor. 4.1, which we call asymptotic Dirichlet series, we evaluate the arithmetic functions in their numerators.

Proposition 5.1. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be prime number decompositions.

Then

[summation over (d|n)][mu](d)[log.sup.2]d = [summation over (d|p)] [mu](d)[log.sup.2]d. (5.1)

If n = p is prime then

[summation over (d|p)][mu](d)[log.sup.2]d = -[log.sup.2]p. (5.2)

If n = [p.sub.i][p.sub.j] for prime numbers [p.sub.i][not equal to][p.sub.j] then

[summation over (d|[p.sub.i][p.sub.j])][mu](d)[log.sub.2] d = 2 log[p.sub.i] log[p.sub.j]. (5.3)

If k [greater than or equal to] 3 in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for different prime numbers [p.sub.i] then

[summation over (d|P)] (d) [log.sub.2] d = 0. (5.4)

Proof. Eq. (5.1) is obvious from the properties of the Mobius function, as is Eq. (5.2). Eq. (5.3) follows from

[summation over (d|[p.sub.i][p.sub.j])][mu](d) [log.sup.2]d = -[log.sup.2] [p.sub.i] - [log.sup.2] [p.sub.j] + [log.sup.2] [p.sub.i][p.sub.j] = 2log[p.sub.i]log [p.sub.j]. (5.5)

Eq. (5.4) for k = 3 follows from expanding [log.sup.2][p.sub.i][p.sub.j] = (log [p.sub.i] + log [p.sub.j]), etc.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.6)

For the general case k, Eq. (5.4) is proved by induction. Assuming its validity for k we can show that for k + 1:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.7)

This follows from verifying that the coefficients of [log.sup.2] [p.sub.i] are zero, as are those of 2log [p.sub.i][p.sub.j]. In essence, Eqs. (5.3,5.4) comprise Golomb's identity (1.1)for [LAMBDA](n), log n.

A comment on the constraint (2[alpha] - D, 2[alpha] + D) = 1 of Golomb's formula is in order. If d|2D and d|2a - D then d|2a + D; if d|2[alpha] - D, but d is not a divisor of 2D, then d is not a divisor of 2[alpha] + D. Thus, for odd D, if d|2[alpha] - D then d does not divide 2[alpha] + D, except for a finite number of divisors of D. Likewise, if d|3(2[alpha] - 1) - D and d does not divide 2D, then d does not divide 3 (2[alpha] - 1) + D. If a = b[delta] with [delta]|D then, for odd D, 2a - D = [delta](2b - [D/[delta]]), 2a + D = [delta](2b + [D/[delta]]) and [summation over (d| [[delta].sup.2]]([4b.sup.2]-[D.sup.2]/[[delta].sup.2])) [mu]d [log.sup.2]d = 0 follows from Lemma 5.1. For even D, if 2a - 1 = b[delta], [delta]|D with [delta] [not equal to] 2 then 9 [(2[alpha] - 1).sup.2] - [D.sup.2] = [[delta].sup.2] [9b.sup.2] - [D.sup.2]/ [[delta].sup.2] and [summation over (d|[[delta].sup.2]] ([9b.sup.2]- [D.sup.2]/[[delta].sup.2])) [mu](d) [log.sup.2] d = 0 follows again from Lemma 5.1. Thus, despite (2a - D, 2 + D) [not equal to] 1,(3 (2[alpha] - 1) - D, 3 [(2[alpha] - 1).sup.2] + D) [not equal to] 1, Golomb's identity is trivially satisfied for such values of a.

Lemma 5.1. The coefficient of the asymptotic Dirichlet series A(w) of Theorem 4.1 is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.8)

where p, [p.sub.i] are prime numbers [not equal to] 2.

Proof. This follows from Prop. 5.1. Using the prime number decomposition of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in conjunction with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.9)

the first and third lines of Eq. (5.8) are immediate consequences of Golomb's identity. The second line is a special case of [summation over (d|p)][mu](d)[log.sup.2] d = -[log.sub.2] p, where p is a prime number.

Theorem 5.1. For odd D > 0, A(w) of Eq. (4.10) in Theor. 4.1 becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.10)

Thus, A(w) has a simple pole at w = 1/2 with the positive residue log 2.

Proof. Summing the Dirichlet series A(w) using Lemma 5.1 yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.11)

from which Eq. (5.10) and the residue follow.

6. Limit of the Integral

In order to study the integral of Theor. 4.1 we now evaluate the contour integral

I(w) [equivalent to] [1/2[pi]i] [[integral].sub.C] Z(s)[q.sub.D](w - s)ds = [I.sub.1] + [I.sub.2] + [I.sub.3] + [I.sub.4], (6.1)

where the contour C is a rectangle with the vertices 1 + [epsilon] - iT, 1 + [epsilon]+iT, -[epsilon] + iT, - [epsilon] - iT and T [not equal to] [gamma], [rho] = [beta] + i[gamma] the general root of [zeta](s) in the critical strip, [I.sub.1] is the integral from 1 + [epsilon] - iT to 1 + [epsilon] + iT, [I.sub.2] from 1 + [epsilon] + iT, to -[epsilon] + iT, [I.sub.3] from - [epsilon] + iT, to -[epsilon] - iT and [I.sub.4] from -[epsilon] - iT to 1 + e - iT, [epsilon] > 0. The limit [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the integral [I.sub.1] in Eq. (4.9) of Theorem 4.1 needs to be analyzed.

