Ramanujan J (2006) 12:267–293 DOI 10.1007/s11139-006-0078-y
An Euler product transform applied to q-series Geoffrey B. Campbell
Received: 18 August 2003 / Accepted: 8 August 2006 C Springer Science + Business Media, LLC 2006
Abstract This paper introduces the concept of a D-analogue. This is a Dirichlet series analogue for the already known and well researched hypergeometric q-series, often called the basic hypergeometric series. The main result in this paper is a transform, based on an Euler product over the primes. Examples given are D-analogues of the q-binomial theorem and the q-Gauss summation. Keywords Dirichlet series and zeta functions . Basic hypergeometric functions in one variable . Dirichlet series and other series expansions . Exponential series 2000 Mathematics Subject Classification Primary—11M41; Secondary—33D15, 30B50
1 Introduction We introduce a new concept into mathematical analysis; that of a D-analogue. This is a Dirichlet series analogue for the already known and well researched hypergeometric q-series, often called the basic hypergeometric series. The basic hypergeometric series, whilst widely used in mathematics, is itself an analogue for the ordinary hypergeometric series developed by Gauss in the early nineteenth century. A comparison shows that the original Gauss series is 2 F1 (a, b; c; z)
= 1+ +
ab a(a + 1)b(b + 1) 2 z+ z c1! c(c + 1)2!
a(a + 1)(a + 2)b(b + 1)(b + 2) 3 z + ···, c(c + 1)(c + 2)3!
(1.1)
G. B. Campbell () Department of Mathematics, La Trobe University, Bundoora, Victoria, 3086, Australia e-mail:
[email protected] Springer
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G. B. Campbell
a particular case due to Gauss being (c)(c − a − b) . (c − a)(c − b)
=
2 F1 (a, b; c; 1)
(1.2)
Now, the q-series analogue of these due to Heine [26, 27] in 19th century is φ(a, b; c; q, z) ≡ 2 φ1 (a, b; c; q, z) =
∞ (a; q)k (b; q)k k=0
(q; q)k (c; q)k
zk ,
(1.3)
where the standard notation for q shifted factorials is used, (a; q)k =
1, k = 0; (1 − a)(1 − aq) · · · (1 − aq k−1 ), k = 1, 2, . . .,
so then (a; q)∞ may be inferred. The corresponding q-analogue of the Gauss summation (1.2) above is 2 φ1 (a, b; c; q, c/ab)
=
(c/a; q)∞ (c/b; q)∞ . (c; q)∞ (c/ab; q)∞
(1.4)
Note the resemblance between (1.2) and (1.4) in occurrence of the a, b, c variables. A new D-analogue version of the Gauss and Heine summations (1.1) and (1.3) is in the series 2 1 (a, b; c; γ , z)
=
∞ σ−γ (a; k)σ−γ (b; k) 1 . σ−γ (c; k) kz k=1
(1.5)
In the right side we apply the sum of nth powers of divisors of k function σn (k) to define σ−γ (a; 1) = 1, and for positive integers k ≥ 2, σ−γ k p|k p j . σ−γ (a; k) = j p|k p j=0 σ−γ a−2
(1.6)
Before stating Theorems 1.1 and 1.2 we remark that their proofs are given in later sections. A striking D-analogue of (1.2) and (1.4) is, Theorem 1.1. For positive integers a, b and c and γ ≥ 0, such that cγ , (c − a − b)γ , (c − a)γ , (c − a − b)γ , are each > 1, 2 1 (a, b; c; γ , (c
Springer
− a − b)γ ) =
ζ (c; γ )∞ ζ (c − a − b; γ )∞ . ζ (c − a; γ )∞ ζ (c − b; γ )∞
(1.7)
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In the right side we apply the Riemann zeta function ζ (a) for positive integers a to define for positive integers n ζ (a; γ )n =
n−1
ζ ((a + k)γ ),
(1.8)
k=0
with the extension to ζ (a; γ )∞ as n → ∞. Our notation for σ−γ (a; k) is suggestive of the relation to a multidimensional divisor function. We shall explore this interpretation for σ−γ (a; k) in later papers. This type of function often appears in the “Dirichlet series generated” coefficients of a D-series, corresponding to the idea in analogy that the q-shifted factorial function (a; q)k occurs in the generated coefficients of powers of z in a q-series such as (1.3). In the present paper, we show representatives of two separate classes of D-analogues. One of these involves the above ζ (a; γ )n function, and the other involves the Jordan totient function Jn (k) extended in a similar way to the manner of our extending ζ (a) into ζ (a; γ )n . Later papers will give further D-series analogues of q-series in which an Euler product is taken over a finite set of primes to arrive at the new D-summations. We therefore examine a new class of Dirichlet series based on the already known and well established q-series; an example these being (1.7). In this paper we feature as examples of our new transform, the D-binomial theorem, and the D-analogue of the q-Gauss 2 F1 summation formula given above as (1.7). In later papers similar summations will be given for analogues of q-Kummer, the q-Dixon sums and other well known formulae and transformations. This paper examines a new class of Dirichlet series transformations based on known q-series identities, these new D-analogues arrived at from applying an Euler product operator over all primes. We also show that with a partial product in the operator, the same rationale will lead to yet another set of D-analogues from the same q-series under transform. t Assume the positive integer prime decomposition m = i=1 piai , and that we ast bi + sociate a set Sm = {x ∈ Z : x = i=1 pi for each bi a non-negative integer} with this number. Then for positive integers n we have new q-binomial theorem analogues Theorem 1.2. For positive integers n, β > 1, γ > 0, ∞ σ−γ (n; k)
kβ
k=1
=
n
ζ (β + kγ ),
(1.9)
k=0
n σ−γ (n; k)λ(k) 1 , = β k p|m p k∈Sm k=0 σ−(β+kγ )
(1.10)
where λ(k) is the Liouville function, λ(k) =
t i=1
(−1)ai
for each k =
t
piai .
(1.11)
i=1
See [9, 25, 31, 33] for classical accounts describing this function. Springer
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Similar summations to (1.7), (1.9) and (1.10) are easily found as further analogues for the q-Gauss, q-Kummer, and the q-Dixon sums, as well as Jackson’s and Watson’s formulae; particular cases of which include the Rogers-Ramanujan identities. In a further research project, we will extend this concept to write down D-analogues for the bilateral q-series and state some analogues of classical q-series transformations. This will include the Jacobi triple product and the quintuple and sextuple product formulae. A further natural extension of these ideas is in the study of D-analogues for the q-gamma and q-beta functions. The likely forms of such D-series analogues would give corresponding results for the classical q-gamma and q-beta functions. It would also be interesting to explore contiguous relations between the new analogues, as these type of equations have traditionally yielded new and beautiful continued fraction formulae.
2 Euler product Dirichlet series analogues of q-series In this paper we obtain new Dirichlet series summations that would not naturally be adduced from the known approaches to the study of Dirichlet series and arithmetical functions. These “D-analogues” of q-series follow by replacing the q-series variables ai , bi , . . . and q, by p −ai γ , p −bi γ , . . . and p −γ , where p is prime, then forming an Euler product over primes on both sides of the resulting summation formula. As in the author’s PhD thesis [19], this transform yields Dirichlet series whose generated coefficients are rational functions of “sum of powers of divisors” functions. Sivaramakrishnan [31] gives a beautiful exposition of these arithmetical functions, albeit not in the same context as our current treatment. His approach is recommended reading as he deals also with the Jordan totient function which we apply in this paper for the partial Euler product transforms. We follow Hardy and Wright [25] by defining a Dirichlet series as a series of the form F(s) =
∞ αk k=1
ks
.
