Around the Lipschitz Summation Formula

Boundary behavior of important functions has been an object of intensive research since the time of Riemann. Kurokawa, Kurokawa-Koyama, and Chapman studied the boundary behavior of generalized Eisenstein series which falls into this category. )e underlying principle is the use of the Lipschitz summation formula. Our purpose is to show that it is a form of the functional equation for the Lipschitz–Lerch transcendent (and in the long run, it is equivalent to that for the Riemann zeta-function) and that this being indeed a boundary function of the Hurwitz–Lerch zeta-function, one can extract essential information. We also elucidate the relation between Ramanujan’s formula and automorphy of Eisenstein series.


Introduction
Boundary behavior of core functions has always been the object of intensive research since it exhibits a peculiar phenomenon that cannot be predicted by the behavior inside the domain. ere are many instances of such unexpected behavior cf. [1][2][3]. Kurokawa [4] and Koyama and Kurokawa [5] studied the following limiting values by the Lipschitz summation formula: where E k (τ) is the generalized Eisenstein series defined by (25). It has been elucidated and generalized by Chapman [6] who also used the Lipschitz summation formula for which he appealed to [7]. Knopp and Robbins in their Remarks 1 and 2 state their own views on the Lipschitz summation formula and Stark's method [8] to the effect that they are not directly related to the functional equation (just as, for the Riemann zeta-function, the partial fraction expansion does not seem to be related). In [9], Murty and Sinha [10] result has been elucidated as a manifestation of one of the equivalent conditions to the functional equation, the Fourier-Bessel expansion, or the perturbed Dirichlet series ( [11], Chapter 4), thereby explaining the genesis of Stark's method.
e Lipschitz summation formula for quadratic fields is also deduced there. We shall turn to this toward the end of Section 4.
We cite the passage from [12] " e relation between modular forms and Dirichlet series with functional equations was discovered by Hecke, whose epoch-making work during the years 1930-1940, based on that discovery and that of the 'Hecke operators', brought out completely new aspects of a theory which many mathematicians would have regarded as a closed chapter long before. " We refer to this as part of the Riemann-Hecke-Bochner correspondence (RHB correspondence) ( [11], p. 4 and 22) which is coined by Knopp [13].
Our main aim in this paper is to prove the general modular relation, eorem 4, for the Lipschitz-Lerch transcendent (57) and deduce the general Lipschitz summation formula, Corollary 4. From this, we show that, in this case again, generalized RHB correspondence or the modular relation is the key for everything.
But prior to this, in Section 2, we state the modular relation for the Lambert series generated by the product of two Riemann zeta-functions with variables different by an odd integer and prove the automorphy of the Eisenstein series by the RHB correspondence, which of course settles the even weight case of (1). For another relation, cf. Bruinier and Funke [14]. e even difference case turns out to be a reminiscent of the Wigert-Bellman divisor problem [15] as alluded to in [9]. In Section 3, we state the results in another form based on the shifted Mellin inversion.
Here, we use a method similar to the one in [16] (pp. 73-75) of using the Ewald expansion ( [11], Chapter 5) as opposed to the Fourier-Bessel expansion alluded to above. Since it is equivalent to the Lerch functional equation ( [17], eorem 5.3, p. 130) which in turn is equivalent to an asymmetric form (3) of the functional equation for the Riemann zeta-function, we thereby show that, in the long run, the genesis is in the functional equation for the Riemann zeta-function.
As a necessary step, we show that the reciprocal Hurwitz formula amounts to a ramified functional equation, Lemma 1. ere are many cases of such ramified functional equations (cf. [18] and references therein). We state one of the earliest occurrences.
In what follows, we always use the notation s � σ + it as the complex variable.