Lemma 6.1. For u > 5/2

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.2)

Proof. Eq. (6.2) follows from using Eq. (2.4) and the residue theorem taking into account the poles at s = 1, [rho] inside the rectangular contour.

For the evaluation of [I.sub.2], [I.sub.4] we need bounds for Z(s), [q.sub.D](w - s) for s = [sigma] [+ or -] iT.

Lemma 6.2. [I.sub.2] = O ([log.sub.3] T), [I.sub.4] = O([log.sup.3] T) for u > 5/2 and all sufficiently large and appropriately chosen T.

Proof. We know [6] that for sufficiently large [absolute value of t] and -1 < [sigma] < 2

-[zeta]'(s)/[zeta](s) = - [summation over (p,[absolute value of t - [gamma]]]<1)] ([1/[s - [rho]]] + [1/[rho]]) + O(log[absolute value of t]). (6.3)

Choosing t = [Imaginary part](s), T = [absolute valule of t] sufficiently large and appropriately we can arrange that [absolute value of t - [gamma]][greater than or equal to] [1/log T], when [absolute value of [gamma] - t] < 1. There are at most O(log T) roots [rho] with [absolute value of [gamma] - t] < 1. As a result [6],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.4)

Hence [absolute value of Z(s)] = O ([log.sup.3] T) for [I.sub.2] and the same bound holds for [I.sub.4] and -[epsilon] [less than or equal to] [sigma] [less than or equal to] 1 + [epsilon].

On all four sides of the rectangular path -[epsilon] [less than or equal to] [sigma] [less than or equal to] 1 + [epsilon], [Real part](w - s) > -[3/2] for u > 3/2 the Dirichlet series [q.sub.D](w - s) remains bounded for u -[sigma] > 1/2, in absolute magnitude, for |t|[right arrow][infinity] according to Lemma 3.2.

Lemma 6.3. For u> 5/2, [I.sub.3] = O ([log.sup.2] T).

Proof. For [I.sub.3] we will use the functional equation (2.8) of Z(s) to evaluate the first term on the rhs of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.5)

[q.sub.D](w - s)Z(1 - s), by substituting the absolutely converging Dirichlet series. This yields a double series [summation over (m,n)] as for [I.sub.1] in Lemma 4.1 involving the integral [[integral].sup.T.sub.- T](m[[[(2n).sup.2] - [D.sup.2])].sup.it] dt. Thus, there is no contribution for [4n.sup.2] - [D.sup.2] = m that would be proportional to T. As earlier, [summation over (m [not equal to][(2n).sup.2] - [D.sup.2])] involves sin(T log[[(2n).sup.2] - [D.sup.2]]m) and is O(1) for u + [epsilon] > 3/2. The same applies to the double sums involving ([2n).sup.2] and [4n.sup.2] - [2D.sup.2] instead of [(2n).sup.2] - [D.sup.2.]

To deal with the term involving -[[zeta]'(s)]/[zeta](s)] 2([gamma] + log 2[pi]) [q.sub.D] (w - s) we use the functional equation of Lemma 2.2 and then substitute the absolutely converging Dirichlet series for [q.sub.D](w - s) and -[zeta]'(s)]/[zeta](s) at [sigma] = -[epsilon]. Again, each double series is obtained as for [I.sub.1] in Lemma 4.1 involving the integral [[integral].sup.T.sub.-T][(m[[(2n).sup.2] - [D.sup.2]).sup.it]dt, [[integral].sup.T.sub.-T][(m[(2n).sup.2]).sup.it] dt, or [[integral].sup.T.sub.-T][[m(4[n.sup.2] - 2[D.sup.2])].sup.it]dt, respectively. Again, there is no contribution for [(2n).sup.2] - [D.sup.2] = m, [(2n).sup.2] - 2[D.sup.2] = m, or [(2n).sup.2] = m, that would be proportional to T. As earlier, [summation over (m[not equal to][(2n).sup.2] - [D.sup.2])], [summation over (m=[(2n).sup.2] - 2[D.sup.2])] or [summation over (m[not equal to][(2n).sup.2]] lead to the bound O(1).

Putting all this together, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.6)

The term [1/2[pi]i] [[integral].sup.-[epsilon]+iT.sub.-[epsilon]-iT] [q.sub.D](w - s) [[([pi]/2).sup.2]/[cos.sup.2]s[pi]/2]ds = O(1) because [([cos.sup.2] s[pi]/2).sup.-2] = O([e.sup.-[absolute value of t][pi])] for t [right arrow] [+ or -][infinity].

The terms involving [absolute value of tan [s[pi]/2]] [right arrow] 1 for t [right arrow][+ or -][infinity], so [absolute value of [d/ds] tan [s[pi]/2]] + O([e.sup.-[absolute value of t][pi]]/2]). Using integration by parts, we get the following estimate for this term from the Dirichlet series for [q.sub.D] (w - s)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the integrated term is O(1) because the Dirichlet series is absolutely convergent, tan s[pi]/2 = O(1) and the integral is O(1). Terms involving [tan.sup.2] s[pi]/2, tan [(s[pi]/2)][[[GAMMA]'(s)/2[pi][GAMMA](s)] are handled in the same way.