(2.1)
The variable s may be real or complex, and F(s) is called the (Dirichlet) generating function of the sequence (αk )∞ k=1 . We say also that the series (2.1) is a “D-series” and we also call it a “D-analogue” if it demonstrably maps from a q-series, an example of which is (1.3), by way of applying an Euler product mapping. We shall encounter “D-analogues” of q-series identities or transformations, which will give new Dirichlet series generating functions in the shape of D-series identities and transformations. We will compare results arisen from a finite product over primes, with results arisen from an infinite product over primes. Under the infinite Euler product transform, a product of q-shifted factorials becomes a product of Riemann zeta functions. In the finite Euler product cases, the results feature the Jordan totient function and the sum of powers of divisors functions. These two functions in the partial Euler product transformed identities play the role of the Riemann zeta function in the results obtained from Euler product transform taken over all primes. Springer
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With either kind of D-analogue, the resulting generating functions are relatively simple. For example, the right side of (1.9) would be easily noticed in browsing references. Yet such functions do not appear in the well-known and often-cited texts such as Hardy and Wright [25], Apostol [9], Sivaramakrishnan [31], or Lang [29]. Interest arises nowadays in Dirichlet series’ link with thermodynamics. For example Ninham et al [30], and Glasser and Zucker [24] have obtained applicable new Dirichlet series from considering properties of integer lattice sums. Their methods in places can be applied to the summations throughout this paper. The work of Huxley [28] on the theory of lattice points and exponential sums overlaps into the realms of the possible directions of future research stemming from the current paper’s D-series. In the author’s [16–18], a transform involving infinite products over all primes yielded finite products of Riemann zeta functions as the generating functions. In this chapter we derive both finite and infinite products over primes and the reader may observe that the method is similar in the earlier papers. For present purposes we make use of the Jordan totient function Jγ (k) = k γ
t
−γ
1 − pi
,
(2.2)
i=1
t where γ ≥ 0, and k = i=1 piai the unique prime decomposition of k (see Sivaramakrishnan [31]). We also make use of the sum of powers of divisors function σ−γ (k) =
d
−γ
=k
−γ
d|k
Suppose m =
t i=1
−(a +1)γ t 1 − pi i . σγ (k) = −γ 1 − pi i=1
(2.3)
piai the unique prime decomposition of m and let
+
Sm = x ∈ Z : x =
t
pibi
for each non-negative integer bi .
(2.4)
i=1
Since we will frequently employ this set to sum upon and form products over, it is prudent to now give examples of this. We therefore have cases of (2.4) as follows: S2 = S4 = S8 = S16 = · · · = {1, 2, 4, 8, 16, . . .}, S3 = S9 = S27 = S81 = · · · = {1, 3, 9, 27, 81, . . .}, S6 = S12 = S18 = S24 = S36 = · · · = {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, . . .}, S10 = S20 = S40 = S50 = S80 = · · · = {1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, . . .}.
In order to set the scene for our transform theorem, we consider several applications of Euler products over the primes, or over the primes dividing the integer m. Springer
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Lemma 2.1. If we take γ > 0, and suitably chosen z, then
1+
p
=
1 − p −(1+1)γ 1 1 − p −(2+1)γ 1 1 − p −(3+1)γ 1 + + + · · · 1 − p −γ p z 1 − p −γ p 2z 1 − p −γ p 3z
1+
p
=
k=1
∞ σ−γ (k)
kz
k=1
and
p|m
=
1 − p −(k+1)γ 1 1 − p −γ p kz
(2.5a) (2.5b)
;
(2.5c)
1 − p −(1+1)γ 1 1 − p −(2+1)γ 1 1 − p −(3+1)γ 1 1+ + + + ··· 1 − p −γ p z 1 − p −γ p 2z 1 − p −γ p 3z
1+
p|m
=
∞
∞ 1 − p −(k+1)γ 1 1 − p −γ p kz k=1
(2.6a) (2.6b)
∞ σ−γ (k) . kz k∈Sm
(2.6c)
These are easy to see once we realize that a typical term in the expansion of (2.5a) or (2.6a) is, by application of (2.3),
−(a +1)γ t 1 − pi i σ−γ (k) 1 = . ai z −γ kz p 1 − pi i i=1
(2.7)
The idea of the series and product expansions of (2.5a) to (2.6c) is fundamental in the following chapters of this paper. Further simple extensions of such results are required for our work. We next present them in (2.8a) to (2.12b). Lemma 2.2. For the conditions of Lemma 2.1 we have for any positive integer a,
1+
p
1 − p −(1+a)γ 1 1 − p −(2+a)γ 1 1 − p −(3+a)γ 1 + + + ··· −γ z −γ 2z 1− p p 1− p p 1 − p −γ p 3z
∞ 1 − p −(k+a)γ 1 = 1+ 1 − p −γ p kz p k=1 a−1 ∞ σ −γ k p|k p = ; kz k=1
Springer
(2.8a) (2.8b)
(2.8c)
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and
p|m
=
1 − p −(1+a)γ 1 1 − p −(2+a)γ 1 1 − p −(3+a)γ 1 1+ + + + ··· 1 − p −γ p z 1 − p −γ p 2z 1 − p −γ p 3z
1+
p|m
∞ 1 − p −(k+a)γ 1 1 − p −γ p kz k=1
σ−γ k p|k pa−1
=
kz
k∈Sm
(2.9a) (2.9b)
.
(2.9c)
Extending this idea a little further, we see that, Lemma 2.3. For the conditions of Lemma 2.1 we have for any positive integers a, b,
p
=
1 − p −(1+a)γ 1 1 − p −(2+a)γ 1 1 − p −(3+a)γ 1 1+ + + + ··· 1 − p −(1+b)γ p z 1 − p −(2+b)γ p 2z 1 − p −(3+b)γ p 3z
1+
p
∞ 1 − p −(k+a)γ 1 1 − p −(k+b)γ p kz k=1
(2.10a) (2.10b)
a−1 ∞ σ 1 −γ k p|k p ; = b−1 k z k σ p p|k k=1 −γ
(2.10c)
and
1+
p|m
=
p|m
1 − p −(1+a)γ 1 1 − p −(2+a)γ 1 1 − p −(3+a)γ 1 + + + ··· 1 − p −(1+b)γ p z 1 − p −(2+b)γ p 2z 1 − p −(3+b)γ p 3z
1+
∞ k=1
1 − p −(k+a)γ 1 1 − p −(k+b)γ p kz
σ−γ k p|k pa−1 1 . = b−1 k z p|k p k∈Sm σ−γ k
(2.11a) (2.11b)
(2.11c)
We observe that (2.10) and (2.11) are easy extensions of (2.8) and (2.9) because we are performing a multiplicative operation on the Dirichlet series coefficients. Indeed, we take this further, so that we may ultimately apply this idea to the generic q-series. We can next show that Springer
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Lemma 2.4. If ai and bi are positive integers then
1 − p −(k+a1 )γ 1 − p −(k+a2 )γ · · · 1 − p −(k+ar )γ 1 1+ (2.12a) −(k+b1 )γ 1 − p −(k+b2 )γ · · · 1 − p −(k+br −1 )γ p kz p k=1 1 − p a1 −1 ∞ σ σ−γ k p|k pa2 −1 · · · σ−γ k p|k par −1 1 −γ k p|k p ; = b1 −1 σ b2 −1 . . . σ br −1 −1 k z −γ k −γ k p|k p p|k p p|k p k=1 σ−γ k
∞
(2.12b) and also that
1 − p −(k+a1 )γ 1 − p −(k+a2 )γ · · · 1 − p −(k+ar )γ 1 1+ (2.13a) −(k+b1 )γ 1 − p −(k+b2 )γ · · · 1 − p −(k+br −1 )γ p kz p|m k=1 1 − p σ−γ k p|k pa1 −1 σ−γ k p|k pa2 −1 · · · σ−γ k p|k par −1 1 . = b1 −1 σ b2 −1 . . . σ br −1 −1 k z −γ k −γ k p|k p p|k p p|k p k∈Sm σ−γ k