An Example of the Riemann-Hecke-Bochner Correspondence
roughout in what follows, we appeal to the Riemann zetafunction defined in the first instance for σ � Res > 1 by is satisfies the asymmetric form of the functional equation: which is a prototype of the Hurwitz formula (cf. (64) for its reciprocal). We fix the integer α throughout. We consider the product of two zeta-functions: where the series is absolutely convergent for is the sum-of-divisors function. We note that φ(s − a) includes the case of ζ(s)ζ(s − α) as φ(s − a) � ∞ n�1 ((σ α (n))/n s ). is will be pursued in Section 3.
e zeta-function φ(s) satisfies the asymmetric functional equation: Noting that the product of cosines amounts to (i) First we treat the case of α � 2ϰ + 1 an odd integer. en by the reciprocal relation for the gamma function, we see that the functional equation (6) amounts to Now by the well-known procedure-Hecke gamma transform (e.g., [16]), we have for c > σ φ and Re where (c) indicates the Bromwich contour σ � c, − ∞ < t < ∞. By a standard procedure of moving the line to the left up to (d), where d < − α < 0 (d � − α − (1/2), say), whereby noting that the horizontal integrals vanish in the limit as |t| ⟶ ∞, we obtain where P(x) � P α (x) is the residual function consisting of the sum of residues of the integrand at − a, . . . , − 1, 0, 1, a + 1. Writing 1 − s for s in (11), we see that the right-hand side of (11) becomes

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Substituting the absolutely convergent series in (4) and factoring out (x/4π 2 ) a , we deduce that Hence, by the Mellin inversion again, we obtain for Rex > 0, i.e., the Bochner modular relation [19].
To compute the residual function, we use Table 1, taking into account the trivial zeros of the Riemann zeta-function at negative even integers. For 1 ≠ α > 0, the residual function is on writing k � 2j − 1. In literature, this is expressed in another form based on the explicit formula for zetavalues cf., e.g., ([20], p. 71 and 91): where the Bernoulli numbers B 2k are b-notation ( [20], p. 90).
If we write − x � 2πiτ, then τ ∈ H and the Lambert series in Liouville's form amounts to the Eisenstein series E k (τ) in Definition 1, where k � a + 1 is even and (14) gives cf. (51) and (52).
Definition 1. For any k ∈ N, Kurokawa introduces the general Eisenstein series: which is not necessarily modular for k odd.
we see that it is nothing but i.e., the automorphy of E 2ϰ+2 (z), cf. (25). us, we have established.

Theorem 1.
e Bochner modular relation (14) entails at one end of the spectrum α � − (2ϰ + 1) Ramanujan's formula (17) and at the other end α � 2ϰ + 1 the automorphy of the Eisenstein series (27), thus abridging analytic number theory and the theory of modular forms.
(ii) Now we turn to the case of a � 2ϰ. We digress from (12) which should be replaced by In the same way as we have deduced (13), we obtain We shall stop here since it would be difficult to express the resulting integrals and Bellman's method [15] yields an asymptotic formula rather than an equality. Partial theory of modular relations for the product of zetafunctions is given in ( [11], Chapter 9, pp. 241-265), which is still in progress.

Remark 1.
at the odd integer difference case (i) reduces to the one-gamma factor case to the RHB correspondence is not coincidental and is expounded in [11] (pp. 81-86), where one can also find a plausible discovery of Ramanujan of the transformation formula for the Dedekind eta-function η. Weil's paper [22] is the most well-known paper that contains the proof of the latter, but prior to this, Chowla gave a proof [23] for the discriminant function, which is the 24th power of η. Ramanujan's formula is stated as I, Entry 15 of Chapter 16 [24], Entry 21 (i), Chapter 14 of Ramanujan's Notebook II [25] (which is and also as IV, Entry 20 of [26]). e most extensive account of information surrounding Ramanujan's formula is [27], while [28] is the most informative account of special values of the zetafunctions. e intersection of references in these two excellent survey papers (which have a lot in common) is a null set. In literature, Ramanujan's formula is stated for α � πx > 0, β � (π/x) > 0 satisfying the following relation: and in terms of Lambert series.
e Lambert series L(z) is defined for |z| < 1 by which is transformed into the Fourier series: the Liouville formula, where b ℓ � d|ℓ a d . e sum-of-divisors function is the case a n � n α . Original Ramanujan's formula looks like having little to do with modular forms. Equation (17) being a rephrased Lambert series in Liouville's form has amenity to the q-expansion, and so to automorphy.

