Limiting Values and Functional and Difference Equations

Boundary behavior of a given important function or its limit values are essential in the whole spectrum of mathematics and science. We consider some tractable cases of limit values in which either a difference of two ingredients or a difference equation is used coupled with the relevant functional equations to give rise to unexpected results. As main results, this involves the expression for the Laurent coefficients including the residue, the Kronecker limit formulas and higher order coefficients as well as the difference formed to cancel the inaccessible part, typically the Clausen functions. We establish these by the relation between bases of the Kubert space of functions. Then these expressions are equated with other expressions in terms of special functions introduced by some difference equations, giving rise to analogues of the Lerch-Chowla-Selberg formula. We also state Abelian results which not only yield asymptotic formulas for weighted summatory function from that for the original summatory function but assures the existence of the limit expression for Laurent coefficients.

These are continued meromorphically over the whole plane with a simple pole at s = 1 for the principal character. L(s, χ) with a non-principal character χ can be continued analytically over σ > 0 and the point s = 1 is a regular point with the Taylor coefficients L (k) (s, χ), k ≥ 0 as well as special values K(k, χ), Cf. §4.
Indeed, in [41, pp. 71-81], almost all (buds of) ingredients are given for the study of generalized Euler constants and is worth being called genesis of the study of generalized Euler constants including the Kronecker limit formula [41, pp. 161-182]. It gives an explicit evaluation of the Laurent constant of the Epstein zeta-function.
The generalized Euler constants γ k (a, M) in (52) for an arithmetic progression is naturally a highlighted subject and after [7], [46], [36], and [24], Shirasaka [69] is a culmination providing the genuine generating function for them, based on the theory of Hurwitz zeta-function. In another direction, [32] and [13] are another summit of the study on periodic Dirichlet series, appealing to the Deninger-Meyer method, based on the theory of Lerch zeta-function. One of our main results is Theorem 4 which give the evaluation of the generalized Euler constants in terms of Hurwitz and Lerch zeta-functions. This reflects the base change of the Kubert space K s as described by Milnor. The other main result of ours is an Abelian theorem in the spirit of [9], [41] and [56] entailing [8]. Under the assumption of the functional equation, this depends on [16] and partly [73] and without it, depends on [56]. Thus these two theorems pave a promenade to the study on Laurent series coefficients of a large class of zeta-and special functions starting from the Abelian theorem.

