Vistas of Special Functions
Abstract
This is a unique book for studying special functions through zeta-functions. Many important formulas of special functions scattered throughout the literature are located in their proper positions and readers get enlightened access to them in this book. The areas covered include: Bernoulli polynomials, the gamma function (the beta and the digamma function), the zeta-functions (the Hurwitz, the Lerch, and the Epstein zeta-function), Bessel functions, an introduction to Fourier analysis, finite Fourier series, Dirichlet L-functions, the rudiments of complex functions and summation formulas. The Fourier series for the (first) periodic Bernoulli polynomial is effectively used, familiarizing the reader with the relationship between special functions and zeta-functions. © 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.
... the partial fraction expansion for the contangent, which is known to be equivalent to the functional equation for the Riemann zeta-function (see [18,Chapter 5]). In this paper we refer to the results in [19,20] freely. ...
... However, it is also a regular posit since the pole of Γ(s) is cancelled by the factor ζ(s, x) + ζ(s, 1 − x) = 0 by (74). See ( [18], pp. 145-147), ( [34], pp. ...
... Hence (18) may be viewed as a consequence of the modular relation (19). The novel view-point stated for the first time in [13] 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. ...
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.
... which is one of the definitions of the Bernoulli polynomial [Vista,p. 4] ...
... which admits the asymptotic expansion [Vista,(3.7), p.56] ...
... We now recover Theorem 4 again by Theorem 5 with R n (x) = a n , a periodic sequence as in Theorem 4, a n = a k , n ≡ k(mod q). Then (6.10) reads (6.12) [Vista,p.171,(8.17)], whence F (s) is meromorphic over the whole plane except at a possible simple pole at s = 1. ...
This paper is intended for a rambling introduction to number-theoretic concepts through built-in properties of (number-theoretic) special functions. We follow roughly the historical order of events from somewhat more modern point of view. §1 deals with Euler's fundamental ideas as expounded in [Ay] and [Lan], from a more advanced standpoint. §2 gives some rudiments of Bernoulli numbers and polynomials as consequences of the partial fraction expansion. §3 states sieve-theoretic treatment of the Euler product. Thus, the events in § §1-3 more or less belong to Euler's era. §4 deals with RSA cryptography as motivated by Euler's function, with its several descriptions being given. §5 contains a slight generalization of Dirichlet's test on uniform converengence of series, which is more effectively used in §6 to elucidate Riemann's posthumous Fragment II than in [AdR]. Thus § §5-6 belong to the Dirichlet-Riemann era. §7 gives the most general modular relation which is the culmination of the Riemann-Hecke-Bochner correspondence between modular forms and zeta-functions. Appendix gives a penetrating principle of the least period that appears in various contexts.
... This is continued meromorphically over the whole plane with a unique simple pole at s = 1 (cf. e.g., [36]). Let {x} denote the fractional part of the real number x. ...
... (cf. e.g., [36], Chapter 4), we may nd the values of ζ ε (s, h, c) at non-positive integers −l ...
... whose proof depend on the functional equation. Although they state [23] that (2.10) may be proved by the θ-transformation method or by the functional equation for the Hurwitz zeta-function, we note that the latter statement lacks an essential ingredient, i.e., Eisenstein's formula ([36] ...
... We shall also use the following lemma [Vista,Lemma 8.3] in an essential way. ...
... For any a 6 0 (mod N ), one has fðaÞ ¼ X We appeal to the following form of the Dirichlet class number formula [Vista,(8.30 ...
... Remark 3. The proof hinges on the pseudogroup structure of the set of all odd characters and the form of the class number formula ð3:3Þ valid for all odd characters, in the spirit of [HKT] (cf. [Vista,Chapter 8]). The ordinary form of the class number formula for an odd primitive character mod N is ...
In this paper, we make structural elucidation of some interesting arithmetical identities in the context of the Parseval identity.