∞
(2.13b)
Now, the usual basic hypergeometric series as given in the classical monograph by Gasper and Rahman [23] is r φs (A1 ,
A2 , . . . , Ar ; B1 , B2 , . . . , Bs ; q, x) A1 , A2 , . . . , Ar ; q, x ≡ r φs B1 , B2 , . . . , Bs
=
∞ k=0
n 1+s−r (A1 ; q)k (A2 ; q)k · · · (Ar ; q)k (−1)q ( 2 ) xk, (q; q)k (B1 ; q)k (B2 ; q)k · · · (Bs ; q)k
(2.14a) (2.14b) (2.14c)
with ( n2 ) = n(n−1) , and q = 0 when r > s + 1. We assume the Bi terms in this are 2 chosen so that no denominator is zero. The series under the product operators in Equations (2.5a), (2.6a), (2.8a), (2.9a), (2.9a), (2.10a), (2.11a), (2.12a) and (2.13a) are basic hypergeometric series like (2.14c) in which r = s + 1, q = p −γ , Ai = p −ai γ , Bi = p −bi γ , x = p −z (with p prime), and |ai − bi−1 | is a positive integer. Therefore, substitutions such as this on (2.14) may lead to an expression like (2.12b) or (2.13b) if a product over all primes is performed. We now consider some formulae using q-shifted factorials which will assist our simplification. Observe that for positive integers a and n, and for primes p,
p −aγ ; p −γ
n
( p −γ ; p −γ )n Springer
=
1 − p −(n+2)γ · · · 1 − p −(n+b−1)γ . (2.15) (1 − p −γ ) (1 − p −2γ ) · · · 1 − p −(a−1)γ
1 − p −(n+1)γ
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Likewise, for b and c positive integers, dividing cases of (2.15) by each other gives 1 − p −(n+1)γ 1 − p −(n+2)γ · · · 1 − p −(n+a−1)γ ( p −bγ ; p −γ )n = ( p −cγ ; p −γ )n 1 − p −(n+1)γ 1 − p −(n+2)γ · · · 1 − p −(n+c−1)γ (1 − p −γ )(1 − p −2γ ) · · · 1 − p −(c−1)γ . × (2.16) (1 − p −γ )(1 − p −2γ ) · · · 1 − p −(b−1)γ Using this method we can build a generated coefficient term corresponding to that on the right side of (2.14c) where, as above, r = s + 1, q = p −γ , Ai = p −ai γ , Bi = p −bi γ , x = p −z (with p prime). It is ( p −a1 γ ; p −γ )n ( p −a2 γ ; p −γ )n ( p −a3 γ ; p −γ )n ( p −ar γ ; p −γ )n · · · ( p −γ ; p −γ )n ( p −b1 γ ; p −γ )n ( p −b2 γ ; p −γ )n ( p −br −1 γ ; p −γ )n 1 − p −(n+1)γ 1 − p −(n+2)γ · · · 1 − p −(n+a1 −1)γ = (1 − p −γ )(1 − p −2γ ) · · · (1 − p −(a1 −1)γ ) 1 − p −(n+1)γ 1 − p −(n+2)γ · · · 1 − p −(n+a2 −1)γ × 1 − p −(n+1)γ 1 − p −(n+2)γ · · · 1 − p −(n+b1 −1)γ (1 − p −γ )(1 − p −2γ ) · · · 1 − p −(b1 −1)γ × (1 − p −γ )(1 − p −2γ ) · · · 1 − p −(a2 −1)γ 1 − p −(n+1)γ 1 − p −(n+2)γ · · · 1 − p −(n+a3 −1)γ × 1 − p −(n+1)γ 1 − p −(n+2)γ · · · 1 − p −(n+b2 −1)γ (1 − p −γ )(1 − p −2γ ) · · · 1 − p −(b2 −1)γ × (1 − p −γ )(1 − p −2γ ) · · · 1 − p −(a3 −1)γ .. .
1 − p −(n+1)γ 1 − p −(n+2)γ · · · 1 − p −(n+ar −1)γ 1 − p −(n+1)γ 1 − p −(n+2)γ · · · 1 − p −(n+br −1 −1)γ (1 − p −γ )(1 − p −2γ ) · · · 1 − p −(br −1 −1)γ . × (1 − p −γ )(1 − p −2γ ) · · · 1 − p −(ar −1)γ
×
(2.17)
We will use this soon to deal with the expression
p −a1 γ , p −a2 γ , . . . , p −ar γ ; p −b1 γ , p −b2 γ , . . . , p −b(r −1) γ ; p −γ , p −z −a γ −a γ p 1 , p 2 , . . . , p −ar γ ; p −γ , p −z ≡ r φr −1 p −b1 γ , p −b2 γ , . . . , p −b(r −1) γ
r φr −1
=
∞ k=0
(2.18a) (2.18b)
( p −a1 γ ; p −γ )k ( p −a2 γ ; p −γ )k · · · ( p −ar γ ; p −γ )k p −kz . (2.18c) ( p −γ ; p −γ )k ( p −b1 γ ; p −γ )k ( p −b2 γ ; p −γ )k · · · p −b(r −1) γ ; p −γ k Springer
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From (2.17) used in (2.18) we obtain the following two lemmas. They are important for helping us to prove the required transform from q-series into D-series, giving us a myriad of results such as Theorems 1.1 and 1.2. Lemma 2.5. For positive integers ai , bi , with z chosen for convergence,
r φr −1
p −a1 γ , p −a2 γ , . . . , p −ar γ ; p −γ , p −z
p −b1 γ , p −b2 γ , . . . , p −b(r −1) γ
p
=
σ−γ k p|k p j a −2 σ−γ k p|k p j ar −2 σ−γ k p|k p j 2 ··· j=0 σ j=0 σ j=0 σ ∞ j j j −γ −γ −γ p|k p p|k p p|k p b1 −2 σ−γ k p|k p j b2 −2 σ−γ k p|k p j br −1 −2 σ−γ k p|k p j k=0 ··· j=0 σ j=0 σ j=0 j j j σ−γ −γ −γ p|k p p|k p p|k p
=
∞ σ−γ (a1 ; k)σ−γ (a2 ; k) · · · σ−γ (ar ; k) 1 . σ (b ; k)σ−γ (b2 ; k) · · · σ−γ (br −1 ; k) k z k=0 −γ 1
a1 −2
1 kz
(2.19)
Lemma 2.6. For positive integers ai bi , with z chosen for convergence,
r φr −1
k∈Sm
=
p −b1 γ , p −b2 γ , . . . , p −b(r −1) γ
p|m
=
p −a1 γ , p −a2 γ , . . . , p −ar γ ; p −γ , p −z
σ−γ k p|k p j a −2 σ−γ k p|k p j ar −2 σ−γ k p|k p j 2 ··· j=0 σ j=0 σ j=0 σ j j j −γ −γ −γ p|k p p|k p p|k p br −1 −2 σ−γ k p|k p j b1 −2 σ−γ k p|k p j b2 −2 σ−γ k p|k p j ··· j=0 σ j=0 σ j=0 j j j σ−γ −γ −γ p|k p p|k p p|k p
a1 −2
σ−γ (a1 ; k)σ−γ (a2 ; k) · · · σ−γ (ar ; k) 1 . σ (b ; k)σ−γ (b2 ; k) · · · σ−γ (br −1 ; k) k z k∈Sm −γ 1
1 kz
(2.20)
The proof of Lemmas 2.5 and 2.6 follows simply from straightforward application of (2.17) and the definition of σ−γ (a; k) given in (1.6). A minor adjustment to the term containing z, and remembrance of the definition of the Liouville function given earlier at (1.11) accounts for the next two lemmas, which stated simply are: Lemma 2.7. For positive integers ai , bi , with z chosen for convergence,
r φr −1
p
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p −a1 γ , p −a2 γ , . . . , p −ar γ ; p −γ , p −z p −b1 γ , p −b2 γ , . . . , p −b(r −1) γ
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σ−γ k p|k p j a −2 σ−γ k p|k p j ar −2 σ−γ k p|k p j 2 ··· j=0 σ j=0 σ j=0 σ ∞ j j j −γ −γ −γ p|k p p|k p p|k p j j b1 −2 σ−γ k p|k p b2 −2 σ−γ k p|k p br −1 −2 σ−γ k p|k p j k=0 ··· j=0 σ j=0 σ j=0 pj pj σ pj
a1 −2
=
−γ
=
∞ k=0
−γ
p|k
−γ
p|k
λ(k) kz
p|k
σ−γ (a1 ; k)σ−γ (a2 ; k) · · · σ−γ (ar ; k) λ(k) . σ−γ (b1 ; k)σ−γ (b2 ; k) · · · σ−γ (br −1 ; k) k z
(2.21)
Lemma 2.8. For positive integers ai , bi , with z chosen for convergence,
r φr −1
k∈Sm
=
p −b1 γ , p −b2 γ , . . . , p −b(r −1) γ
p|m
=
p −a1 γ , p −a2 γ , . . . , p −ar γ ; p −γ , p −z
σ−γ k p|k p j a −2 σ−γ k p|k p j ar −2 σ−γ k p|k p j 2 ··· j=0 σ j=0 σ j=0 σ j j j −γ −γ −γ p|k p p|k p p|k p b1 −2 σ−γ k p|k p j b2 −2 σ−γ k p|k p j br −1 −2 σ−γ k p|k p j ··· j=0 σ j=0 σ j=0 j j j σ−γ −γ −γ p|k p p|k p p|k p
a1 −2
σ−γ (a1 ; k)σ−γ (a2 ; k) · · · σ−γ (ar ; k) λ(k) . σ (b ; k)σ−γ (b2 ; k) · · · σ−γ (br −1 ; k) k z k∈Sm −γ 1
λ(k) kz
(2.22)