Lerch zeta-function
Here we assemble various facts about the Lerch zeta-function, cf. also Proposition 2 below and the subsequent passage. The Lerch zeta-function is the polylogarithm function of order s with complex exponential argument absolutely convergent for σ = Re s > 1. This is a special case of (9) with τ = x ∈ R, σ > 1 and is a boundary function thereof and we sometimes refer to it as the boundary Lerch zeta-function.
Cf. [93] for boundary functions in which a looser condition of almost convergence is assumed. In our case it is analyticity and it produces rich results.
In the last a few decades, the most fundamental and influential work related to the Lerch zeta-function are [22], [43] [61], [94] which are partly incorporated in [14]. Milnor [61] gives a very clear description of the 2-dimensional vector space of Kubert functions and above all things elucidates the functional equation (11), (12) for the Hurwitz and the Lerch zeta-function as the relation between basis elements. Yamamoto [94] established the vector space structure of periodic arithmetic functions C(M) or the corresponding Dirichlet series D(M) and evaluated the special values L(k, χ) by providing sound basics for the Lerch zeta-function. The main results are the expressions of short character sums in terms of these special values, Cf. [85]. The method of discrete Fourier transform DFT (or finite Fourier series) has been also developed therein. Cf. e.g. [64], [85], [88].
As a continuation of these, the second author [36] applied the Deninger-Meyer method to evaluate the three types of quantities L(k, χ), L (k) (1, χ) and the generalized Euler constants γ k (a, M), cf. §4 below.
The boundary Lerch zeta-function (6) has its counterpart, the Hurwitz zeta-function This is continued meromorphically over the whole plane with a simple pole at s = 1. These are connected by the Hurwitz formula (i. e. the functional equation for the Hurwitz zeta-function), with x = 1 being the limiting case (14): for σ > 1, 0 < x ≤ 1, while its reciprocal is By (12), the boundary Lerch zeta-function 1−s (x) is continued meromorphically over the whole plane with s = 0 a plausible singular point. However, it is also a reguler posit since the pole of Γ(s) is cancelled by the factor ζ(s, x) + ζ(s, 1 − x) = 0 by (74). Cf. [37, pp. 145-147], [14, pp. 65-84], [43] etc. Both of these reduce to the Riemann zeta-function for x ∈ Z for the former and x = 1 for the latter: valid for σ > 1 in the first instance. This is continued meromorphically over the whole place with a simple pole at s = 1 by way of the functional equation 4 of 23 which is a direct special case of (19) and indeed is equivalent to (12). Most of known zeta-functions satisfy the functional equation of this kind and the relations equivalent to it, the modular relations, have been developed in [38]. We remark that both the Lerch and Hurwitz zeta-functions are special cases of the Lipschitz-Lerch transcendent L(ξ, s, x) ([14, Chapter 3]) which in turn is the boundary function of the Hurwitz-Lerch zeta-function Φ(z, s, x) In the paper [13] ubiquity of the Lerch zeta-function, especially the monologarithm 1 (x) (21) of the complex exponential argument, has been pursued.
In [85] the ubiquity and omnipotence is shown of the (boundary) Lerch zeta-function by proving above all things that even Yamamoto's decisive results on short character sums [94], in the long run, are consequences of the modular relation, i.e. the functional equation for the Lerch zeta-function. Also mentioned is the closed expressions for the Laurent coefficients, called generalized Euler constants, cf. §4 below.
Below we assemble various known facts about Lerch and Hurwitz zeta-function that lie scattered around in literature.
For s = 1, the series on the right of (6) is uniformly convergent in an interval not containing an integer and defines the polylogarithm function of order 1, which is indeed, the boundary function given as the genuine Fourier series-the monologarithm function (21) below. Cf. [51]. We refer to (9) as the Lerch zeta-function with (6) as its boundary function.
Thus the Lerch zeta-function incorporates three aspects-modular function (9), the boundary function as the zeta-function (6) and the monologarithm function (6) with s = 1.
For fixed s ∈ C, the Lerch zeta-function (9) is a one-valued analytic function on the τ-plane with slit along negative imaginary axis and its translations by integers, i.e.
( κ a )B a x κ−a be the κ-th Bernoulli polynomial of degree κ with B 0 (x) = 1 and let [x] be the integer part of x .
For κ ∈ N, the κ-th periodic Bernoulli polynomialB κ (x) is defined bȳ indicates the fractional part. The following is well-known and taken as a heaven-sent fact that it has the Fourier expansion, cf.
where the prime on the summation sign means that n = 0 is excluded and the summation is taken in symmetric sense.
In [14, (3.7), pp. [48][49][50] there is a list of equivalent expressions for the Hurwitz and reverse Hurwitz formulas. One of them reads which in particular gives the Fourier series for ζ(0, x) = −B 1 (x). More generally, cf. [14, (1.1)]. Hence (18) may be viewed as a consequence of the modular relation (19). The novel view-point stated for the first time in [85] is that it is the very definition (6), as the boundary function of the "q-expansion," of the Lerch zeta-function that gives the Fourier expansion (18) and (69) when substituted in (70) and (71) and that for finding radial limits, use of the Lerch zeta-function is in the very nature of things since it does is the limit function.
For the latter, as has been noticed [13], there are many instances of radial limits in which the odd part, the first periodic Bernoulli polynomial, of the polylogarithm function of order 1 appears as a result of eliminating the real part, the log sin function. Cf. e.g.
[? ], [87]. The polylogarithm function of order 1 is indeed the monologarithm function, the ordinary logarithm function extended to the circle of convergence (|z| = 1, z = 1) The series is absolutely convergent for |z| < 1 and uniformly convergent for |z| = 1, z = 1. We assemble the identities for 1 (s) of which use has been made.
is its real part, the first Clausen function (or the loggamma function) and the imaginary part is (70) with κ = 1, which reads indicates the greatest integer function.