¶ In the continuous case, following Romanoff [R] and Wintner [Wi], we study the Hilbert space of square-integrable functions L2(0,1) and provide a new complete orthonormal basis-the Clausen system-, which gives rise to a large number of intriguing arithmetical identities as manifestations of the Parseval identity. Especially, we shall refer to the identity of Mikolás-Mordell.
¶ Secondly, we give a new look at enormous number of elementary mean square identities in number theory, including H. Walum's identity [Wa] and Mikolás' identity (1.16). We show that some of them may be viewed as the Parseval identity. Especially, the mean square formula for the Dirichlet L-function at 1 is nothing but the Parseval identity with respect to an orthonormal basis constructed by Y. Yamamoto [Y] for the linear space of all complex-valued periodic functions.
... A classical theorem rooted in the works of Christian Goldbach (1690-1764) says that ∞ n=2 (ζ(n) − 1) = 1 (see [1], [9], and [12, p. 142]; a short proof is given in the Appendix). The terms ζ(n)−1 of the above series decrease like the powers of 1 2 . ...
... have been studied and evaluated in a number of papers -see, for instance, [9], [12, pp. 213-219, (469), (474), (517), (518)] and also [13], [14]. ...
In this paper we demonstrate the importance of a mathematical constant which is the value of several interesting numerical series involving harmonic numbers, zeta values, and logarithms. We also evaluate in closed form a number of numerical and power series.
... A classical theorem rooted in the works of Christian Goldbach (1690-1764) says that (see [1], [9], and [12, p. 142]; a short proof is given in the Appendix). The terms ( ) 1 n ζ − of the above series decrease like the powers of 1 2 . ...
... Integration of this series yields [12, p. 173 ∑ have been studied and evaluated in a number of papers -see, for instance, [9], [12, pp. 213-219, (469), (474), 517), (518)] and also [13], [14]. ...
In this paper we demonstrate the importance of a mathematical constant which is the value of several interesting numerical series involving harmonic numbers, zeta values, and logarithms. We also evaluate in closed form a number of numerical and power series.
... Therefore we cannot expect general results on the arithmetic properties of the values of analytic functions, unless we restrict to special functions. Among many references on special functions, let us quote [30,16]. ...
... See [30,Chap. 6]. Among the results proved by Siegel in 1929 is the transcendence of the so-called Continued Fraction Constant ...
... In this section we shall state the Ewald expansion for the Epstein zeta-function. Since this topic has been thoroughly investigated in [Vista,Chapter 6] and [Vi2, Chapter 6], we refer to them for basic notions and use the notation given therein. ...
... The following example supplements the passages in [Vista,. ...
The incomplete gamma function expansion for the perturbed Epstein zeta function is known as Ewald expansion in physics and in chemistry.
In this paper we state a special case of the main formula by Haruo Tsukada (giving the modular relation) whose specifications will give almost all existing Ewald
expansions in the H-function hierarchy. An Ewald
expansion for us are those formulas given by or its variants and
especially the incomplete gamma expansion. We shall treat the case of a single gamma factor which includes both the Riemann
as well as the Hecke type of functional equations and unify them in the main theorem ( theorem 2.1). This result reveals the H-function hierarchy: the
confluent hypergeometric function series entailing the Ewald
expansions. Further some special cases of this theorem entails various well known results, e.g., Bochner-Chandrasekharan theorem, Atkinson-Berndt theorem etc.
... We may also prove (5.2) using the integral representation [Vi1,p. 56] ...
... Now that we have a trail from the functional equation to the Fourier series of polynomials and Clausen functions, we may claim that this theorem of Dirichlet could be derived from the functional equation. Note also that in the proof of [Ber2,Theorem 3.1] about the excess of residues over non-residues in the whole interval, Berndt appeals to the partial fraction expansion of the cotangent function, which has been proved to be equivalent to the functional equation (via a certain known formula) [Vi1,Chapter 5]. We hope to turn to closer study of this aspect elsewhere [KLW]. ...