3 Some further required notation In this section we create the notation for us to be able to see the features in analogy between the q-analogues and their D-analogues. The choice of notation mimics, where possible, the q-series notation. Firstly we use the notation given in Gasper and Rahman [23] for a product of q-shifted factorials: Definition 3.1. (a1 , a2 , . . . , ar ; q)n = (a1 ; q)n (a2 ; q)n · · · (ar ; q)n .
(3.1)
Then, let us create the following definitions and notations. Firstly we enlarge the definition given previously in (1.8), Definition 3.2. If ζ (a) is the Riemann zeta function and γ chosen such that the functions all exist, define for positive integers n,
ζ (a; γ )n =
n−1 k=0
ζ ((a + k)γ ) =
p
1 ( p −aγ ;
p −γ )n
,
ζ (a1 , a2 , . . . , ar ; γ )n = ζ (a1 ; γ )n ζ (a2 ; γ )n · · · ζ (ar ; γ )n .
(3.2) (3.3) Springer
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This definition will bring a required notation for the D-analogue of the normal gamma function (z) or its q-analogue q (z) found in the q-series literature such as Gasper and Rahman [23]. Definition 3.3. If σk (a) is the sum of kth powers of the divisors of positive integer a as in (1.6) then for positive integers n, σ−γ k p|k p j , σ−γ (a; k) = j j=0 σ−γ p|k p a−2
defined as 1 at a = 1,
(3.4)
and σ−γ (a1 , a2 , . . . , ar ; k) = σ−γ (a1 ; k)σ−γ (a2 ; k) · · · σ−γ (ar ; k).
(3.5)
With these definitions, Lemmas 2.5 and 2.6 become respectively, Theorem 3.1. For positive integers ai , bi , with z chosen for convergence,
r φr −1
p
∞ σ−γ (a1 , a2 , . . . , ar ; k) 1 p −a1 γ , p −a2 γ , . . . , p −ar γ ; p −γ , p −z = . −b1 γ −b2 γ −b(r −1) γ p ,p ,..., p σ−γ (b1 , b2 , . . . , br −1 ; k) k z k=0
(3.6)
Theorem 3.2. For positive integers ai , bi , with z chosen for convergence,
r φr −1
p|m
σ−γ (a1 , a2 , . . . , ar ; k) 1 p −a1 γ , p −a2 γ , . . . , p −ar γ ; p −γ , p −z = . p −b1 γ , p −b2 γ , . . . , p −b(r −1) γ σ (b , b , . . . , br −1 ; k) k z k∈Sm −γ 1 2 (3.7)
Similarly, Lemmas 2.7 and 2.8 become respectively, Theorem 3.3. For positive integers ai , bi , with z chosen for convergence,
r φr −1
p
∞ σ−γ (a1 , a2 , . . . , ar ; k) λ(k) p −a1 γ , p −a2 γ , . . . , p −ar γ ; p −γ , p −z = . −b1 γ −b2 γ −b(r −1) γ p ,p ,..., p σ−γ (b1 , b2 , . . . , br −1 ; k) k z k=0
(3.8) Theorem 3.4. For positive integers ai , bi , with z chosen for convergence, −a γ −a γ σ−γ (a1 , a2 , . . . , ar ; k) λ(k) p 1 , p 2 , . . . , p −ar γ ; p −γ , p −z = . r φr −1 p −b1 γ , p −b2 γ , . . . , p −b(r −1) γ σ (b , b , . . . , br −1 ; k) k z p|m k∈Sm −γ 1 2
(3.9) Springer
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Theorems 3.1 to 3.4 above demonstrate the methodology of our work in this paper. For example, Theorem 3.1 is nothing more than a hypergeometric-style generalization of the classical result due to Euler,
p
1 1 1 1 1 1 1 + s + 2s + 3s + · · · = 1 + s + s + s + · · · . p p p 2 3 4
(3.10)
So, it would seem that in order to reach the goal of defining the D-series analogue of q-series, we need terminology for the expressions on the left sides of (3.6) to (3.9). Clearly, in the these instances we have gone some way towards this, but so far we have found results pertaining to D-analogues of r φr −1 q-series. We shall content ourselves with this limitation for now, as it still gives us a wealth of new analogue summation formulae to work with. Continuing in this manner, we invent the following notation: Definition 3.4. For positive integers ai , bi , with z > 0 and chosen for convergence, define r r −1 (a1 , a2 , . . . , ar ; b1 , b2 , . . . , br −1 ; γ , z)
≡ r r −1 =
a1 , a2 , . . . , ar ; γ , z
b1 , b2 , . . . , br −1
∞ σ−γ (a1 , a2 , . . . , ar ; k) 1 . σ (b , b , . . . , br −1 ; k) k z k=1 −γ 1 2
(3.11)
Definition 3.5. For positive integers ai , bi , with z > 0 and chosen for convergence, define r r −1 (m
| a1 , a2 , . . . , ar ; b1 , b2 , . . . , br −1 ; γ , z) m | a1 , a2 , . . . , ar ; γ , z
≡ r r −1 =
b1 , b2 , . . . , br −1
σ−γ (a1 , a2 , . . . , ar ; k) 1 . σ (b , b , . . . , br −1 ; k) k z k∈Sm −γ 1 2
(3.12)
Definition 3.6. For positive integers ai , bi , with z > 0 and chosen for convergence, define r r −1 (a1 , a2 , . . . , ar ; b1 , b2 , . . . , br −1 ; γ , −\z)
≡ r r −1
m | a1 , a2 , . . . , ar ; γ , −\z
b1 , b2 , . . . , br −1
∞ σ−γ (a1 , a2 , . . . , ar ; k) λ(k) = . σ (b , b , . . . , br −1 ; k) k z k=0 −γ 1 2
(3.13) Springer
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Definition 3.7. For positive integers ai , bi , with z > 0 and chosen for convergence, define r r −1 (m
| a1 , a2 , . . . , ar ; b1 , b2 , . . . , br −1 ; γ , −\z) m | a1 , a2 , . . . , ar ; γ , −\z
≡ r r −1 =
b1 , b2 , . . . , br −1
σ−γ (a1 , a2 , . . . , ar ; k) λ(k) . σ (b , b , . . . , br −1 ; k) k z k∈Sm −γ 1 2
(3.14)
Note the use of −\z to account for a negative value in the q-series, which becomes pertinent in the D-series analogue. Hence we see the importance of knowing whether the real part of z is positive or negative, and in the interesting instance where z = 0, the coefficients will require a more sophisticated analysis to determine whether the series is defined. We shall leave this issue to another paper. Note that as m in (3.12) or (3.14) nbecomes more highly composite in the sense that m ∈ {2, 2.3, 2.3.5, 2.3.5.7, . . . , i=1 pi , . . .}, as n → ∞, then (3.12) tends to (3.11), and (3.14) tends to (3.13). The negative z in each first two lines of definitions 3.6 and 3.7 indicates that the resultant D-series came from transforming a q-series power series in z < 0 leading to the Liouville function in the λ(k)k −z term after the Euler product transform. We have shown in the above workings how an oscillating termed q-series can be dealt with in the Euler product transform to a D-series. We have, until now, required our parameters ai (for 1 ≤ i ≤ r ) and bi (for 1 ≤ i ≤ (r − 1)) to be positive integers. We next show what the transform becomes for some of the ai and bi as negative integers under certain paired conditions. To do this we now let our Euler product transform apply in the following set of circumstances. Let us write A1 , A2 , . . . , Ar > 0; and Ar +1 , Ar +2 , . . . , Ar +s < 0; whilst B1 , B2 , . . . , Br −1 > 0; and Br , Br +1 , . . . , Br +s−1 < 0. Next let us write Ar +1 = −C1 , Ar +2 = −C2 , . . . , Ar +s = −Cs ;
and
Br = −D1 , Br +1 = −D2 , . . . , Br +s−1 = −Ds , so then
Springer
A1 , A2 , . . . , Ar > 0;
(3.15a)
B1 , B2 , . . . , Br −1 > 0;
(3.15b)
C1 , C2 , . . . , Cs > 0;
(3.15c)
D1 , D2 , . . . , Ds > 0.
