Euler--Maclaurin and Poisson analogues of the summations $\sum_{a < n \leq b} χ(n) f(n)$, $\sum_{a < n \leq b} d(n) f(n)$, $\sum_{a < n \leq b} d(n) χ(n) f(n)$ have been obtained in a unified manner, where $(χ(n))$ is a periodic complex sequence; $d(n)$ is the divisor function and $f(x)$ is a sufficiently smooth function on $[a,b]$. We also state a generalised Abel's summation formula, generalised Euler's summation formula and Euler's summation formula in several variables.
We discover new Voronoi formulae for automorphic forms on GL($n$) for $n\geq 4$. There are $[n/2]$ different Voronoi formulae on GL($n$), which are Poisson summation formulae weighted by Fourier coefficients of the automorphic form with twists by some hyper-Kloosterman sums.
In terms of the derivative operator, integral operator and Saalschütz's theorem, two families of summation formulae involving generalized harmonic numbers are established.
In terms of the derivative operator and Watson-type $_3F_2$-series identities, three families of summation formulae involving generalized harmonic numbers are established.
By means of the derivative operator and Whipple-type $_3F_2$-series identities, two families of summation formulae involving generalized harmonic numbers are established.
By simple elementary method,we obtain with ease,a highly simple expression for the remainder term of the divisor problem and use it to obtain an Euler-Maclaurin analogue of summation involving divisor function.We also obtain a relation connecting the remainder term of the divisor problem and the remainders of approximate functional equations for Riemann zeta function and its square,for positive values of arguments.
In this paper we establish a new summation method by expanding $\prod_{k}(1-\frac{z}{a_{k}})^{-1}$ with two approaches: the Taylor expansion and the infinite partial fraction decomposition. Here we focus on the case when $a_{k}$ is arithmetic sequence. By this summation we obtain many equalities involve Hurwitz zeta function and Gammma function.
A generalization of the classical Lipschitz summation formula is proposed. It involves new polylogarithmic rational functions constructed via the Fourier expansion of certain sequences of Bernoulli--type polynomials. Related families of one--dimensional hyperfunctions are also constructed.
This paper presents a family of rapidly convergent summation formulas for various finite sums of analytic functions. These summation formulas are obtained by applying a series acceleration transformation involving Stirling numbers of the first kind to the asymptotic, but divergent, expressions for the corresponding sums coming from the Euler-Maclaurin summation formula. While it is well-known that the expressions obtained from the Euler-Maclaurin summation formula diverge, our summation formulas are all very rapidly convergent and thus computationally efficient.
In 1776, L. Euler proposed three methods, called prima methodus, secunda methodus and tertia methodus, to calculate formulae for double zeta values. However strictly speaking, his last two methods are mathematically incomplete and require more precise reformulation and more sophisticated arguments for their justification. In this paper, we reformulate his formulae, give their rigorous proofs and also clarify that the formulae can be derived from the extended double shuffle relations.
A general Voronoï summation formula for the (metaplectic) double cover of $\text{GL}_2$ is derived via the representation theoretic framework à la Ichino--Templier. The identity is also formulated classically and used to establish Voronoï summation formulae for half-integral weight modular forms and Maaß forms.
Let $(α_n(a,k),β_n(a,k))$ be a WP-Bailey pair. Assuming the limits exist, let \[ (α_n^*(a),β_n^*(a))_{n\geq 1} = \lim_{k \to 1}\left(α_n(a,k),\frac{β_n(a,k)}{1-k}\right)_{n\geq 1} \] be the \emph{derived} WP-Bailey pair. By considering a particular limiting case of a transformation due to George Andrews, we derive new basic hypergeometric summation and transformation formulae involving derived WP-Bailey pairs. We then use these formulae to derive new identities for various theta series/products which are expressible in terms of certain types of Lambert series.