The terms involving [q.sub.D](w - s) [[GAMMA]'(s)/[GAMMA](s)] are also treated by integration by parts where the integrated term is O(log T) and the integral is integrated by parts leading to [d/ds] [[GAMMA]'(s)]/[GAMMA](s)] = O([T.sup.-1]) at s = -[epsilon] [+ or -] iT. Hence the remaining integral is O(log T). The term involving [q.sub.D](w - s) [d/ds] [[GAMMA]'(s)/[GAMMA](s)] is treated by integration by parts leading to the same estimate. The term [q.sub.D](w - s) [([GAMMA]'(s)/[GAMMA](s)).sup.2] is also treated by integrating by parts leading to an O([log.sup.2]T) estimate.

The product [q.sub.D](w - s) [[zeta]'(1 - s)/[zeta](1 - s)] is another absolutely converging Dirichlet series without constant term, i.e. unity. All these terms in Eq. (6.6) are also treated by integration by parts leading to the estimate O(log T). Putting all this together proves Lemma 6.3.

Theorem 6.1. For odd D > 0, u > 5/2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.7)

Proof. We apply the limit T [right arrow] [infinity] on I(w) to Eqs. (6.1),(6.2) using the bounds on [I.sub.2],[I.sub.3],[I.sub.4] from Lemmas 6.2, 6.3 to obtain Eq. (6.7) because the first line of Eq. (6.2) and the first term of the second line drop out. The last line follows from Theor. 4.1 and Lemma 4.1.

This is our main result. Due to the limit T [right arrow] [infinity] in Theor. 6.1 the twin prime distributions depend on the asymptotic properties of the roots of the Riemann zeta function. This feature contrasts with the analytic proof of the prime number theorem, where the remainder term is linked to the roots of the Riemann zeta function producing the staircase-like corrections of the smooth asymptotic limit from all zeta function roots, whereas the leading asymptotic term has nothing to do with them, originating from the simple pole of the Riemann zeta function.

7. Twin Primes For Even D

Definition 7.1. The corresponding constraint generating Dirichlet series is defined as

[Q.sub.d](s) = [summation over(a[D/6])] [1/[[[3.sup.2][(2[alpha]-1).sup.2]] - [D.sup.2].sup.s], [sigma] > 1/2 (7.1)

with [D/6] the integer part of D/6.

Lemma 7.1. The expansion corresponding to Lemma 3.1 becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.2)

Definition 7.2. The subtracted constraint Dirichlet series is defined as

q(s) = [Q.sub.D](s) - [3.sup.-2s][(1 - [2.sup.-2s])[zeta](2s) - [summation over (a [less than or equal to] [D/6])] [(2a - 1).sup.-2s] (7.3)

Equation (3.7) of Lemma 3.2 for this case becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.4)

with -2[D/6] < c < -2[D/6] + 1. Lemma 3.2 is valid for this case with the general term grouped as

[1/[[3.sup.2][(2[alpha] - 1).sup.2]] - [D.sup.2].sup.s]] - [3.sup.-2s] [(2[alpha]).sup.-2s] (7.5)

Theorem 7.1. For even D> 0, [Real part](w) = u > [sigma] + [3/2], [sigma] > 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.6)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.7)

This is Theor. 4.1 for even D.

Lemma 7.2. The general coefficient of the asymptotic Dirichlet series A(w) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.8)

Theorem 7.2. For even D>0, A(w) ofEq. (4.10)in Theor. 4.1 becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.9)

Thus, A(w) has a simple pole at w = 1 /2 with the residue log3.

Theor. 6.1 remains valid, except for replacing the twin Dirichlet series in Eq. (6.7) by the lhs ofEq. (7.6) in Theor. 7.1.

8. Discussion and Conclusion

The twin prime formulas link the twin primes with the roots of the Riemann zeta function, which may be viewed as a step in Riemann's program linking prime numbers to these roots. Their dependence on the limit T [right arrow] o demonstrates that the twin prime distributions depend on the asymptotic properties of these roots. This qualitative feature differs fundamentally from the prime number theorem for ordinary prime numbers where all roots contribute to the remainder term.

References

[1] Golomb, S. W., 1970, "The Lambda Method in Prime Number Theory," J. Number Theory 2, pp. 193-198.

[2] Murty, M. R., 2001, Problems in Analytic Number Theory. Springer, New York, NY.

[3] Hardy, G. H. and Wright, E. M., 1988, An Introduction to the Theory of Numbers, Clarendon Press, Oxford, 5th ed.

[4] Titchmarsh, E. C. and Heath-Brown, D. R., 1986, The Theory of the Riemann Zeta Function, Clarendon Press, Oxford.

[5] Arfken, G. B. and Weber, H. J., 2005, Mathematical Methods for Physicists, ElsevierAcademic Press, 6th ed., Amsterdam.

[6] Ivic, A., 1985, The Riemann Zeta-Function, Dover, New York, NY.

H. J. Weber

Department of Physics

University of Virginia

Charlottesville, VA 22904, U.S.A.

Golomb used his arithmetic formula [1]

2[LAMBDA](2a - 1)[LAMBDA](2a + 1) - [summation over (d|4[a.sup.2] - [D.sup.2])] [mu](d)[log.sup.2]d, a [greater than or equal to] [a.sub.D], (1.1)

where [LAMBDA](n) is the von Mangoldt function and [mu](n) the Mobius function [2], [3], as coefficients of a power series which naturally converts to a Lambert series. Due to the lack of an Abelian theorem for the latter, no further progress toward a solution of the twin prime problem was possible. However, the formula may be used to construct twin prime Dirichlet series. Such a method is applied here to twin primes p,p' = p + 2D for odd D > 0 first and then even D > 0. Based on the corresponding Golomb identity

2[LAMBDA](2a - D)[LAMBDA](2a + D) = [summation over (d|4[a.sup.2] - [D.sup.2])] [mu](d)[log.sup.2]d, a [greater than or equal to] [a.sub.D], (1.2)

for the generalized twin prime numbers p = 2a - D, p' = 2a + D. The numbers [a.sub.D] > 0 characterize the first twins that are a distance 2D apart. While the running median 2a between p = 2a - D and p' = 2a + D for odd D is an even number independent of D, whereas aD depends on D and is more irregular. For example, [a.sub.D] = 2 for D = 1; [a.sub.D] = 4 for D = 3; [a.sub.D] = 4 for D = 5; [a.sub.D] = 5 for D = 7; [a.sub.D] = 7 for D = 9, etc.