(3.15d)
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Under these conditions, the usual basic hypergeometric series as given in Gasper and Rahman [23] and in equations (2.14a) to (2.14c) of Section 2 reads as r +s φr +s−1 (A 1 ,
A2 , . . . , Ar , −C1 , −C2 , . . . , −Cs ;
B1 , B2 , . . . , Br −1 , −D1 , −D2 , . . . , −Ds ; q, x) A1 , A2 , . . . , Ar , −C1 , −C2 , . . . , Cs ; q, x ≡ r +s φr +s−1 B1 , B2 , . . . , Br −1 , −D1 , −D2 , . . . , −Ds ∞
(A1 ; q)k (A2 ; q)k · · · (Ar ; q)k (−C1 ; q)k (−C2 ; q)k · · · (−Cs ; q)k xk, (q; q) k (B1 ; q)k (B2 ; q)k · · · (Br −1 ; q)k (−D1 ; q)k (−D2 ; q)k · · · (−Ds ; q)k k=0 ∞ (A1 ; q)k (A2 ; q)k · · · (Ar ; q)k C12 ; q 2 k C22 ; q 2 k · · · Cs2 ; q 2 k 2 2 = 2 2 2 2 k=0 (q; q)k (B1 ; q)k (B2 ; q)k · · · (Br −1 ; q)k D1 ; q k D2 ; q k · · · Ds ; q k
=
×
(D1 ; q)k (D2 ; q)k · · · (Ds ; q)k k x . (C1 ; q)k (C2 ; q)k · · · (Cs ; q)k
(3.16)
We see that this r +s φr +s−1 q-series will lead us to its own form of D-analogues when the above methods of finite and infinite Euler products are applied. Consider firstly as a simplest case, the Euler product applied to this with only one each of the C and D terms. We therefore frame the following definition of Theorem 3.5. For positive integers ai , bi , ci , di , with z chosen for convergence,
r +s φr +s−1
p
=
p −a1 γ , p −a2 γ , . . . , p −ar γ , − p −c1 γ , − p −c2 γ , . . . , − p −cs γ ; p −γ , p −z p −b1 γ , p −b2 γ , . . . , p −b(r −1) γ , − p −d1 γ , − p −d2 γ , . . . , − p −ds γ
∞ σ−γ (a1 , a2 , . . . , ar ; k)σ−2γ (c1 , c2 , . . . , cs ; k)σ−γ (d1 , d2 , . . . , ds ; k) 1 , σ (b , b , . . . , br −1 ; k)σ−2γ (d1 , d2 , . . . , ds ; k)σ−γ (c1 , c2 , . . . , cs ; k) k z k=1 −γ 1 2
(3.17) and
r +s φr +s−1
p|m
=
p −a1 γ , p −a2 γ , . . . , p −ar γ , − p −c1 γ , − p −c2 γ , . . . , − p −cs γ ; p −γ , p −z p −b1 γ , p −b2 γ , . . . , p −b(r −1) γ , − p −d1 γ , − p −d2 γ , . . . , − p −ds γ
σ−γ (a1 , a2 , . . . , ar ; k)σ−2γ (c1 , c2 , . . . , cs ; k)σ−γ (d1 , d2 , . . . , ds ; k) 1 . σ (b , b , . . . , br −1 ; k)σ−2γ (d1 , d2 , . . . , ds ; k)σ−γ (c1 , c2 , . . . , cs ; k) k z k∈Sm −γ 1 2 (3.18)
Similarly, we have for − p −z in place of p −z , Springer
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Theorem 3.6. For positive integers ai , bi , ci , di , with z chosen for convergence,
r +s φr +s−1
p −a1 γ , p −a2 γ , . . . , p −ar γ ,− p −c1 γ , − p −c2 γ , . . . ,− p −cs γ ; p −γ ,− p −z p −b1 γ , p −b2 γ , . . . , p −b(r −1) γ , − p −d1 γ , − p −d2 γ , . . . , − p −ds γ
p
=
∞ σ−γ (a1 , a2 , . . . , ar ; k)σ−2γ (c1 , c2 , . . . , cs ; k)σ−γ (d1 , d2 , . . . , ds ; k) λ(k) , σ (b , b , . . . , br −1 ; k)σ−2γ (d1 , d2 , . . . , ds ; k)σ−γ (c1 , c2 , . . . , cs ; k) k z k=1 −γ 1 2
(3.19) and
r +s φr +s−1
p −a1 γ , p −a2 γ , . . . , p −ar γ ,− p −c1 γ ,− p −c2 γ , . . . , − p −cs γ ; p −γ , − p −z p −b1 γ , p −b2 γ , . . . , p −b(r −1) γ , − p −d1 γ , − p −d2 γ , . . . , − p −ds γ
p|m
=
σ−γ (a1 , a2 , . . . , ar ; k)σ−2γ (c1 , c2 , . . . , cs ; k)σ−γ (d1 , d2 , . . . , ds ; k) λ(k) . σ (b , b , . . . , br −1 ; k)σ−2γ (d1 , d2 , . . . , ds ; k)σ−γ (c1 , c2 , . . . , cs ; k) k z k∈Sm −γ 1 2 (3.20)
Theorems 3.5 and 3.6 show how to take account of negative parameters as they might arise in the q-series under the product operator for an Euler product in the context of this paper. Our notation always requires equal numbers of ci and di terms in the q-series, due to the necessary pairing of numerator and denominator terms for cancellations. We are now in a position to apply these theorems to known basic hypergeometric series summations and transforms to obtain new results which are demonstrably Dirichlet series analogues of the original q-summation formulae. However, finally, let us state the Definition 3.8. Either side of (3.17), (3.18), (3.19) and (3.20) are defined respectively as: r +s r +s−1
r +s r +s−1
r +s r +s−1
r +s r +s−1
a1 , a2 , . . . , ar , −\c1 , −\c2 , . . . , −\cs ;
γ, z
b1 , b2 , . . . , br −1 , −\d1 , −\d2 , . . . , −\ds m | a1 , a2 , . . . , ar , −\c1 , −\c2 , . . . , −\cs ;
, γ, z
b1 , b2 , . . . , br −1 , −\d1 , −\d2 , . . . , −\ds a1 , a2 , . . . , ar , −\c1 , −\c2 , . . . , −\cs ;
γ , −\z
b1 , b2 , . . . , br −1 , −\d1 , −\d2 , . . . , −\ds
(3.21a) ,
,
γ , −\z , b1 , b2 , . . . , br −1 , −\d1 , −\d2 , . . . , −\ds
m | a1 , a2 , . . . , ar , −\c1 , −\c2 , . . . , −\cs ;
(3.21b)
(3.21c)
(3.21d)
where the −\ preceding any variable denotes that it comes from a negative valued parameter in the q-series. Springer
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In future applications of the Euler product transforms of this section we will be forced to delineate between the negative and positive values of a parameter even when this was previously not an issue of consideration in the q-series itself.