Limit values in Riemann's fragment II
We elucidate the main ingredients from the first author's paper [86] and its sequel [87]. [86] condenses the 67 pages long paper [21] into 17 pages.
In order to remove singularities, it is rather common to use a well-known device of taking the odd part or an alternate sum described by ∑ 2 n a n = ∑ n a n − ∑ 2|n a n (24) resp. ∑ n (−1) n a n = ∑ 2|n a n − ∑ 2 n a n = 2 ∑ 2|n a n − ∑ n a n , by (24), where n runs over a finite range or the series are absolutely convergent. All sums in Definition 1 are odd parts. Thus the following result follows if only we may take the limit z → ξ ∈ R, which is assured by Theorem 2 below. All the theorems that Riemann considers in the second fragment are rephrases of the results of §40 of Jacobi and we state them as the following Definition 1. Let τ ∈ H. The elliptic modular functions k = k(τ), K = K(τ), k = k (τ) are defined respectively by and where in the last two sums, following Riemann, p runs through odd integers.

Theorem 1. ([86, Theorem 3])
Let ξ = M Q be a rational number with M even and Q > 1 and let z = ye πiξ , y ∈ [0, 1). Then we have and log 2K and for Q odd. Similar but more involved results hold for Q even.
To state the Dirichlet-Abel theorem [86, Theorem 1] we need rudiments of discrete Fourier transform, DFT. Its theory is stated in many literature, [64], [88], [92, pp. 89-109] etc. The theory of DFT for arithmetic functions has been developed in [52] in the case of periodic functions. Cf. also [37, §8.1], where a is an integer variable. Then the set {ε j (a)|1 ≤ j ≤ M} forms a basis of the vector space C(M) of all periodic functions f with period M. We define the DFTf (or the bth Fourier coefficient) of Then the Fourier inversion or Fourier expansion formula holds true: Note that (34) is the expression of f with respect to the basis {ε j }. An important relation reads with the Gauss sum Another instance is the following. Let χ a be the characteristic function χ a mod M ([13, p. 73]). Let Since the series in (39) It follows that D(s, f ) can be continued meromorphically over the whole plane and that it is entire if and only iff (M) = 0 which is defined bŷ where ζ(s, a, M) indicates the partial zeta-function. Hence in parallel to (40), we have another expression Proof. The first equality follows from (43) while the second follows from (40).
, and a fortiori, there are M different Then the Dirichlet series is uniformly convergent in σ > 0 and x ∈ I. If, further, R k are all continuous on I, then F(s) is also continuous on I and where ψ is the Euler digamma function (1) and 1 (x) is the first polylogarithm function (21). For a periodic (41) and γ 0 ( f ) by (75).
We note that if f is odd (cf. (48) below), thenf (M) = 1 2 f odd + f odd = 0, which explains why the odd parts are far easier to treat and the left-hand side of (47) reduces to D(1, f ). Enough to recall the case of the class number formula for an odd Dirichlet character. For a more general result than (47), cf. Theorem 4 below. [32], [53]) We let be odd, resp. even part of f : which amounts to (50) on clearing the denominators and multiplying by π M s−1 2 . The Dirichlet series φ(s), ψ j (s) as in [38] are said to satisfy the ramified functional equation, a special case of [38, (7.18)

Generalized Euler constants
The genesis of generalized Euler constants lies in the Laurent expansion of ζ(s) around s = 1: where γ = γ 0 indicates the Euler constant. As one of the driving forces in number theory has been generalizing to an arithmetic progression, it is natural to consider the generalized Euler constants γ k (a, M) for an arithmetic progression a mod M. The kth one is defined in the first instance by A concise survey of literature about them may be found in [85]. One of two most proper ways of introducing them is through the Laurent expansion of the partial zeta-function defined in (42): Here ζ(s, x) indicates the Hurwitz zeta-function (10) whose Laurent expansion were obtained by Wilton [91], Berndt [5], Balakrishnan [1] et al.
This is a genuine generating function for them: holds and moreover, The other method, as can be predicted e.g. from the Hurwitz formula (11), depends on the Lerch zeta-function and the method is called the Deninger-Meyer method as described below.
Since (38) it follows that as is indicated by [39]. Hence finding the derivatives and the generalized Euler constants lead to the same thing.
In Deninger [22], cf. Meyer [60, pp. 526-557], it has been shown that the best and most proper approach to the determination of Laurent coefficients of a zeta-function is to use the reciprocal of the Hurwitz formula, i.e. to appeal to the functional equation (12) ( [14, pp. 48-51]), expanding the analytic function on the right-hand side into power series around s = 0, which involves higher derivatives of the Hurwitz zeta-function. We call this the Deninger-Meyer method. The Hurwitz zeta-function and its derivatives can be characterized as a principal solution to the difference equation (89), cf. §5 below. This establishes the closed formula for the derivative of the Dirichlet L-function L(s, χ) at s = 1 leading to the generalized Lerch-Chowla-Selberg formula for a real quadratic field beyond that of an imaginary quadratic field. Thus, this gives the evaluation of L (k) (1, χ) for k = 0, 1. Higher order derivatives are computed in [36] and then more extensively in [12]. Cf. [14, pp. 177-191]. This procedure is a reminiscence of Stark's method of evaluating the value at s = 0 rather at s = 1 of L-functions, [18], [76], [77] etc. Let D = d dt [66,p. 192,Problem 5] gives where S(n, k) are Stirling numbers of the second kind defined by Here ∆ indicates the difference operator [19, p. 13], [66, p. 202] defined by (63) and in general