We calculate mean values of $\operatorname{GL}_n$-automorphic $L$-functions twisted by primitive even Dirichlet characters of prime-power conductor, at arbitrary points within the critical strip, by derivation of special Voronoi summation formulae. Our calculation is novel in that the twisted sum can be expressed in terms of the average itself, and also that it sees the derivation of various new summation formulae in the setting of prime-power modulus. One consequence, as we explain, is to show the analytic continuation and additive summation formulae for hyper-Kloosterman Dirichlet series ass…
We extend formulae of Mertens and Mirsky on the asymptotic behaviour of the standard Euler function to the Euler functions of principal rings of integers of imaginary quadratic number fields, giving versions in angular sector and with congruences.
We discuss two theorems in analytic number theory and combinatory analysis that have seen increased use in recent years. A corollary to a Tauberian theorem of Ingham allows one to quickly prove asymptotic formulas for arithmetic sequences, so long as the corresponding generating function exhibits exponential growth of a certain form near its radius of convergence. Two common methods for proving the required analytic behavior are modular transformations and Euler-Maclaurin summation. However, these results are sometimes stated without certain technical conditions that are necessary for the comp…
In this paper, we study the formulae for a product of two product Euler polynomials. From this study, we derive some formulae for the integral of the product of two or more Ruler polynomials.
In this paper, we work out some explicit formulae for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. As applications of these formulae, we give new closed form representations of several quadratic Euler sums through Riemann zeta function and linear sums. The given representations are new.
In this paper, we define a parametric variant of generalized Euler sums and call them the (alternating) parametric Euler $T$-sums. By using the contour integration method and residue theorem, we establish several explicit formulae for the linear parametric Euler $T$-sums. Furthermore, by applying the results, we obtain explicit formulae for the Hoffman's (alternating) double $t$-values and Kaneko-Tsumura's (alternating) double $T$-values.
This paper is concerned with the generalized Euler polynomial matrix $\E^{(α)}(x)$ and the Euler matrix $\E$. Taking into account some properties of Euler polynomials and numbers, we deduce product formulae for $\E^{(α)}(x)$ and determine the inverse matrix of $\E$. We establish some explicit expressions for the Euler polynomial matrix $\E(x)$, which involving the generalized Pascal, Fibonacci and Lucas matrices, respectively. From these formulae we get some new interesting identities involving Fibonacci and Lucas numbers. Also, we provide some factorizations of the Euler polynomial matrix in …
We generalize the standard Poisson summation formula for lattices so that it operates on the level of theta series, allowing us to introduce noninteger dimension parameters (using the dimensionally continued Fourier transform). When combined with one of the proofs of the Jacobi imaginary transformation of theta functions that does not use the Poisson summation formula, our proof of this generalized Poisson summation formula also provides a new proof of the standard Poisson summation formula for dimensions greater than 2 (with appropriate hypotheses on the function being summed). In general, ou…
The Apostol-Dedekind sum with quasi-periodic Euler functions is an analogue of Apostol's definition of the generalized Dedekind sum with periodic Bernoulli functions. In this paper, using the Boole summation formula, we shall obtain the reciprocity formula for this sum.