4 Application to known q-series identities We start this section with two theorems which will be the basis of all of our results on the transformed q-series to D-series. They demonstrate that a valid q-series identity or transform maps onto a valid Dirichlet series identity or transform in many cases. We shall frequently employ Definition 3.2 here, and a counterpart to this wherein we use the Jordan totient function, namely t −γ Definition 4.1. Consider the Jordan totient function Jγ (m) = m γ i=1 (1 − pi ), t where γ ≥ 0, and m = i=1 pibi is the unique prime decomposition of m. Define then for positive integers m, n, a1 , a2 ,. . . , ar , J (m | a1 , a2 , . . . , ar ; γ )n =
r n−1 r J(ai + j)γ (m) = ( p −ai γ ; p −γ )n . (4.1) (ai + j)γ m i=1 j=0 i=1 p|m
It is relatively easy to see from this and (2.3) that r n−1 r (1 − p −(ai + j)γ ) J (m | a1 , a2 , . . . , ar ; γ )n ( p −ai γ ; p −γ )n = = J (m | b1 , b2 , . . . , br −1 ; γ )n (1 − p −(bi + j)γ ) i=1 p|m ( p −bi γ ; p −γ )n i=1 j=0 p|m ai −1 ai ai +1 a1 +n−2 r σ σ−γ σ−γ · · · σ−γ −γ p|m p p|m p p|m p p|m p , = bi −1 σ bi σ bi +1 · · · σ b1 +n−2 −γ −γ −γ p|m p p|m p p|m p p|m p i=1 σ−γ
(4.2)
and we shall put this fact to good use in our analysis concerning the finite Euler product transforms. Theorem 4.1. Suppose that for each prime p and positive integers ai , bi , ci , di , ei , f i , gi , h i , we have a q-series identity of the generic form r +s φr +s−1
p −a1 γ , p −a2 γ , . . . , p −ar γ , − p −c1 γ , − p −c2 γ , . . . , − p −cs γ ;
p −γ , p −z
p −b1 γ , p −b2 γ , . . . , p −br −1 γ , − p −d1 γ , − p −d2 γ , . . . , − p −ds γ (4.3a)
=
( p −e1 γ , p −e2 γ , . . . , p −er γ ; p −γ )m 1 ( p −g1 γ , p −g2 γ , . . . , p −gs γ ; p −2γ )m 2 · · · . ( p − f1 γ , p − f2 γ , . . . , p − fr γ ; p −γ )n 1 ( p −h 1 γ , p −h 2 γ , . . . , p −h s γ ; p −2γ )n 2 · · · (4.3b) Springer
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Then the two Euler product transforms applied over primes to both sides of this yield respectively, the two D-analogue summation formulae a1 , a2 , . . . , ar , −\c1 , −\c2 , . . . , −\cs ; γ , z (4.4a) r +s r +s−1 b1 , b2 , . . . , br −1 , −\d1 , −\d2 , . . . , −\dr =
ζ ( f 1 , f 2 , . . . , f s1 ; γ )n 1 ζ (h 1 , h 2 , . . . , h s2 ; 2γ )n 2 · · · , ζ (e1 , e2 , . . . , er1 ; γ )m 1 ζ (g1 , g2 , . . . , gr2 ; 2γ )m 2 · · ·
and r +s r +s−1
=
m | a1 , a2 , . . . , ar , −\c1 , −\c2 , . . . , −\cr ; γ , z
(4.4b)
b1 , b2 , . . . , br −1 , −\d1 , −\d2 , . . . , −\dr
J (m | e1 , e2 , . . . , er1 ; γ )m 1 J (m | g1 , g2 , . . . , gr1 ; 2γ )m 2 · · · . J (m | f 1 , f 2 , . . . , fr2 ; γ )n 1 J (m | h 1 , h 2 , . . . , h r2 ; 2γ )n 2 · · ·
(4.5a) (4.5b)
For the sake of completeness and brevity we also state Theorem 4.2. Theorem 4.1 with − p −z replacing p −z in (4.3a), together with − \ z replacing z in each of (4.4a) and (4.5a) is true. These are the required theorems to enable us to write down at a glance, many D-series analogues of q-series. The remainder of this paper will show some simple examples, all of which appear to be new results. 5 The D-binomial theorem In this section we present some simply arrived at applications of the transforms in Theorems 4.1 and 4.2 of the previous section. In later papers we shall go into greater detail as regards the significance of these new results. But for now we shall content ourselves with statement and derivation of some new but seemingly fundamental summation formulae. The q-binomial theorem is (see Gasper and Rahman [23, page 7]), 1 φ0 (a; −; q, z)
=
(az; q)∞ , (z; q)∞
|z| < 1, |q| < 1.
(5.1)
Applying our work of the previous sections to this, we can expect to find a D-series equivalent involving Riemann zeta functions, and another involving Jordan totient functions. In fact, four D-series arise from applying the transforms of Theorems 4.1 and 4.2 since the real part of z can be either positive or negative in (5.1) but for the D-series we treat these cases as distinct. Applying these theorems to (5.1) gives the required analogues of the q-binomial theorem. The continuity of z in (5.1) is retained in the transforms to Theorem 5.1. (The D-binomial theorem) For positive integers a, zγ > 1, (z + a)γ > 1, ζ (z; γ )∞ , (5.2) 1 0 (a; −; γ , zγ ) = ζ (a + z; γ )∞ Springer
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=
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ζ (a + z; γ )∞ ζ (2z; 2γ )∞ , ζ (z; γ )∞ ζ (2(a + z); 2γ )∞ J (m | a + z; γ )∞ , J (m | z; γ )∞
1 0 (m
| a; −; γ , zγ ) =
1 0 (m
| a; −; γ , −\zγ ) =
J (m | a + z; γ )∞ J (m | 2z; 2γ )∞ , J (m | z; γ )∞ J (m | 2(a + z); 2γ )∞
(5.3) (5.4) (5.5)
We have in particular from (5.2), letting a = 1, then a = 2 with β = zγ , ∞ 1 = ζ (β) β k k=1
and
∞ σ−γ (k) k=1
kβ
= ζ (β)ζ (β + γ )
(5.6)
trivially as known results. Then we have for successively a = 3, a = 4, from (5.2), ∞ σ−γ (k)σ−λ k p|k p 1 = ζ (β)ζ (β + γ )ζ (β + 2γ ), β k σ−γ p k=1 p|k 2 ∞ σ−γ (k)σ−γ k p σ p k −γ p|k p|k 1 kβ 2 σ−γ k=1 p|k p σ−γ p|k p = ζ (β)ζ (β + γ )ζ (β + 2γ )ζ (β + 3γ ),
(5.7)
(5.8)
and so on, valid where each of the Riemann zeta functions converges. (5.7) and (5.8) are not in the extensive literature on Dirichlet summations and the Riemann zeta function. This latter example turns out to enumerate a tiling pattern arising in the study of quasicrystals.