For most even D, such as 2, 4, 8, 10, 14, 16,... the median 3(2a - 1) between the twin primes p = 3 (2a - 1) - D and p' = p = 3 (2a - 1) + D that are a distance 2D apart is again a linear function of the running natural number [alpha]. Golomb's identity for these cases is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

For 6|D the median is more irregular. Therefore, these twin prime numbers and those not of the form (2[alpha] - D, 2[alpha] + D); (3 (2[alpha] - 1) - D, 3 (2[alpha] - 1) + D) will not be considered here.

The strategy will be to decompose the relevant Golomb identity into two factors whose generating Dirichlet series, one acting as the prime number sieve and the other to implement the constraint n = 4[a.sup.2] - [D.sup.2] in [summation over (d|n)] (d) [log.sup.2] d for the twins for odd D and n = 4[(2a - 1).sup.2] - [D.sup.2] for even D, are then used in a product formula for Dirichlet series to construct the twin prime Dirichlet series.

2. Twin Prime Generating Dirichlet Series

Definition 2.1. The generating Dirichlet series of one factor of the term on the rhs of Golomb's identity (1.1) is defined as

Z(s) [equivalent to] [[infinity].summation over (n=2}][1/[n.sup.s]][summation over (d|n)][mu](d)[log.sup.2]d, [Real part](s) [equivalent to] [sigma] > 1. (2.1)

The series converges absolutely for [sigma] > 1. Multiplying termwise the Dirichlet series

[[d.sup.2]/d[s.sup.2]] [1/[zeta](s)] = [[infinity].summation over (n=2)][mu](n)[log.sup.2]n/[n.sup.s] (2.2)

and [zeta](s), which is justified by absolute convergence of both series for [sigma] > 1, gives Eq. (2.1). Carrying out the differentiations yields

Z(s) = [zeta](s) [[d.sup.2]/d[s.sup.2]] [1/[zeta](s)] = 2[([zeta]'(s)/[zeta](s)).sup.2] - [[zeta]"(s)/[zeta](s)] = - [d/ds] [[zeta]'/[zeta]](s) + [([zeta]'(s)/[zeta](s)).sup.2]. (2.3)

Among the poles of the meromorphic function Z(s) are the roots p of the Riemann zeta function in the critical strip 0 < [sigma]< 1, which is clear from Eq. (2.3). The next two lemmas display analytic properties of Z(s) that are needed in Section 6.

Lemma 2.1. A pole expansion of Z(s) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

with [gamma] = 0.57721566... the Euler-Mascheroni constant and p denoting the zeros of [zeta](s) in the critical strip 0 < [sigma] < 1.

Proof. The pole expansions [2], [4] of the meromorphic functions [GAMMA]'(s)/[GAMMA]'(s), [zeta](S)/[zeta](S),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

in Eq. (2.3) lead to the corresponding pole expansion of Z(s).

Thus, Z(s) has a simple pole at s = 1 with the residue

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

at s = [rho] with the residue

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.7)

at s = -2n for n = 1, 2,... and double poles at s = [rho], all with coefficients 2, and s = -2n for n = 1, 2,...

Lemma 2.2. The functional equation for Z(s) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

Proof. Differentiating the functional equation [2], [4], [6] of [zeta]'(s)/[zeta](s),

-[zeta]'(1 - s)/[zeta](1 - s) = -log2[pi] - [[pi]/2] tan [s[pi]/2] + [[GAMMA]'(s)/[GAMMA](s)] + [[zeta]'(s)/[zeta](s)] (2.9)

we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.10)

and thus the corresponding functional relation for Z(s).

3. Constraint Generating Dirichlet Series

Definition 3.1. The generating Dirichlet series that implements the constraint in Golomb's identity is defined as

[Q.sub.D](s) = [[infinity].summation over (a>[D/2])] [1/[(4[a.sup.2] - [D.sup.2]).sup.s]], (3.1)

with [D/2] the integer part of D/2. It converges absolutely for [sigma]> 1/2.

Lemma 3.1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.2)

where [D/2] is the integer part of D/2.

Proof. Using a binomial expansion and interchanging the summations, which is justified by absolute convergence for [sigma]> 1 /2, we obtain Eq. (3.2).

Therefore, [Q.sub.D](s) has a simple pole at s = 1/2 with residue 1/4 and at s = 1/2(1 - 2v) with residue [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for v = 1,2,... and is regular elsewhere.

Definition 3.2. The subtracted constraint Dirichlet series is defined as

q(s) = [Q.sub.D](s) - [2.sup.-2s][[zeta](2s) - [[summation over (a[less than or equal to][D/2])][a.sup.-2s] = [summation over (a>[D/2])] {[1/[(4[a.sup.2] - [D.sup.2]).sup.s] - [1/[(4[a.sup.2]).sup.s]). (3.3)

Corollary 3.1. [q.sub.D](s) can be resumed as the contour integral

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

where the contour C runs parallel to the imaginary axis from c - i[infinity] to c + i[infinity] with -[D/2] - 1 < c < -[D/2], and [D/2] the integral part of D/2.