We consider the zeta functions satisfying the functional equation with multiple gamma factors and prove a far-reaching theorem,an intermediate modular relation, which gives rise to many (including many of the hitherto found) arithmetical Fourier series as a consequence of the functional equation. Typical examples are the Diophantine Fourier series considered by Hardy and Littlewood and one considered by Hartman and Wintner, which are reciprocals of each other, in addition to our previous work. These have been thoroughly studied by Li, Ma and Zhang. Our main contribution is to the effect that the modular relation gives rise to the Fourier series for the periodic Bernoulli polynomials and Kummer’s Fourier series for the log sin function, thus giving a foundation for a possible theory of arithmetical Fourier series based on the functional equation.
... In the case of the ordinary Eisenstein series, their Fourier expansions have been derived by the beta-function integral (7), cf. [1,5,7,12]. This familiar Eisenstein series has also been studied and used in many places, cf. ...
... This approach was taken by Chowla and Selberg [5]. The presentation of the Chowla?Selberg integral formula and its generalization has been developed by Terras [12], in [7,11]. The revival of the use of the beta-function integral was made by Berndt [1] since it was (probably first time) used by Hardy [6] in 1908. ...
In this note, we get the Fourier expansion for the non-holomorphic Eisenstein series by slight modification of Maass’ original method, which enables us to prove as a bonus, two integral representations for the modified Bessel function of the third kind. This kind reveals a hidden inner structure of the non-holomorphic Eisenstein series and the Bessel diffenrential equation. We also explain a work of Motohashi on the Kronecker limit formula for the Epstein zeta-function from our point of view.
... 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]). ...
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. 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 also state Abelian results which yield asymptotic formulas for weighted summatory function from that for the original summatory function
... The proof can be found in [10, pp. 39-43], [33, p. 91], [24,Exercise 5.4], but we indicate it here. Rewriting (2.1) in the form ...
In this paper we shall establish the counterpart of Szmidt, Urbanowicz and Zagier’s formula [Zbl 0829.11011] in the sense of the Hecke correspondence. The motivation is the derivation of the values of the Riemann zeta-function at positive even integral arguments from the partial fraction expansion for the hyperbolic cotangent function (or the cotangent function). Since the last is equivalent to the functional equation, we may view their elegant formula as one for the Lambert series, and comparing the Laurent coefficients, we may give a functional equational approach to the short-interval character sums with polynomial weight. In view of the importance of these short-interval character sums, we assemble some handy formulations for them that are derived from Szmidt, Urbanowicz and Zagier’s formula and Yamamoto’s method [Zbl 0371.10028], which also gives the conjugate sums. We shall also state the formula for the values of the Dirichlet L-function with imprimitive characters.
... Hence, using the reciprocal relation We also recall the Clausen functions (i) We note that (1.12) is Example 3.5 of [2] in which z 0. In the form of (1.12), one cannot clearly see that it is a result of applying the functional equations (1.3) and (1.7). More importantly, by [8], [1], it is known that the functional equations are equivalent in a loose sense, i.e. they are equivalent if we take some known formulas for granted. Thus we may think of (1.12) as the result of a duplicate use of the functional equation-a double reflection (with respect to ¼ 1=2), which makes things return back to the nearly original form (for another example, see [5] [6]. ...
Espinosa and Moll [2], [3] studied “the Hurwitz transform” meaning an integral over [0, 1] of a Fourier series multiplied by the Hurwitz zeta function , and obtained numerous results for those which arise from the Hurwitz formula. Ito's recent result [4] turns out to be one of the special cases of Espinosa and Moll's theorem. However, they did not give rigorous treatment of the relevant improper integrals.
¶ In this note we shall appeal to a deeper result of Mikolás [9] concerning the integral of the product of two Hurwitz zeta functions and derive all important results of Espinosa and Moll. More importantly, we shall record the hidden and often overlooked fact that some novel-looking results are often the result of “duplicate use of the functional equation”, which ends up with a disguised form of the original, as in the case of Johnson's formula [5]. Typically, Example 9.1 ((1.12) below) is the result of a triplicate use because it depends not only on our Theorem 1, which is the result of a duplicate use, but also on (1.3), the functional equation itself.