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This tiling pattern is enumerated recursively by a divisor function related explicitly to the coefficient terms that occur in the D-analogue of the binomial coefficients. For a proof of this see Baake et al. [14]. This fascinating new connection between the theory of tilings arising in the study of quasicrystals may be well worth further investigation. A similar rationale when applied instead to (5.3) gives ∞ λ(k) k=1
kβ
=
ζ (2β) ζ (β)
and
∞ σ−γ (k)λ(k)
kβ
k=1
=
ζ (2β)ζ (2β + 2γ ) , ζ (β)ζ (β + γ )
(5.9)
where the first of these is a known result, and the second of these is accessible from known methods. Then we have the new results for the conditions such that each Riemann zeta function converges, ∞ σ−γ (k)σ−γ k p|k p λ(k) ζ (2β)ζ (2β + 2γ )ζ (2β + 4γ ) , (5.10) = β k ζ (β)ζ (β + γ )ζ (β + 2γ ) σ−γ p k=1 p|k 2 ∞ σ (k)σ p σ p k k −γ −γ −γ p|k p|k λ(k) kβ 2 σ−γ k=1 p|k p σ−γ p|k p =
ζ (2β)ζ (2β + 2γ )ζ (2β + 4γ )ζ (2β + 6γ ) . ζ (β)ζ (β + γ )ζ (β + 2γ )ζ (β + 3γ )
(5.11)
If we think in terms of the application of Theorems 4.1 and 4.2, the following theorem emerges. Note that the identities (5.6) to (5.11) may also be summarized in (5.12) and (5.14).
Theorem 5.2. For positive integers n and for values taken where the zeta functions converge, ∞ σ−γ (n; k)
n
ζ (β + kγ ),
(5.12)
n σ−γ (n; k) m β+kγ = , kβ J (m) k∈Sm k=0 β+kγ
(5.13)
k=1
kβ
=
k=0
∞ σ−γ (n; k)λ(k)
=
n ζ (2β + 2kγ )
, ζ (β + kγ )
n σ−γ (n; k)λ(k) = σ−(β+kγ ) p . kβ p|m k∈Sm k=0 k=1
Springer
kβ
(5.14)
k=0
(5.15)
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We note here that M¨obius inversion of the Dirichlet series translates these for the same criteria into Theorem 5.3; a fact which of itself is not remarkable, yet a reminder that any of our Dirichlet series can be expressed as M¨obius inverted versions, including the D-analogues of many well known q-series formulae which would result from Theorems 4.1 and 4.2. Theorem 5.3. For the conditions of Theorem 5.2, ∞ σ−γ (n; k)μ(k) k=1
kβ
=
n
1 , ζ (β + kγ ) k=0
n σ−γ (n; k)μ(k) Jβ+kγ (m) = , β k m β+kγ k∈Sm k=0 ∞ σ−γ (n; k)λ(k)μ(k) k=1
kβ
=
n ζ (β + kγ ) , ζ (2β + 2kγ ) k=0
n σ−γ (n; k)λ(k)μ(k) 1 . = β k σ p|m p k∈Sm k=0 −(β+kγ )
(5.16)
(5.17)
(5.18)
(5.19)
We therefore see that the D-analogue of the q-binomial theorem leads us to the D-series for the finite product of n + 1 Riemann zeta functions whose arguments are in arithmetic progression. This also implies we can write down the D-series for the reciprocal of that product. We have given Theorem 5.1 the seemingly obvious name of the D-binomial theorem. There is also a connection between the Jordan totient function and the Dirichlet character, as is shown in Apostol [9], and although this may be of interest to researchers, it is beyond the scope of our work for this paper. For n equal to 1, then 2 in (5.13), 1 mβ , = β k Jβ (m) k∈Sm
(5.20)
σ−γ (k) m β+γ , = β k Jβ (m)Jβ+γ (m) k∈Sm
(5.21)
trivially as results that may be of some interest to number theorists, as the Jordan totient function relates to the Euler totient function and the Dirichlet character. Then we have for successively n = 3, n = 4, from (5.13), σ−γ (k)σ−γ k p|k p 1 m β+2γ = , k β Jβ (m)Jβ+γ (m)Jβ+2γ (m) σ−γ k∈Sm p|k p
(5.22) Springer
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σ−γ (k)σ−γ (k) k p|k p σ−γ k p|k p 2 1 kβ 2 σ−γ k∈Sm p|k p σ−γ p|k p =
m β+3γ , Jβ (m)Jβ+γ (m)Jβ+2γ (m)Jβ+3γ (m)
(5.23)
and so on. (5.20) to (5.23) are not in the extensive literature on Dirichlet summations, arithmetical functions and Euler products. A similar rationale when applied instead to (5.15) gives
λ(k) = σ−β p , (5.24) kβ p|m k∈Sm
σ−γ (k)λ(k) = σ−β p σ−(β+γ ) p , (5.25) kβ p|m p|m k∈Sm
k σ (k)σ p −γ −γ p|k λ(k) = σ−β p σ−(β+γ ) p σ−(β+2γ ) p , kβ σ−γ p|m p|m p|m k∈Sm p|k p σ−γ (k)σ−γ k p|k p σ−γ k p|k p 2 λ(k) kβ 2 σ−γ p σ p k∈Sm −γ p|k p|k
= σ−β p σ−(β+γ ) p σ−(β+2γ ) p σ−(β+3γ ) p . p|m
p|m
p|m
(5.26)
(5.27)
p|m
We note at this point that there are interesting special cases ot Theorem 5.2 if we allow γ → 0. We note that limγ →0 σ−γ (k) = d(k); the numer of divisors of k. Likewise, limγ →0 Jβ+kγ (m) = Jβ (m). Furthermore, J1 (m) = ϕ(m); the Euler totient function of m. Then if we allow γ to tend toward zero we obtain: Theorem 5.4. For positive integers n and for values taken where the Dirichlet series on the left sides converge, ∞ n−1 d k p|k p j 1 = ζ (β)n+1 , β j k d p p|k k=1 j=0
β n+1 n−1 d k p|k p j 1 m = , β j k Jβ (m) p|k p k∈Sm j=0 d ∞ n−1 d k p|k p j λ(k) ζ (2β) n+1 = , j kβ ζ (β) p|k p k=1 j=0 d Springer
(5.28)
(5.29)
(5.30)
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n+1 n−1 d k p|k p j λ(k) = σ−β p . j kβ p|k p p|m k∈Sm j=0 d
(5.31)
The convergence conditions and limits of such series are apparently quite worthy of closer examination, particularly when we consider a case of (5.29) such as n−1 d k p|k p j 1 m n+1 . = j k ϕ(m) p|k p k∈Sm j=0 d
(5.32)