Proof. Adapting a variant of the integral representation of the zeta function due to Kloosterman [4],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)

where the contour runs parallel to the imaginary axis through the abscissa c. The integral representation can also be obtained by folding the contour to the left, running from -[infinity] back to -[infinity] enclosing the point-[D/2]-1 in an anticlockwise sense and applying the residue theorem.

Lemma 3.2. The Dirichlet series [q.sub.D](s) is regular for [sigma] > -1/2, O(1) for [absolute value of t][right arrow][infinity] and, with its general term groupedas [[(2a).sup.2] - [[D.sup.2]].sup.-s] - [(2a.sup.-2s,] converges absolutely for [sigma] > -1/2.

Proof. For a large compared to [absolute value of t], the convergence is the same as for t = 0, i.e. s = [sigma], which is a well known property of Dirichlet series. A binomial expansion of the general term of [q.sub.D]([sigma]),

[(2[alpha).sup.-2[sigma]] [[(1 - [D.sup.2]/[(2[alpha]).sup.2]).sup.-[sigma]] - 1]= O([[alpha].sup.-2[sigma]-2]), (3.6)

shows the absolute convergence and regularity for [sigma]> -1/2.

The convergence of the integral representation of [q.sub.D](s) maybe improved by including more terms of Stirling's asymptotic series [5]. This observation leads to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.7)

Now we deform the contour to enclose the real axis in the left-hand plane along two rays from the origin with opening angle [theta]/[absolute value of t] around [pi] for fixed [Imaginary part](s) = t [not equal to] 0 and -[pi] < [phi] < [pi], i.e. z = -[re.sup.i[phi][absolute value of t]] 0 is used to show that the general term is O(1) for t [right arrow][+ or -][infinity]. The conic area bounded by the rays is closed off by a small circular path around z = -[D/2] - 1. Stirling's series for the logarithmic derivative of the Gamma function applies outside the cone area guaranteeing convergence of the integral in Eq. (3.7). Each term in

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.8)

remains bounded, in absolute value, as we let t [right arrow][+ or -][infinity] for [sigma]> -1/2.

4. Twin Prime Dirichlet Series

The following product formula for absolutely converging Dirichlet series is one of our main tools; it is obtained from summing a well-known formula from which mean values of Dirichlet series are usually derived. We mention it for ease of reference and because details of its proof are needed in Sect. 6.

Lemma 4.1. (Product formula.) Let the Dirichlet series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.1)

be single-valued, regular and absolutely convergent for [sigma] > [[sigma].sub.a] and [sigma] > [[sigma].sub.b], respectively. Then the product series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.2)

is regular and converges absolutely for [Real part](w) [equivalent to] u > [sigma] + [[sigma].sub.b], [sigma] > [[sigma].sub.a] + 1. The limit of the integral exists and is a regular function of the variable w [equivalent to] u + iv.

Proof. Substituting the Dirichlet series and using [2], [4]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.3)

we obtain the first term on the rhs of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.4)

which converges absolutely because u [greater than or equal to] [[sigma].sub.a] + [[sigma].sub.b] + [epsilon].

To deal with [summation over (n[not equal to]m)] we split the summation m[not equal to] n into the ranges n [less than or equal to] m/2, m/2 < n < 2m, n [greater than or equal to] 2m. Since [absolute value of log n/m] [greater than or equal to] 1/log 2 for n [less than or equal to] m/2 we obtain the estimate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.5)

provided u - [sigma] > [[sigma].sub.b] and [sigma] > [[sigma].sub.a]. For n [greater than or equal to] 2m, we get the same estimate. For m/2 < n < 2m, we split the range m/2 < n < 2m into m/2 < n < m and m < n < 2m with n [not equal to] m and use

1/log(1 - [1/n]) = n + O(1), [1/log(1 - [2/n]) = [n/2] + O(1), ..., n[n-1.summation over (j=1)] [1/j] = O(n log n) (4.6)

to obtain the estimate

[summation over ([m/2]<n<m)] [1/[absolute value of log [n/m]] = O(m log m), (4.7)

and the same estimate for the range m < n < 2m. Putting all this together, we find for [sigma] > [[sigma].sub.a] + 1, u - [sigma] > [[sigma].sub.b] that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.8)

Using termwise differentiation with respect to the variable w in the integrals in conjunction with the estimate log n = O([n.sub.[epsilon]]) shows the absolute convergence of the termwise differentiated product series and the regularity of the product series.

As a first step toward constructing the twin prime Dirichlet series we apply the product formula to Z(s)[q.sub.D](s - w).

Theorem 4.1. For odd D>0, [Real part](w) = u>[sigma] + 3/2, [sigma] > 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.10)

where the limit of the integral on the rhs is a regular function for u > 5/2 and the asymptotic series A(w) converges for u > 1 /2.