The Ramanujan-Guinand Formula and Its Consequences
In Section 2, we established that at both ends of the spectrum, the Bochner modular relation amounts to Ramanujan's formula and the automorphy of Eisenstein series, respectively. In this section, we partially follow [29], reproduced in [11] (pp. 86-92), and elucidate the mechanism hidden in the correspondence φ(s) ⟷ φ(s − a) by differentiation of the Ramanujan-Guinand formula.
In [14], they mention duality between the space of weak Maass forms of (negative) weight k ∈ (1/2)Z and the space of holomorphic cusp form of (positive) weight 2 − k.
To this end, we introduce the Mellin inversion with shifted argumentI a (x). Let a ≥ 0 be a fixed integer to be taken as the number of times of differentiation throughout. Let φ(s) be the zeta-function defined by (4) with α � 2ϰ + 1 and ϰ a nonnegative integer: the other case being included in eorem 2 below. e argument of its proof goes in the lines of Section 2. For Rex > 0 let where c > 1 + a. e Hecke gamma transform reads e special case, is the Lambert series appearing in Ramanujan's formula (17). Differentiating I(x) a-times with respect to x, whereby we perform differentiation under integral sign, we have the additional factor which is (− 1) a (Γ(s + a)/Γ(s)), whence we deduce the remarkable formula: i.e., a-times differentiation of the Lambert series (36) is effected by shifting the argument of φ(s) by a in (36) and multiplying by (− 2π) a . In view of (38), the a-times differentiated form of Ramanujan's formula (17) amounts to a counterpart of the modular relation for I a (x). (34), we have the modular relation for ϰ ≥ 0 and 0 ≤ a ≤ 2ϰ + 1,

Theorem 2 (Ramanujan-Guinand formula). For the Mellin transform I a (x) with shifted argument as defined by
where P a (x) is the residual function.
Proof. Proof depends on the following equation: which is a variant of the functional equation (6) in the following form: Moving where P(x) � P a (x) denotes the sum of residues of the integrand at its poles at s � a − 2ϰ − 1, a − 2ϰ, a − 2ϰ + 1, a − 2ϰ + 3, . . . , 0, a + 1.
Substitute (40) and change the variable s ⟷ a − 2ϰ − s in the integral J a (x). en Substituting Mathematical Problems in Engineering we find that Since the gamma factor can be computed as follows for 0 ≤ a ≤ 2ϰ + 1, where for a � 2ϰ + 1, the right-hand side is to mean Γ(s + 2ϰ + 1), we conclude from (42) and (45) that where the sum reduces to 1 for a − 2ϰ + 1. Finally, we note that the integral on the right-hand side of (47) becomes by the change of variable s ⟷ s + k Hence, (47) leads to (39), completing the proof.

Corollary 1
(i) e case a � 0 is Ramanujan's formula (17) with the residual function valid for Rex > 0.
(ii) e case a � 2ϰ is Guinand's formula, cf. ( [29], (iii) e case a � 2ϰ + 1 reads where ese lead to (21), thence to the automorphy (27). (iv) e special case of (50) with ϰ � 1, i.e., once differentiated form of Ramanujan's formula, yields Terras' formula [30,31]: Remark 2. e Mellin inversion with shifted argument (34) is an additive version of the "pseudomodular relation principle," which is a processed modular relation with the processing gamma factor Γ(s + a) ( [11], p. 50). In this special case, the Main Formula (38) is the manifestation of the statement of Razar that the differentiation of Lambert series essentially corresponds to the shift of the argument of the associated Dirichlet series.
is can be regarded as once differentiated form of Guinand's formula. We note that, in the case of ϰ � 0, the twice differentiated Guinand's formula, with a suitable modification, coincides with the automorphic property of the Eisenstein series E 2 : Mathematical Problems in Engineering Equation (51), once differentiated form of Guinand's formula (�(2n + 1) times differentiated form of Ramanujan's formula�the negative case thereof ) leads to (27), automorphy of G 2ϰ+2 .