Proposition 2. [94]
For τ ∈ H and n ∈ N −n (τ) = where the integral is taken along a line joining 0 and τ. The boundary functions are whose special case being whereB κ (x) is the κ-th periodic Bernoulli polynomial in (17) and A κ (x) is the counterpart, the κ-th Clausen function. The first Clausen function is given by (22) and higher order ones are obtained by integration: where the sign function sgn(n) is 1 for n > 0, −1 for n < 0 and 0 for n = 0.
Proof. It suffices to prove (66), which is an immediate consequence of (61).
However, using the analytic function s (x), the situation is much more transparent. Indeed, from (68) we have natural expressions: and which are in conformity with the functional equations (parity relation) Since we have the formula the parity relation (72) entails This is a generalization of trivial zeros of the Riemann zeta-function.
We turn to the case of Dirichlet series D(s, f ) with periodic coefficients f (n) described toward the end of §3.
i.e. 1 a M is a generating function of σ a k and a fortiori, the case k = 0 leading to (47).
This is a prototype of the equivalence theorem in [30] to the effect that the finite expressions for the Dirichlet class number formula is equivalent to (79).
Funakura [27] deduces as his main theorem, the Kronecker limit formula for D(s, f ) which is compactified as Corollary 2 below. His proof depends on the first equality of Proposition 1 and therein he uses Gauss' formula to transform the result into the required form. For its elucidation we refer to [85].
which may be expressed as and as (81) and (82) follow from the results stated toward the end of (2).
The pioneer results of Briggs [7] and Lehmer [46] are immediate and natural consequences of (47) or (78) with k = 0 as well as a corollary to Funakura's theorem, Corollary 2: Theorem 4 shows ubiquity of 1 (x). However, the main formula (78) is in its first stage. For the Deninger-Meyer method involves its second stage of expressing the derivatives of the Lerch zeta-function by those of the Hurwitz zeta-function.
We may compute the higher order Laurent coefficients of D(s, f ) around s = 1 by (40) by applying the Deninger-Meyer method, i.e. by finding the coefficient (12).
where a, b, c, d run through integers ≥ 0 whose sum is k + 1. In actual calculation, one can omit those terms which correspond to d = 0, 2|c by (74). Corresponding to (85), we have Hence