Proof. Since Z(s) converges absolutely for [sigma] > 1, the product [Z(s)2.sup.-2w-2s][[zeta](2w - 2s) [summation over ([alpha][less than or equal to][D/2]) [a.sup.-2w+2s]] converges absolutely for u - [sigma] > 1/2, i.e. u > 3/2. Lemma 4.1 implies that the integral in Eq. (4.9) is O(T) and the limit T [right arrow] [infinity] exists representing a regular function for [sigma] > 5/2. The integral over [[Z(s)2.sub.-2(w-s)][[zeta](2(w - s)) - [summation over ([alpha][less than or equal to][D/2]) in Eq. (4.9) is O(T). We apply the product formula and obtain the first term on the rhs of Eq. (4.9) for u> 5/2.

5. The prime pair generating and asymptotic Dirichlet series

In order to analyze the generating Dirichlet series Z(s) introduced in Section 2 and find the Dirichlet series on the rhs ofEqs. (7.6), (4.10) in Theor. 4.1, which we call asymptotic Dirichlet series, we evaluate the arithmetic functions in their numerators.

Proposition 5.1. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be prime number decompositions.

Then

[summation over (d|n)][mu](d)[log.sup.2]d = [summation over (d|p)] [mu](d)[log.sup.2]d. (5.1)

If n = p is prime then

[summation over (d|p)][mu](d)[log.sup.2]d = -[log.sup.2]p. (5.2)

If n = [p.sub.i][p.sub.j] for prime numbers [p.sub.i][not equal to][p.sub.j] then

[summation over (d|[p.sub.i][p.sub.j])][mu](d)[log.sub.2] d = 2 log[p.sub.i] log[p.sub.j]. (5.3)

If k [greater than or equal to] 3 in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for different prime numbers [p.sub.i] then

[summation over (d|P)] (d) [log.sub.2] d = 0. (5.4)

Proof. Eq. (5.1) is obvious from the properties of the Mobius function, as is Eq. (5.2). Eq. (5.3) follows from

[summation over (d|[p.sub.i][p.sub.j])][mu](d) [log.sup.2]d = -[log.sup.2] [p.sub.i] - [log.sup.2] [p.sub.j] + [log.sup.2] [p.sub.i][p.sub.j] = 2log[p.sub.i]log [p.sub.j]. (5.5)

Eq. (5.4) for k = 3 follows from expanding [log.sup.2][p.sub.i][p.sub.j] = (log [p.sub.i] + log [p.sub.j]), etc.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.6)

For the general case k, Eq. (5.4) is proved by induction. Assuming its validity for k we can show that for k + 1:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.7)

This follows from verifying that the coefficients of [log.sup.2] [p.sub.i] are zero, as are those of 2log [p.sub.i][p.sub.j]. In essence, Eqs. (5.3,5.4) comprise Golomb's identity (1.1)for [LAMBDA](n), log n.

A comment on the constraint (2[alpha] - D, 2[alpha] + D) = 1 of Golomb's formula is in order. If d|2D and d|2a - D then d|2a + D; if d|2[alpha] - D, but d is not a divisor of 2D, then d is not a divisor of 2[alpha] + D. Thus, for odd D, if d|2[alpha] - D then d does not divide 2[alpha] + D, except for a finite number of divisors of D. Likewise, if d|3(2[alpha] - 1) - D and d does not divide 2D, then d does not divide 3 (2[alpha] - 1) + D. If a = b[delta] with [delta]|D then, for odd D, 2a - D = [delta](2b - [D/[delta]]), 2a + D = [delta](2b + [D/[delta]]) and [summation over (d| [[delta].sup.2]]([4b.sup.2]-[D.sup.2]/[[delta].sup.2])) [mu]d [log.sup.2]d = 0 follows from Lemma 5.1. For even D, if 2a - 1 = b[delta], [delta]|D with [delta] [not equal to] 2 then 9 [(2[alpha] - 1).sup.2] - [D.sup.2] = [[delta].sup.2] [9b.sup.2] - [D.sup.2]/ [[delta].sup.2] and [summation over (d|[[delta].sup.2]] ([9b.sup.2]- [D.sup.2]/[[delta].sup.2])) [mu](d) [log.sup.2] d = 0 follows again from Lemma 5.1. Thus, despite (2a - D, 2 + D) [not equal to] 1,(3 (2[alpha] - 1) - D, 3 [(2[alpha] - 1).sup.2] + D) [not equal to] 1, Golomb's identity is trivially satisfied for such values of a.

Lemma 5.1. The coefficient of the asymptotic Dirichlet series A(w) of Theorem 4.1 is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.8)

where p, [p.sub.i] are prime numbers [not equal to] 2.

Proof. This follows from Prop. 5.1. Using the prime number decomposition of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in conjunction with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.9)

the first and third lines of Eq. (5.8) are immediate consequences of Golomb's identity. The second line is a special case of [summation over (d|p)][mu](d)[log.sup.2] d = -[log.sub.2] p, where p is a prime number.

Theorem 5.1. For odd D > 0, A(w) of Eq. (4.10) in Theor. 4.1 becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.10)

Thus, A(w) has a simple pole at w = 1/2 with the positive residue log 2.

Proof. Summing the Dirichlet series A(w) using Lemma 5.1 yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.11)

from which Eq. (5.10) and the residue follow.