We provide a multidimensional weighted Euler--MacLaurin summation formula on polytopes and a multidimensional generalization of a result due to L. J. Mordell on the series expansion in Bernoulli polynomials. These results are consequences of a more general series expansion; namely, if $χ_{τ\mathcal{P}}$ denotes the characteristic function of a dilated integer convex polytope $\mathcal{P}$ and $q$ is a function with suitable regularity, we prove that the periodization of $qχ_{τ\mathcal{P}}$ admits an expansion in terms of multivariate Bernoulli polynomials. These multivariate polynomials are re…
In the sixth chapter of his notebooks Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula to the partial sums of the series. This method is now called the Ramanujan summation process. In this paper we calculate the Ramanujan sum of the exponential generating functions $\sum_{n\geq 1}\log n e^{nz}$ and $\sum_{n\geq 1}H_n^{(j)} e^{-nz}$ where $H_n^{(j)}=\sum_{m=1}^n \frac{1}{m^j}$. We find a surprising relation between the two sums when $j=1$ from w…
Lucy Slater used Bailey's $_6ψ_6$ summation formula to derive the Bailey pairs she used to construct her famous list of 130 identities of the Rogers-Ramanujan type. In the present paper we apply the same techniques to Chu's $_{10}ψ_{10}$ generalization of Bailey's formula to produce quite general Bailey pairs. Slater's Bailey pairs are then recovered as special limiting cases of these more general pairs. In re-examining Slater's work, we find that her Bailey pairs are, for the most part, special cases of more general Bailey pairs containing one or more free parameters. Further, we also find ne…
The bilateral series corresponding to many of the third-, fifth-, sixth- and eighth order mock theta functions may be derived as special cases of $_2ψ_2$ series \[ \sum_{n=-\infty}^{\infty}\frac{(a,c;q)_n}{(b,d;q)_n}z^n. \] Three transformation formulae for this series due to Bailey are used to derive various transformation and summation formulae for both these mock theta functions and the corresponding bilateral series. \\ New and existing summation formulae for these bilateral series are also used to make explicit in a number of cases the fact that for a mock theta function, say $χ(q)$, and …
Leonhard Euler likely developed his summation formula in 1732, and soon used it to estimate the sum of the reciprocal squares to 14 digits --- a value mathematicians had been competing to determine since Leibniz's astonishing discovery that the alternating sum of the reciprocal odd numbers is exactly $π/4$. This competition came to be known as the Basel Problem, and Euler's approximation probably spurred his spectacular solution in the same year. Subsequently he connected his summation formula to Bernoulli numbers, and applied it to many other topics, masterfully circumventing that it almost a…
We have established novel integral representations of the Riemann zeta-function and Dirichlet eta-function based on powers of trigonometric functions and digamma function, and then use these representations to find close forms of Laurent series expansions of these same powers of trigonometric functions and digamma function. The so obtained series can be used to find numerous summation rules for certain values of the Riemann zeta and related functions and numbers, such as e.g. Bernoulli and Euler numbers.
In this note, we provide refined estimates of the following sums involving the Euler totient function: $$\sum_{n\le x} ϕ\left(\left[\frac{x}{n}\right]\right) \qquad \text{and} \qquad \sum_{n\le x} \frac{ϕ([x/n])}{[x/n]}$$ where $[x]$ denotes the integral part of real $x$. The above summations were recently considered by Bordellès et al. and Wu.
An elliptic $BC_n$ generalization of the classical two parameter Bailey Lemma is proved, and a basic one parameter $BC_n$ Bailey Lemma is obtained as a limiting case. Several summation and transformation formulas associated with the root system $BC_n$ are proved as applications, including a $_6ϕ_5$ summation formula, a generalized Watson transformation and an unspecialized Rogers--Selberg identity. The last identity is specialized to give an infinite family of multilateral Rogers--Selberg identities. Standard determinant evaluations are then used to compute $B_n$ and $D_n$ generalizations of t…
This note provides truncated formulae with explicit error terms to compute Euler products over primes in arithmetic progressions of rational fractions. It further provides such a formula for the product of terms of the shape $F(1/p, 1/p^s)$ when $F$ is a two-variable polynomial with coefficients in $\mathbb{C}$ and satisfying some restrictive conditions.
In this paper we present a generalization of Faulhaber's formula to sums of arbitrary complex powers $m\in\mathbb{C}$. These summation formulas for sums of the form $\sum_{k=1}^{\lfloor x\rfloor}k^{m}$ and $\sum_{k=1}^{n}k^{m}$, where $x\in\mathbb{R}^{+}$ and $n\in\mathbb{N}$, are based on a series acceleration involving Stirling numbers of the first kind. While it is well-known that the corresponding expressions obtained from the Euler-Maclaurin summation formula diverge, our summation formulas are all very rapidly convergent.
