A CLOSER LOOK AT SOME NEW IDENTITIES

We obtain infinite products related to the concept of visible from the origin point vectors.
Among these is
∏k=3∞(1−Z)φ,(k)/k=11−Zexp(Z32(1−Z)2−12Z−12Z(1−Z)),  |Z|<1,
 in which φ3(k) denotes for fixed k, the number of positive integer solutions of (a,b,k)=1 where
a<b<k, assuming (a,b,k) is the gcd(a,b,k).


INTRODUCTION.
In several of the author's recent papers (see ) a class of new elementary infinite product identities was introduced.These were gven the name vpv identities, meaning visible (from the origin) pom/vec/or tdenttttes, due to the fact that they involved summation over so called visible lattice points in various dimensional spaces For an introduction to the idea of visible lattice points see Apostol [2], where their distribution Is calculated The identities which occurred most frequently in Campbell [7] were often related to each other by vpv lattice sums dividing up space into radial regions from the origin These identities when grouped were called compamon tden/ttles, because of their interdependence We used the operators defined by where is the set of positive integers less th and relatively prime to k Whilst these operators are usel for the 2-D sums, as shown in the author's papers [5-6, 9], we will not require them for the higher dimensional sums considered here Recently Mosseri [13], and Baake et al [3], have considered the so-called Optical Fourier Transform of the 2-D visible lattice points as an optical experiment The subject of lattice point methods applied to physics and chemstu is also examined n Frankel et al [10], and Glasser and Zucker [11]   Some of the methods of these papers are applicable to vpv sums and products In Campbell [5][6][7][8][9] the following companion identities were given.
(l-y)Z,(1 x'y)'' (l-Y) (x. ) ( 2) 1' (1-) (l-xey) '' (1 )'"'-"', (l 3) (1-y)(l-.)(1xy)l/t(1-xtya) ' (1 y)'"'-*', ( vaid respectively for I1 and y] < 1, Ixl and I1 < 1, ixl and lYl < 1. These led to ideas which li to several fields of research For example, new products over the primes were presented in Campbell [6-7,9], new Jacobi theta nction identities in [6] Paper [6] also contained results on Dirichlet summations such as those connected with Ramanujan arithmetical nctions, and was related to a method of Meinardus for obtaining asymptotic estimates of coeNcients from infinite products (see Andrews [1], Hardy and Wright [11]) In [7], some her identities denttes of ths type were given and the existence of yet others was indicated In the present note we gve results derived from considerations similar to those in [7] The underlying concept in the author's papers [5-9] is expressed in LEnA Consider an infinite region radiating out of the origin in any Euclidean vector space The set of all lattice point vectors from the origin in that region is precisely the set of positive nteger multiples of the visible point vectors (vpv's) in that region.
lmmedmte a pror consequences of this may be written.For example, val,d tbr )' and : < I, Yl and I.< < ', respectively It .4straightforward to generalize these results to any number of variables For example, wOx'yCz w,th (I 7) valid fbr tw[, x y] and ]z < whilst (1 8)is true if absolute values of z. ,,z.xz.yz, wxz, xyz, ,,,yz, wxyz are all less than unity Another consequence of the above considerations is THEOREM 2. Under the same conditions as (1 5) and (1 6) respectively, 1- a.b<This is true since logarithmic derivatives of both sides of (1 9) and (1.10) yield (1 5) and (1 6) respectively This establishes the theorem to within a constant factor in each product, and this constant may be determined from allowing both x and y to approach unity 2.
VPV IDENTITIES AND BOUNDARY LIMIT CASES.
The dentities of sectmn have interesting cases in which variables other than = approach unity (often the boundary of convergence) given certain modificauons In Campbell [5] we find the identity where p is the Euler totient function.An equivalent result was stated in Borwein and Borwein [4], I'I (1 y)"a) -.

=e (22)
where p is the Mobius function.We now establish THEOREM 2.1 where opt(k) denotes for fixed k the number of positive integer solutions of (a,b,k)= where a<b<k The function p (k) is a natural generalization of the Euler totient function p which is the number of positive integers less than and relatively prime to k In the recent paper by the author [7], it was shown that for absolute values of z, yz, xyz, all less than unity, l-I (1 x,,/,z,) -1, (a.b. )=! b.
A DIRICItLET SERIES VERSION OF A GENERAL VPV TItEOREM.
In the author's paper [7] we find the fundamental identity in vpv theory, THEOREM 3 If i= 1, 2, 3, ,n then for each Ix, 1< 1, and b, s C, with ',, b, 1, I-I "' x : x'" (3 ) Ths result came from iemma much in the manner shown above for theorem 2 lne of reasoning applies also to Dirichlet summations over the vpv lattices We now state THEOREM32 If/= 1,2, ,n then for eachb, C, Jlb,>l, Ths result appears also to be fundamental n some sense, but is not seen in any of the literature, except Ibr the case with n 2 given recently in Campbell [6] The latter case is so near the surface it s probably known The proof of (3 2) follows trivially from applying lemma to the left side series after multiplying both sides by y, ,,,:.z"a(' """ ") ((b' + b: +... + b, ).
(3 3) It may of course be argued that (3.2) is simply a limiting case of (3.1) Indeed generalization of the Euler totient function much in the style of our p (k) will give the generating Dirichlet series via (3 2)