6. Limit of the Integral

In order to study the integral of Theor. 4.1 we now evaluate the contour integral

I(w) [equivalent to] [1/2[pi]i] [[integral].sub.C] Z(s)[q.sub.D](w - s)ds = [I.sub.1] + [I.sub.2] + [I.sub.3] + [I.sub.4], (6.1)

where the contour C is a rectangle with the vertices 1 + [epsilon] - iT, 1 + [epsilon]+iT, -[epsilon] + iT, - [epsilon] - iT and T [not equal to] [gamma], [rho] = [beta] + i[gamma] the general root of [zeta](s) in the critical strip, [I.sub.1] is the integral from 1 + [epsilon] - iT to 1 + [epsilon] + iT, [I.sub.2] from 1 + [epsilon] + iT, to -[epsilon] + iT, [I.sub.3] from - [epsilon] + iT, to -[epsilon] - iT and [I.sub.4] from -[epsilon] - iT to 1 + e - iT, [epsilon] > 0. The limit [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the integral [I.sub.1] in Eq. (4.9) of Theorem 4.1 needs to be analyzed.

Lemma 6.1. For u > 5/2

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.2)

Proof. Eq. (6.2) follows from using Eq. (2.4) and the residue theorem taking into account the poles at s = 1, [rho] inside the rectangular contour.

For the evaluation of [I.sub.2], [I.sub.4] we need bounds for Z(s), [q.sub.D](w - s) for s = [sigma] [+ or -] iT.

Lemma 6.2. [I.sub.2] = O ([log.sub.3] T), [I.sub.4] = O([log.sup.3] T) for u > 5/2 and all sufficiently large and appropriately chosen T.

Proof. We know [6] that for sufficiently large [absolute value of t] and -1 < [sigma] < 2

-[zeta]'(s)/[zeta](s) = - [summation over (p,[absolute value of t - [gamma]]]<1)] ([1/[s - [rho]]] + [1/[rho]]) + O(log[absolute value of t]). (6.3)

Choosing t = [Imaginary part](s), T = [absolute valule of t] sufficiently large and appropriately we can arrange that [absolute value of t - [gamma]][greater than or equal to] [1/log T], when [absolute value of [gamma] - t] < 1. There are at most O(log T) roots [rho] with [absolute value of [gamma] - t] < 1. As a result [6],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.4)

Hence [absolute value of Z(s)] = O ([log.sup.3] T) for [I.sub.2] and the same bound holds for [I.sub.4] and -[epsilon] [less than or equal to] [sigma] [less than or equal to] 1 + [epsilon].

On all four sides of the rectangular path -[epsilon] [less than or equal to] [sigma] [less than or equal to] 1 + [epsilon], [Real part](w - s) > -[3/2] for u > 3/2 the Dirichlet series [q.sub.D](w - s) remains bounded for u -[sigma] > 1/2, in absolute magnitude, for |t|[right arrow][infinity] according to Lemma 3.2.

Lemma 6.3. For u> 5/2, [I.sub.3] = O ([log.sup.2] T).

Proof. For [I.sub.3] we will use the functional equation (2.8) of Z(s) to evaluate the first term on the rhs of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.5)

[q.sub.D](w - s)Z(1 - s), by substituting the absolutely converging Dirichlet series. This yields a double series [summation over (m,n)] as for [I.sub.1] in Lemma 4.1 involving the integral [[integral].sup.T.sub.- T](m[[[(2n).sup.2] - [D.sup.2])].sup.it] dt. Thus, there is no contribution for [4n.sup.2] - [D.sup.2] = m that would be proportional to T. As earlier, [summation over (m [not equal to][(2n).sup.2] - [D.sup.2])] involves sin(T log[[(2n).sup.2] - [D.sup.2]]m) and is O(1) for u + [epsilon] > 3/2. The same applies to the double sums involving ([2n).sup.2] and [4n.sup.2] - [2D.sup.2] instead of [(2n).sup.2] - [D.sup.2.]

To deal with the term involving -[[zeta]'(s)]/[zeta](s)] 2([gamma] + log 2[pi]) [q.sub.D] (w - s) we use the functional equation of Lemma 2.2 and then substitute the absolutely converging Dirichlet series for [q.sub.D](w - s) and -[zeta]'(s)]/[zeta](s) at [sigma] = -[epsilon]. Again, each double series is obtained as for [I.sub.1] in Lemma 4.1 involving the integral [[integral].sup.T.sub.-T][(m[[(2n).sup.2] - [D.sup.2]).sup.it]dt, [[integral].sup.T.sub.-T][(m[(2n).sup.2]).sup.it] dt, or [[integral].sup.T.sub.-T][[m(4[n.sup.2] - 2[D.sup.2])].sup.it]dt, respectively. Again, there is no contribution for [(2n).sup.2] - [D.sup.2] = m, [(2n).sup.2] - 2[D.sup.2] = m, or [(2n).sup.2] = m, that would be proportional to T. As earlier, [summation over (m[not equal to][(2n).sup.2] - [D.sup.2])], [summation over (m=[(2n).sup.2] - 2[D.sup.2])] or [summation over (m[not equal to][(2n).sup.2]] lead to the bound O(1).

Putting all this together, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.6)

The term [1/2[pi]i] [[integral].sup.-[epsilon]+iT.sub.-[epsilon]-iT] [q.sub.D](w - s) [[([pi]/2).sup.2]/[cos.sup.2]s[pi]/2]ds = O(1) because [([cos.sup.2] s[pi]/2).sup.-2] = O([e.sup.-[absolute value of t][pi])] for t [right arrow] [+ or -][infinity].

The terms involving [absolute value of tan [s[pi]/2]] [right arrow] 1 for t [right arrow][+ or -][infinity], so [absolute value of [d/ds] tan [s[pi]/2]] + O([e.sup.-[absolute value of t][pi]]/2]). Using integration by parts, we get the following estimate for this term from the Dirichlet series for [q.sub.D] (w - s)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the integrated term is O(1) because the Dirichlet series is absolutely convergent, tan s[pi]/2 = O(1) and the integral is O(1). Terms involving [tan.sup.2] s[pi]/2, tan [(s[pi]/2)][[[GAMMA]'(s)/2[pi][GAMMA](s)] are handled in the same way.