Difference equations
According to [22, p. 173] there is a solution to the difference equation given by the Gaussian representation such that R k (x) is convex for large argument > A > 0 and is the uniquely determined solution to (89) having the value 0 at x = 1 and a fortiori on choosing The well-known Lerch formula is a special case of (93). The R-function or higher order derivatives of the Hurwitz zeta-function is introduced using the Dufresnoy-Pisot theorem which is a generalization of the Bohr-Mollerop theorem [11] while in [36] Nörlund's principal solution [62] is used which is also used in [24] to introduce a generalized gamma function to express the generalized Euler constants. Its logarithmic derivative appears in Ramanujan's second notebook [6]. Higher-order derivatives of Dirichlet L-function has been studied in [13], [14, §8.1] using the Deninger R k -function. In a similar setting, Vignéras [83] uses the Dufresnoy-Pisot theorem to introduce the Barnes multiple gamma function (cf. [74, pp. 49-50]). Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 17 February 2020 doi:10.20944/preprints202002.0245.v1 After [7] and [46] it was Dilcher [23], [24] and Kanemitsu [36] who developed the theory of generalized Euler constants. Most of these results have been summarized and elucidated in [69]. There are further generalizations and we refer to [17] and references there given. We may use the Deninger-Meyer method to find Laurent coefficients of a more general class of Dirichlet series including e.g. [26], [39] in terms of principal solutions to a difference equation, which will be conducted elsewhere.
As an analogy to log Γ(x), Dilcher [24, (4.3)] introduces the kth generalized gamma function log Γ k (x) as a principal solution to the DE By Theorem 5, we have apparently so that we could build a theory of log Γ k (x) on that of the Hurwitz zeta-function (for the case of the theory of gamma function on the Riemann zeta-function, cf. [37,Chapter 5]).
The general formula for the difference operator of order α ∈ N with difference y ≥ 0 is given by If f has the α-th derivative f (α) , then This has been most prominently applied by Landau [42] and has been constantly used by later authors in the context of Riesz means.

Abel-Tauber process
Our aim is to formulate a theorem which improves the asymptotic formulas in [10] and generalizes [2] to the case of a general Dirichlet series f (s) satisfying a functional equation, where we let f (s) = ∞ ∑ n=1 a n λ s n (100) be a Dirichlet series absolutely convergent in some half-plane where {a n } is a complex sequence and {λ n } is an increasing sequence with λ 1 > 0. Instead of studying the behavior of each value of a n , it is customary and more effective to consider the summatory function with empty sum being 0. Asymptotic formulas for the summatory function (101) may be obtained by various methods. The most typical way is the use of the Cauchy residue theorem and application of the functional equation, if any, satisfied by f (s). Historically, Voronoȋ's expression for the summatory function of the divisor function led to the estimate O(x 1 3 +ε ) [84]. One of the most comprehensive treatment has been given by [16]. Its generalization to the class of Dirichlet series with periodic coefficients has been done in [73].
holds true.
In the following lemma we use the notation. ϕ(s) = f (s) as in (100) and ψ(s) = ∑ ∞ n=1 b n µ s n and they satisfy the functional equation (105) with the multiple gamma factor and (101) and Q 0 (x) is essentially the sum of residues of ϕ which gives rise to the main term M(x) in Theorem 6.

Lemma 2. ([16, Theorem 4.1]) Suppose that the functional equation
is satisfied with r ≥ 0 and that the only singularities of the function ϕ(s) are poles. Then we have where for every η ≥ 0, and where y = x + O x 1−η− 1 2A , q =maximum of the real parts of singularities of ϕ, l =maximum of order of a pole with real part q, and u = β − r If in addition, a n ≥ 0, then the last term in can be suppressed.

Theorem 6. (Abel-Tauber theorem) Suppose
P l denoting a polynomial of degree l, as a result of the functional equation (Lemma 2) or otherwise. Then for 0 ≤ b < 1 − u(< 1) and k a non-negative integer Proof. By the first equality in (102), Substituting the first equality in (108) and applying integration by parts to the integral involving E(t), we have x log k λ 1 . Hence (110) follows from which (109) follows. To prove (110), we note that Hence for σ > 1 For simplicity we take b = 0 and substitute (109). It suffices to consider the case M(t) = ct log l t in which case M (t) = c(log l t + l log l−1 t) and so its contribution to the main term in (109) is We use the case b = 0. Then this simply becomes the same as in [9, p. 41] and so which shows (111) almost verbatim to that of Theorem 1 of [9]. whence Remark 1. As remarked by [9], the reult is true for A(x) = x a P(log x) + O(x u ), a − 1 < u < a. (109) gives asymptotic formulas for a large class of arithmetic functions including those studied by [10]. The results of Theorem 6 are still true if (108) is replaced by the PNT type estimate, which corresponds to the case A < 0 in [16, p. 112]. Theorem 6 generalized in this way may serve to unify the unorganized tables of asymptotic formulas given in some collections of formulas in number theory. The proof of Lemma 2 depends on the Riesz sums and then the differencing (98). In order to cover the case of periodic Dirichlet series, we need to incorporate the situation of ramified functional equations as in Smith [73], which will be conducted elsewhere.