... We also note that the reciprocal relation for the gamma function is equivalent to the Hurwitz formula, while (1.13) is its consequence (cf. [Vista,Chapter 5]). ...
In this paper we shall prove that the combination of the general distribution property and the functional equation for the Lipschitz-Lerch transcendent capture the whole spectrum of deeper results on the relations between the values at rational arguments of functions of a class of zeta-functions. By Theorem 1 and its corollaries, we can cover all the previous results in a rather simple and lucid way. By considering the limiting cases, we can also deduce new striking identities for Milnor's gamma functions, among which is the Gauss second formula for the digamma function.
... Equating (2.10) and (2.13), we complete the proof. We note that the reciprocal relation is "equivalent to" the Hurwitz formula while (1.15) is its consequence, cf.[15, Chapter 5]. Hence (1.32) is a consequence of three times applications of the functional equation (plus Eisenstein formula). ...
In this paper, we shall give a complete structural description of generalizations of the classical Eisenstein formula that
expresses the first periodic Bernoulli polynomial as a finite combination of cotangent values, as a relation between two bases
of the vector space of periodic Dirichlet series. We shall also determine the limiting behavior of them, giving rise to Gauss’
famous closed formula for the values of the digamma function at rational points on the one hand and elucidation of Eisenstein-Wang’s
formulas in the context of Kubert functions on the other.
W shall reveal that most of the relevant previous results are the combinations of the generalized Eisenstein formula and the
functional equation.
KeywordsEisenstein formula-Hurwitz zeta-function-polylogarithm function-Gauss formula
MSC(2000)11M41-11A25-11R29
... See [3] and also the recent book [4]. We begin by recalling here Apostol's definition as follows: ...
Using the Padé approximation of the exponential function, we obtain recurrence relations between Apostol–Bernoulli and between Apostol–Euler polynomials. As applications, we derive some new lacunary recurrence relations for Bernoulli and Euler polynomials with gap of length 4 and lacunary relations for Bernoulli and Euler numbers with gap of length 6.
... The Bernoulli numbers n B are given by the generating function [9] ...
We show that the formula recently derived by Coffey for the Stieltjes
constants in terms of the Bernoulli numbers is mathematically equivalent to the
much earlier representation derived by Briggs and Chowla.
... ("Anyons" are particles which are neither Fermions nor Bosons, having a fractional spin.) For further properties of the Riemann zeta and related functions we refer to [1], [2], [3], [11], [15] , [17], [18], [19] and [20]. The plan of the paper is as follows. ...
On the one hand the Fermi-Dirac and Bose-Einstein functions have been
extended in such a way that they are closely related to the Riemann and other
zeta functions. On the other hand the Fourier transform representation of the
gamma and generalized gamma functions proved useful in deriving various
integral formulae for these functions. In this paper we use the Fourier
transform representation of the extended functions to evaluate integrals of
products of these functions. In particular we evaluate some integrals
containing the Riemann and Hurwitz zeta functions, which had not been evaluated
before.
Using probability theory we derive an expression for the sum of a series of definite integrals involving upper incomplete Gamma functions. In the proof, a normal variance mixture distribution with Beta mixing distributions plays a crucial role. We also give an interesting application of our result, namely, a new summation formula for some derivatives of the Bessel functions of the first kind and the Struve functions with respect to the order.
Chowla's (inverse) problem is a deduction of linear independence over the rationals of circular functions at rational arguments from L.(1, x) ≠ 0, while determinant expressions for the (relative) class number of (subfields of) a cyclotomic field are referred to as the Maillet-Demyanenko determinants. In Wang, Chakraborty and Kanemitsu (to appear), Chowla's problem and Maillet-Demyanenko determinants (CPMD) in the case of Bernoulli polynomial entries (odd part) are unified as different-looking expressions of the (relative) class number on the grounds of the base change formula for periodic Dirichlet series, Dedekind determinant and the Euler product. Our aim here is to show that the genesis of the new theory of discrete Fourier transform as well as the Dedekind determinant is the characters of a finite Abelian group and its convolution map, thus revealing that CPMD boils down to analysis of the class number by group characters. We settle the case of Clausen function (log sine function) entries (even part) as an example. Other cases are similar.