6 The q-Gauss sum D-analogue The results of Section 5 are new and easy consequences of Theorems 4.1 and 4.2. There are many other D-analogues that can simply be written down, such as for example, deriving from the q-Gauss sum. The classical ordinary hypergeometric sum is the Gauss 2 F1 (a, b; c; 1) given in (1.2) whose q-analogue, first given by Heine [26] and [27] in the 19th century, is (1.4). For an account of this see Gasper and Rahman [23, page 10]. A Dirichlet series analogue arrived at by applying Theorem 4.1 is Theorem 1.1, which we restate here with the full veracity of Theorem 4.1 as Theorem 6.1. For positive integers a, b and c and γ ≥ 0, such that cγ , (c − a − b)γ , (c − a)γ , (c − a − b)γ , are each >1, 2 1 (a, b; c; γ (c
− a − b)γ ) =
ζ (c, c − a − b; γ )∞ , ζ (c − a, c − b; γ )∞
(6.1)
J (m | c − a, c − b; γ )∞ . J (m | c, c − a − b; γ )∞
(6.2)
and 2 1 (m
| a, b; c; γ (c − a − b)γ ) =
This is equivalent to ∞ σ−γ (a; k)σ−γ (b; k) k=1
σ−γ (c; k)
1 ζ (c; γ )∞ ζ (c − a − b; γ )∞ = , k (c−a−b)γ ζ (c − a; γ )∞ ζ (c − b; γ )∞
(6.3)
and σ−γ (a; k)σ−γ (b; k) J (m | c − a; γ )∞ J (m | c − b; γ )∞ 1 = ; (6.4) (c−a−b)γ σ (c; k) k J (m | c; γ )∞ J (m | c − a − b; γ )∞ −γ k∈Sm Springer
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which are by definition the same as b−2 c−2 j ∞ a−2 σ−γ k p|k p j σ−γ k p|k p j σ−γ 1 p|k p (c−a−b)γ j j j k σ p σ p σ p k −γ −γ p|k p|k p|k k=1 j=0 j=0 j=0 −γ =
∞ ζ ((c + k)γ )ζ ((c − a − b + k)γ ) k=0
ζ ((c − a + k)γ )ζ ((c − b + k)γ )
,
(6.5)
and b−2 c−2 j a−2 σ−γ k p|k p j σ−γ k p|k p j σ−γ 1 p|k p (c−a−b)γ j j j k k σ p σ p σ p −γ −γ p|k p|k p|k k∈Sm j=0 j=0 j=0 −γ =
∞ (1 − p −(c−a+k)γ )(1 − p −(c−b+k)γ ) . (1 − p −(c+k)γ )(1 − p −(c−a−b+k)γ ) k=0 p|m
(6.6)
The convergence conditions for (6.6) are clear from the fact that the summation derives from a finite Euler product transform. The conditions for (6.5) are rather more complicated to analyze, but we can easily verify that particular cases are correct, and clearly (6.5) is a limiting case of (6.6). These resulting particular cases are all new. We now give some examples. Firstly if in Theorem 6.1 we allow a = n, b = n, c = 2n + 1, Corollary 6.1. If γ > 1, 2n−1 j ∞ n−2 σ−γ k p|k p j σ−γ 1 p|k p j j kγ p|k p p|k p k=1 j=0 σ−γ j=n−1 σ−γ k =
ζ (γ )ζ (2γ )ζ (3γ ) · · · ζ (nγ ) , ζ ((n + 1)γ )ζ ((n + 2)γ )ζ ((n + 3)γ ) · · · ζ (2nγ )
(6.7)
and 2n−1 j n−2 σ−γ k p|k p j σ−γ 1 p|k p j j kγ p|k p p|k p k∈Sm j=0 σ−γ j=n−1 σ−γ k n n+1 n+2 2n−1 σ−γ · · · σ−γ σ−γ p|m p σ−γ p|m p p|m p p|m p = 2 3 n−1 σ−γ p|m p σ−γ p|m p σ−γ p|m p · · · σ−γ p|m p (6.8)
If a = n, b = n and c = 3n, in Theorem 6.1 we have
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Corollary 6.2. If γ > 1, 2 2n−1 j ∞ n−2 σ−γ k p|k p j σ−γ 1 p|k p nγ j j k σ p σ p k −γ p|k p|k k=1 j=0 j=3n−2 −γ =
ζ (nγ )ζ ((n + 1)γ )ζ ((n + 2)γ ) · · · ζ ((2n − 1)γ ) , ζ (2nγ )ζ ((2n + 1)γ )ζ ((2n + 2)γ ) · · · ζ ((3n − 1)γ )
(6.9)
and 2 2n−1 j n−2 σ−γ k p|k p j σ−γ 1 p|k p nγ j j k σ p σ p k −γ p|k p|k k∈Sm j=0 j=3n−2 −γ n+1 n+2 2n−1 σ−γ · · · σ−γ σ−γ ( p|m p n )σ−γ p|m p p|m p p|m p . = 2 3 n−1 σ−γ p|m p σ−γ p|m p σ−γ p|m p · · · σ−γ p|m p (6.10)
The cases of Theorem 6.1 with a = n, b = 2n, c = 4n, are
Corollary 6.3. If nγ > 1, 2n−2 4n−2 j ∞ n−2 σ−γ k p|k p j σ−γ k p|k p j σ−γ 1 p|k p nγ j j j k k σ p σ p σ p −γ −γ p|k p|k p|k k=1 j=0 j=0 j=0 −γ =
ζ (nγ )ζ ((n + 1)γ )ζ ((n + 2)γ ) · · · ζ ((2n − 1)γ ) , ζ (2nγ )ζ ((2n + 1)γ )ζ ((2n + 2)γ ) · · · ζ ((4n − 1)γ )
(6.11)
and 2n−2 4n−2 j n−2 σ−γ k p|k p j σ−γ k p|k p j σ−γ 1 p|k p j j j k nγ p|k p p|k p p|k p k∈Sm j=0 σ−γ j=0 σ−γ j=0 σ−γ k σ−γ ( p|m p 3n−1 )σ−γ ( p|m p 3n )σ−γ ( p|m p 3n+1 ) · · · σ−γ ( p|m p 4n−2 ) = . σ−γ ( p|m p)σ−γ ( p|m p 2 )σ−γ ( p|m p 3 ) · · · σ−γ ( p|m p 2n−2 ) (6.12)
Similarly for positive integers (n − 2) and a, the cases of theorem 6.1 with a = a, b = (n − 1)a, c = (n + 1)a, are
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Corollary 6.4. If aγ > 1, ⎞⎛ ⎞⎛ ⎞ ⎛ j j (n−1)a−2 (n+1)a−2 ∞ a−2 σ−γ k σ−γ k p|k p j σ−γ p|k p p|k p 1 ⎝ ⎠⎝ ⎠⎝ ⎠ aγ k j j j σ−γ σ−γ k p|k p k=1 j=0 σ−γ j=0 j=0 p|k p p|k p =
ζ (aγ )ζ ((a + 1)γ )ζ ((a + 2)γ ) · · · ζ ((2a − 1)γ ) , ζ (naγ )ζ ((na + 1)γ )ζ (((na + 2)γ ) . . . ζ ((n + 1)a − 1)γ )
(6.13)
and ⎞⎛ ⎞⎛ ⎞ ⎛ j j (n−1)a−2 (n+1)a−2 a−2 σ−γ k σ−γ k p|k p j σ−γ p|k p p|k p 1 ⎝ ⎠⎝ ⎠⎝ ⎠ aγ k j j j σ−γ σ−γ k p|k p k∈Sm j=0 σ−γ j=0 j=0 p|k p p|k p (n+1)a−2 σ−γ ( p|m p na−1 )σ−γ ( p|m p na )σ−γ ( p|m p na+1 ) · · · σ−γ p|m p . = σ−γ ( p|m p)σ−γ ( p|m p 2 )σ−γ ( p|m p 3 ) · · · σ−γ ( p|m p 2a−2 ) (6.14)
Finally we note that these analogues, although rather cumbersome at times in appearance, seem to have the potential to bring out new and interesting identities, and a parallel theory to the ordinary hypergeometric series, and basic hypergeometric series. Obviously there are many new identities to be found from the transforms of our paper. There is also the plausible, but as yet unrealized prospect, of discovering new q-series identities by applying properties of the Dirichlet series in a kind of converse to Theorems 4.1 and 4.2. However, we must be content with the introduction brought by this paper as it implies a potentially vast new area of research.
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