Lipschitz Summation Formula
Knopp-Robbins [7] in their Remark 1 state their view on the Lipschitz summation formula to the effect that it is conceptually simpler than Riemann's original method of using the theta series. However, at least the special case of the Lipschitz summation formula ( eorem 3) which is applied by Chapman to establish the limit relation has already been used extensively and can be readily deduced from the partial fraction expansion for the cotangent function. Since it is known that the partial fraction expansion is equivalent to the functional equation for the Riemann zeta-function, we may say that Chapman's result is a consequence of the functional equation. By Corollary 4 below, we shall show that the Lipschitz summation formula itself is equivalent to the functional equation, thereby enhancing the above statement. Pasles and Pribitkin [32] extend the Lipschitz summation formula to the two-variable case to which we hope to return elsewhere.
In what follows, we shall generalize eorem 3 in a wider framework, deducing Corollary 4 to eorem 4 as a general modular relation. Let be the Hurwitz-Lerch zeta-function, and let denote the boundary function-the Lipschitz-Lerch transcendent (( [9], pp. 59-62), ( [2], pp. 128-131), [33]). is is in close correspondence with the case of L s (z) and ℓ s (x) considered in [34]. As a consequence of the main formula in Appendix Section, we may deduce, from the reciprocal Hurwitz formula (64), a generalized Lipschitz summation formula (78) which is indeed a form of the functional equation.
satisfy the ramified functional equation, a special case of (A.2) with r � 1 or

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Proof. From Euler's formula, we have We transform the reciprocal Hurwitz formula (0 < x < 1): is the first factor of the right-hand side member and f 2 (s) is the second. Substituting (63), the second factor becomes In view of this, we transform the first factor accordingly Substituting (65) and (66), we transform (64) into where ψ i 's are defined in (59) and (60). Clearing the denominators in (67), It remains to note that the factor − Γ(s/2)Γ(1 − (s/2)) of the second summand may be transformed into Γ(− (s/2))Γ(1 + (s/2)), which is indeed the case in view of the reciprocity formula.
Mathematical Problems in Engineering , and so eorem 4 amounts to

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where Γ(s, c) is the incomplete gamma function defined as follows: Proof. We transform H-functions in Corollary 2 in a concrete form. e procedure is the same for the three H-functions, i.e., duplication formula, H ⟶ G formula, and the explicit formula for the G-function, and we use the known results on them freely, cf. [11,35], etc. We have and similarly to (74) Hence, (71) amounts to  ϕ(x, s, z) where 0 < z, x < 1, and t � − z(u − 1).
Proof. By (72), we have the left-hand side of (71) is equal to Equation (78) is sometimes referred to as the Lipschitz summation formula ( [36][37][38] etc.). e character analogue of the Lipschitz summation formula is known ( [2], pp. 128-131), and so we may naturally treat a more general case which will be conducted elsewhere.

Ramified Functional Equations
ere are some instances of the ramified functional equations in literature.
In [43][44][45], they are stated in the case of zeta-functions with periodic coefficients which satisfy the ramified functional equations as a result of representations in bases consisting of the Hurwirz and Lerch zeta-functions [18]: suppose f(n) be a periodic function with period M, be the associated Dirichlet series absolutely convergent σ � Res > 1 and that be odd, resp. even part of f: f � f even + f odd . en which amounts to (61) on clearing the denominators and multiplying by (π/M) (s− 1)/2 . Wang and Banerjee [46] treat the product of Hurwitz zeta-functions which satisfy a ramified functional equation as a result of the Hurwitz formula: which we suppose have finite abscissa of absolute convergence σ φ , σ ψ i (1 ≤ i ≤ I), respectively. We assume the existence of the meromorphic function χ, which satisfies, for a real number r, the functional equation: We introduce the processing gamma factor: and suppose that for any real numbers u 1 , u 2 (u 1 < u 2 ), lim |v|⟶∞ Γ(u + iv − s | Δ)χ(u + iv) � 0, (A.6) uniformly in u 1 ≤ u ≤ u 2 . We choose L 1 (s) so that the poles of lie on the left of L 2 (s), and those of n j�1 Γ a j + A j s − A j w lie on the right of L 2 (s). Further, they squeeze a compact set S such that s k ∈ S(1 ≤ k ≤ L). Under these conditions, we define the χ-function, key-function, X(z, s | Δ) by