In this paper, we shall establish a hierarchy of functional equations (as a G-function hierarchy) by unifying zeta-functions that satisfy the Hecke functional equation and those corresponding to Maass forms in the framework of the ramified functional equation with (essentially) two gamma factors through the Fourier–Whittaker expansion. This unifies the theory of Epstein zeta-functions and zeta-functions associated to Maass forms and in a sense gives a method of construction of Maass forms. In the long term, this is a remote consequence of generalizing to an arithmetic progression through perturbed Dirichlet series.
Chowla’s (inverse) problem (CP) is to mean a proof of linear independence of cotangent-like values from non-vanishing of L(1,χ) =∑n=1∞χ(n)n. On the other hand, we refer to determinant expressions for the (relative) class number of a cyclotomic field as the Maillet–Demyanenko determinants (MD). Our aim is to develop the theory of discrete Fourier transforms (DFT) with parity and to unify Chowla’s problem and Maillet–Demyanenko determinants (CPMD) as different-looking expressions of the relative class number via the Dedekind determinant and the base change formula.
The Voronoĭ summation formula is known to be equivalent to the functional equation for the square of the Riemann zeta function in case the function in question is the Mellin tranform of a suitable function. There are some other famous summation formulas which are treated as independent of the modular relation. In this paper, we shall establish a far-reaching principle which furnishes the following. Given a zeta function Z(s) satisfying a suitable functional equation, one can generalize it to Zf(s) in the form of an integral involving the Mellin transform F(s) of a certain suitable function f(x) and process it further as Z˜f(s). Under the condition that F(s) is expressed as an integral, and the order of two integrals is interchangeable, one can obtain a closed form for Z˜f(s). Ample examples are given: the Lipschitz summation formula, Koshlyakov’s generalized Dedekind zeta function and the Plana summation formula. In the final section, we shall elucidate Hamburger’s results in light of RHBM correspondence (i.e., through Fourier–Whittaker expansion).
As has been pointed out by Chakraborty et al (Seeing the invisible: around generalized Kubert functions. Ann. Univ. Sci. Budapest. Sect. Comput. 47 (2018), 185-195), there have appeared many instances in which only the imaginary part—the odd part—of the Lerch zeta-function was considered by eliminating the real part. In this paper we shall make full use of (the boundary function aspect of) the q-expansion for the Lerch zeta-function, the boundary function being in the sense of Wintner (On Riemann's fragment concerning elliptic modular functions. Amer. J. Math. 63 (1941), 628-634). We may thus refer to this as the ‘Fourier series-boundary q-series', and we shall show that the decisive result of Yamamoto (Dirichlet series with periodic coefficients. Algebraic Number Theory. Japan Society for the Promotion of Science, Tokyo, 1977, pp. 275-289) on short character sums is its natural consequence. We shall also elucidate the aspect of generalized Euler constants as Laurent coefficients after a brief introduction of the discrete Fourier transform. These are rather remote consequences of the modular relation, i.e. the functional equation for the Lerch zeta-function or the polylogarithm function. That such a remote-looking subject as short character sums is, in the long run, also a consequence of the functional equation indicates the ubiquity and omnipotence of the Lerch zeta-function—and, a fortiori, the modular relation
(S. Kanemitsu and H. Tsukada. Contributions to the Theory of Zeta-Functions: the Modular Relation Supremacy. World Scientific, Singapore, 2014).
In this paper, we aim to present new extensions of incomplete gamma, beta, Gauss hypergeometric, confluent hypergeometric function and Appell-Lauricella hypergeometric functions, by using the extended Bessel function due to Boudjelkha [4]. Some recurrence relations, transformation formulas, Mellin transform and integral representations are obtained for these generalizations. Further, an extension of the Riemann-Liouville fractional derivative operator is established.