Quellenangaben
Some authors are concerned with analogues of the generalized Euler constants in the Gaussian field Q(i) with the ring Z(i) of Gaussian integers. [29] is concerned with a bi-dimensional analogue of the Euler constant defined by and in the spirit of [35]. [26] is concerned with the Laurent expansion and he deduced from the Kronecker limit formula where η(τ) indicates the Dedekind eta-function whose 24th power is (a constant multiple of) the discriminant function where the prime on the summation sign means that the pair (m, n) = (0, 0) is excluded and first proved by Legendre. The same result is stated on [41, p. 176]. The celebrated Lerch-Chowla-Selberg formula gives a relation between two independent special functions-elliptic and gamma functions, and is a consequence of the combination of the Kronecker limit formula and the decomposition of the Dedekind zeta-function (which is, in the long run, a consequence of class field theory). It is shown that the Kronecker limit formula follows from the Fourier-Bessel expansion (a modular relation) and so does (123), and so Legendre's result (125) might, in the long run, correspond to a pre-class field theory for lemniscate functions. We will come to this elsewhere.
According to Wintner [93, p. 634], "By the theory of elliptic modular functions, all expansions considered by Jacobi in his §40 must be identical consequences of the corresponding expansions of the (principal) logarithm of the fundamental invariant (121), that is, of the expansion " Wintner [93, p. 628] also mentions "Riemann's posthumous fragment, based on $ 40 of Jacobi's Fundament Nova, consists of two parts, I and II, · · · " These imply that all the results in §3 are to be limit values of the discriminant function (or Dedekind eta-function). Thus, it seems probable that in Definition 1, one expresses the Lambert series in terms of Lerch zeta-functions and then apply the Dirichlet-Abel theorem to deduce all the results at a stretch. We might also obtain pre-Lerch-Chowla-Selberg type formulas in the spirit of (122) if we consider the even part. We will study this elsewhere.
[9], [39], [41] give Abelian theorems to the effect that if an asymptotic formula for the summatory function (101) is known, then the one for ∑ λ n ≤x n b−1 a n log k n can be derived where k ≥ 0 is an integer, which then implies the Laurent expansion for f (s) in (100). [39] uses the Laplace transform and so does [10]. The asymptotic formula is of the from A(x) = cx a + O(x u log K x), u < a. Briggs [9] enhances [8] and arrive at an Abelian theorem with A(x) = CxP(log x) + O(x u ) with P a polynomial of degree l or with K as large as we please. The latter is a weak form of the PNT and this setting has been fully generalized in [56] in which the asymptotic formula for A(x) is derived from analytic properties of the generating Dirichlet series (100). It also includes the case where (100) has logarithmic singularities, covering the best known form of the PNT. [Ber2], [3] are similar to [9].

Elucidation of some identities
We clarify the underlying reason for identities (2) and (3). Proof of (2) depends on (54), (11) and (67). Indeed, substractin (2) with x replaced by 1 − x from (2), we obtain after transformation Now taking the limit s → 0 on using (11) and (67), we conclude (2). Remarkably enough, this in conjunction with (47) leads to a generalization of the reverse Eisenstein formula [25], [54], [57, pp. 318-319]: This is not coincidental and clarifies the reason why the cotangent function appears in the Dedekind sums and the Dirichlet class number formula, [14, pp. 72-74]. According to [61], the cotangent function Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 17 February 2020 doi:10.20944/preprints202002.0245.v1 is an odd basis of the Kubert space K 0 and cot π a M , 1 ≤ a ≤ M/2 form a basis of a vector space of odd functions in C(M). [71, Lemma 2.1] which plays a crucial role in the proof of the reciprocity laws for Dedekind0like sums is a special case of (47).
We must recall the warning in [28] that formal use of (65) for linear independence over Q leads to triviality since what appear are only derivatives of cotangents.
Elucidation of the genesis of the partial fraction expansion (3) for the cotangenet function has been done in many literature and we refer to [37, pp. 87-104], [38,Chapter 4] etc. It is shown that (3) is a special case of the Fourier-Bessel series equivalent to the functional equation of Riemann's type with a single gamma factor.
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