The terms involving [q.sub.D](w - s) [[GAMMA]'(s)/[GAMMA](s)] are also treated by integration by parts where the integrated term is O(log T) and the integral is integrated by parts leading to [d/ds] [[GAMMA]'(s)]/[GAMMA](s)] = O([T.sup.-1]) at s = -[epsilon] [+ or -] iT. Hence the remaining integral is O(log T). The term involving [q.sub.D](w - s) [d/ds] [[GAMMA]'(s)/[GAMMA](s)] is treated by integration by parts leading to the same estimate. The term [q.sub.D](w - s) [([GAMMA]'(s)/[GAMMA](s)).sup.2] is also treated by integrating by parts leading to an O([log.sup.2]T) estimate.

The product [q.sub.D](w - s) [[zeta]'(1 - s)/[zeta](1 - s)] is another absolutely converging Dirichlet series without constant term, i.e. unity. All these terms in Eq. (6.6) are also treated by integration by parts leading to the estimate O(log T). Putting all this together proves Lemma 6.3.

Theorem 6.1. For odd D > 0, u > 5/2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.7)

Proof. We apply the limit T [right arrow] [infinity] on I(w) to Eqs. (6.1),(6.2) using the bounds on [I.sub.2],[I.sub.3],[I.sub.4] from Lemmas 6.2, 6.3 to obtain Eq. (6.7) because the first line of Eq. (6.2) and the first term of the second line drop out. The last line follows from Theor. 4.1 and Lemma 4.1.

This is our main result. Due to the limit T [right arrow] [infinity] in Theor. 6.1 the twin prime distributions depend on the asymptotic properties of the roots of the Riemann zeta function. This feature contrasts with the analytic proof of the prime number theorem, where the remainder term is linked to the roots of the Riemann zeta function producing the staircase-like corrections of the smooth asymptotic limit from all zeta function roots, whereas the leading asymptotic term has nothing to do with them, originating from the simple pole of the Riemann zeta function.

7. Twin Primes For Even D

Definition 7.1. The corresponding constraint generating Dirichlet series is defined as

[Q.sub.d](s) = [summation over(a[D/6])] [1/[[[3.sup.2][(2[alpha]-1).sup.2]] - [D.sup.2].sup.s], [sigma] > 1/2 (7.1)

with [D/6] the integer part of D/6.

Lemma 7.1. The expansion corresponding to Lemma 3.1 becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.2)

Definition 7.2. The subtracted constraint Dirichlet series is defined as

q(s) = [Q.sub.D](s) - [3.sup.-2s][(1 - [2.sup.-2s])[zeta](2s) - [summation over (a [less than or equal to] [D/6])] [(2a - 1).sup.-2s] (7.3)

Equation (3.7) of Lemma 3.2 for this case becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.4)

with -2[D/6] < c < -2[D/6] + 1. Lemma 3.2 is valid for this case with the general term grouped as

[1/[[3.sup.2][(2[alpha] - 1).sup.2]] - [D.sup.2].sup.s]] - [3.sup.-2s] [(2[alpha]).sup.-2s] (7.5)

Theorem 7.1. For even D> 0, [Real part](w) = u > [sigma] + [3/2], [sigma] > 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.6)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.7)

This is Theor. 4.1 for even D.

Lemma 7.2. The general coefficient of the asymptotic Dirichlet series A(w) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.8)

Theorem 7.2. For even D>0, A(w) ofEq. (4.10)in Theor. 4.1 becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.9)

Thus, A(w) has a simple pole at w = 1 /2 with the residue log3.

Theor. 6.1 remains valid, except for replacing the twin Dirichlet series in Eq. (6.7) by the lhs ofEq. (7.6) in Theor. 7.1.

8. Discussion and Conclusion

The twin prime formulas link the twin primes with the roots of the Riemann zeta function, which may be viewed as a step in Riemann's program linking prime numbers to these roots. Their dependence on the limit T [right arrow] o demonstrates that the twin prime distributions depend on the asymptotic properties of these roots. This qualitative feature differs fundamentally from the prime number theorem for ordinary prime numbers where all roots contribute to the remainder term.

References

[1] Golomb, S. W., 1970, "The Lambda Method in Prime Number Theory," J. Number Theory 2, pp. 193-198.

[2] Murty, M. R., 2001, Problems in Analytic Number Theory. Springer, New York, NY.

[3] Hardy, G. H. and Wright, E. M., 1988, An Introduction to the Theory of Numbers, Clarendon Press, Oxford, 5th ed.

[4] Titchmarsh, E. C. and Heath-Brown, D. R., 1986, The Theory of the Riemann Zeta Function, Clarendon Press, Oxford.

[5] Arfken, G. B. and Weber, H. J., 2005, Mathematical Methods for Physicists, ElsevierAcademic Press, 6th ed., Amsterdam.

[6] Ivic, A., 1985, The Riemann Zeta-Function, Dover, New York, NY.

H. J. Weber

Department of Physics

University of Virginia

Charlottesville, VA 22904, U.S.A.

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Author: | Weber, H.J. |
---|---|

Publication: | Global Journal of Pure and Applied Mathematics |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Apr 1, 2010 |

Words: | 4883 |

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