This note is about generalizing an exam problem given at Technical
University of Cluj-Napoca on 16 July 2018.
We show that the generalised Stieltjes constants may be represented by infinite series involving logarithmic terms. Some relations involving the derivatives of the Hurwitz zeta function are also investigated
We provide a rigorous formulation of Entry 17(v) in Ramanujan's Notebooks and show how this relates to the first Stieltjes constant.
This paper studies algebraic and analytic structures associated with the
Lerch zeta function, extending the complex variables viewpoint taken in part
II. The Lerch transcendent is obtained from the Lerch zeta
function by the change of variable . We show
that it analytically continues to a maximal domain of holomorphy in three
complex variables as a multivalued function defined over the base
manifold . and compute the monodromy functions defining
the multivaluedness. For positive integer values s=m and c=1 this function is
closely related to the classical m-th order polylogarithm We study
its behavior as a function of two variables for special values where
s=m is an integer. For it gives a one-parameter deformation of the
polylogarithm, and satisfies a linear ODE with coefficients depending on c, of
order m+1 of Fuchsian type. We determine its (m+1)-dimensional monodromy
representation, which is a non-algebraic deformation of the monodromy of
The Stieltjes constants have attracted considerable attention in recent years
and a number of authors, including the present one, have considered various
ways in which these constants may be evaluated. The primary purpose of this
paper is to belatedly highlight the fact that Deninger actually ascertained the
first generalised Stieltjes constant at rational arguments as long ago as 1984
and that all of the higher constants (at rational arguments) were determined in
principle by Chakraborty, Kanemitsu and Kuzumaki in 2009. Equivalent results
were obtained by Musser in his 2011 thesis.
The authors of the above papers simply referred to the constants as the
Laurent coefficients which explains why various electronic searches conducted
by this author for Stieltjes constants did not readily highlight these
particular sources.
In this paper the author has employed a slightly different argument to obtain
a simpler expression for the results originally derived by Chakraborty et al.
in 2009.
This is the first of four papers that study algebraic and analytic structures associated to the Lerch zeta function. This paper studies “zeta integrals” associated to the Lerch zeta function using test functions, and obtains functional equations for them. Special cases include a pair of symmetrized four-term functional equations for combinations of Lerch zeta functions, found by A. Weil, for real parameters (a,c) with 0<a,c<1. It extends these functions to real a and c, and studies limiting cases of these functions where at least one of a and c take the values 0 or 1. A main feature is that as a function of three variables (s,a,c), in which a and c are real, the Lerch zeta function has discontinuities at integer values of a and c. For fixed s, the function ζ(s,a,c) is discontinuous on part of the boundary of the closed unit square in the (a,c)-variables, and the location and nature of these discontinuities depend on the real part Re(s) of s. Analysis of this behavior is used to determine membership of these functions in L p ([0,1] 2 ,dadc) for 1≤p<∞, as a function of Re(s). The paper also defines generalized Lerch zeta functions associated to the oscillator representation, and gives analogous four-term functional equations for them. Part II, see Forum Math. 24, No. 1, 49–84 (2012; Zbl 1253.11086).
We shall develop the theory of Barnes multiple zeta- function from a slightly different point of view using the No ̈rlund generalized Bernoulli polynomials and apply it to the integral of the multiple gamma function. We will provide explicit formula for log Γ3-function by which we may supplement [24, p.307], in which the general case is treated only up to r = 2 (G-function). This allows us to convert sums involving Hurwitz zeta-function into the sums involving multiple gamma function, thus covering all possible formulas in this direction.
The aim of this paper is two folds. First, we shall prove a general reduction theorem to the Spannenintegral of products of (generalized) Kubert functions. Second, we apply the special case of Carlitz’s theorem to the elaboration of earlier results on the mean values of the product of Dirichlet L-functions at integer arguments. Carlitz’s theorem is a generalization of a classical result of Nielsen in 1923. Regarding the reduction theorem, we shall unify both the results of Carlitz (for sums) and Mordell (for integrals), both of which are generalizations of preceding results by Frasnel, Landau, Mikolás, and Romanoff et al. These not only generalize earlier results but also cover some recent results. For example, Beck’s lamma is the same as Carlitz’s result, while some results of Maier may be deduced from those of Romanoff. To this end, we shall consider the Stiletjes integral which incorporates both sums and integrals. Now, we have an expansion of the sum of products of Bernoulli polynomials that we may apply it to elaborate on the results of afore-mentioned papers and can supplement them by related results.
In this paper we shall make complete structural elucidation of the explicit formula for the (discrete) mean square of Dirichlet L-function at integral arguments, save for the case s=1, this being completely settled in [1] recently. We shall treat the cases of negative and positive integers arguments separately, the former case being a preliminary and inclusive in the second. It will turn out that in respective cases the characteristic difference properties of Bernoulli polynomials and of the Hurwitz zeta-function are essential and telescoping the resulting difference equations, we obtain the results, revealing the underlying simple structure (known before 1905 at least).
In this note, we shall derive the functional equation for the Hurwitz zeta-function ¿(s,u) from that for the Riemann zeta-function ¿(s), on using an integral expression for ¿(s,u) which in turn depends on the functional equation for ¿(s).
In this paper we establish a class of arithmetical Fourier series as a manifestation of the intermediate modular relation, which is equivalent to the functional equation of the relevant zeta-functions. One of the examples is the one given by Riemann as an example of a continuous non-differentiable function. The novel interest lies in the relationship between important arithmetical functions and the associated Fourier series. E.g., the saw-tooth Fourier series is equivalent to the corresponding arithmetical Fourier series with the Möbius function. Further, if we squeeze out the modular relation, we are led to an interesting relation between the singular value of the discontinuous integral and the modification summand of the first periodic Bernoulli polynomial. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
We continue our study on arithmetical Fourier series by considering two Fourier series which are related to Diophantine analysis.
The first one was studied by Hardy and Littlewood in connection with the classification of numbers and the second one was
studied by Hartman and Wintner by Lebesgue integration theory.
KeywordsDiophantine-Fourier series-Diophantine Dirichlet series-Riesz sum
MR(2000) Subject Classification11F66-11M26-11M41
In this paper, we consider multiplication formulas and their inversion formulas for Hurwitz–Lerch zeta functions. Inversion
formulas give simple proofs of known results, and also show generalizations of those results. Next, we give a generalization
of digamma and gamma functions in terms of Hurwitz–Lerch zeta functions, and consider its properties. In all the sections,
various results are always proved by multiplication and inversion formulas.
Hurwitz–Lerch zeta functions-Transcendental numbers-Digamma function-Gamma functionMathematics Subject Classification (2000)11M35-11J81
In 1949 Chowla and Selberg gave a very useful formula for the Epstein zeta-function associated with a positive definite binary quadratic form. Several generalizations of this formula are given here. The method of proof is new and is based on a theorem that we formerly proved for “generalized” Dirichlet series. An easy proof of Kronecker’s second limit formula is also given.
In this paper we shall unify the results obtained so far in various scattered literature, for Dirichlet characters and the associated Dirichlet L-functions, under the paradigm of periodic arithmetic functions and the associated Dirichlet series. Notably we shall determine the Laurent coefficients of the series in question to cover Funakura’s result and proceed on to prove the Ayoub-Berndt-Carlitz-Chowla-Müller-Redmond theorem.
Professor L. Carlitz has been kind enough to point out that the functions βn(a, (α) which were used in [1] to evaluate the Lerch zeta function π(x, a, s) for negative integer values of s have occurred elsewhere in the literature in other connections, for example in [2] and [3]. As Carlitz points out, formula (3.3) of [1] leads to the result which, for integer values of the variable a, makes apparent the relation of the functions βn(a, (α) with the Mirimanoff polynomials discussed by Vandiver in[3]. There is a misprint in the next to last equation on p. 164 of [1]. The coefficient of a2/2 in the expression for π(x, a, -2) should read i cot Πx + 1 instead of i cot Πx + 1/4.