In the present article, we study Bell based Euler polynomial of order α and investigate some useful correlation formula, summation formula and derivative formula. Also, we introduce some relation of string number of the second kind. Moreover, we drive several important formulae of bell based Euler polynomial by using umbral calculus.
In this paper, we will deal with some new formulae for two product Genocchi polynomials together with both Euler polynomials and Bernoulli polynomials. We get some applications for Genocchi polynomials. Our applications possess a number of interesting properties to study in Theory of Analytic numbers which we express in the present paper.
A method is described to sum multi-dimensional arithmetic functions subject to hyperbolic summation conditions, provided that asymptotic formulae in rectangular boxes are available. In combination with the circle method, the new method is a versatile tool to count rational points on algebraic varieties defined by multi-homogeneous diagonal equations.
In this paper we construct a new q-Euler numbers and polynomials. By using these numbers and polynomials, we give the interesting formulae related to alternating sums of powers of consecutive q-integers following an idea due to Euler.
In this paper, we show that Riemann hypothesis (concerning zeros of the zeta function in the critical strip) is equivalent to the analytic continuation of Euler products obtained by restricting the Euler zeta product to suitable subsets $M_k$, $k\geq 1$ of the set of prime numbers. Each of these Euler product defines so a partial zeta function $ζ_{k}(s)$ equal to a Dirichlet series of the form $\sum ε(n)/n^s$, with coefficients $ε(n)$ equal to 0 or 1 as n belongs or not to the population of integers generated by $M_k$. We show that usual formulas of the arithmetic adapt themselves to such popu…
The Stieltjes constants $γ_k$ appear in the regular part of the Laurent expansion of the Riemman and Hurwitz zeta functions. We demonstrate that these coefficients may be written as certain summations over mathematical constants and specialized hypergeometric functions $_pF_{p+1}$. This family of results generalizes a representation of the Euler constant in terms of a summation over values of the trigonometric integrals Si or Ci. The series representations are suitable for acceleration. As byproducts, we evaluate certain sine-logarithm integrals and present the leading asymptotic form of the p…
Let $(a;q)_{\infty}$ be the $q$-Pochhammer symbol and $\mathrm{li}_2(x)$ be the dilogarithm function. Let $\prod_{α,β,γ}$ be a finite product with every triple $(α,β,γ)\in(\mathbb{R}_{>0})^3$ and $S_{αβγ}\in\mathbb{R}$. Also let the triple $(A,B,v)\in\left(\mathbb{R}_{>0}\times\mathbb{R}^2\right)\cup\left(\{0\}^2\times\mathbb{R}_{>0}\right)\cup\left(\{0\}\times\mathbb{R}_{<0}\times\mathbb{R}\right)$. In this work, we let $z=e^v$, denote by $H_{-1}(u)=vu-Au^2+\sum_α\mathrm{li}_2(e^{-αu})\sum_{β,γ} β^{-1}S_{αβγ}$ and consider the Eulerien series \[\mathcal{H}(z;q)=\sum_{m=0}^{\infty}\frac{q^{Am^…
By making use of the multiplicate form of the extended Carlitz inverse series relations, we establish two general `dual' theorems of Jackson's summation formula for well--poised $_8ϕ_7$-series. Their duplicate forms under the partition pattern $n=\lfloor{\frac{n}2}\rfloor+\lfloor{\frac{n+1}2}\rfloor$ are explored and yield numerous $q$-series identities whose limiting cases as $q\to1$ result in classical $π$-related Ramanujan--like series of convergence rate ``$\frac1{16}$" including one for $1/π^2$ discovered by Guillera (2003). The triplicate dual formulae under the partition pattern $n=\lfl…
In this paper, we establish the following two congruences: \begin{gather*} \sum_{k=0}^{(p+1)/2}(3k-1)\frac{\left(-\frac{1}{2}\right)_k^2\left(\frac{1}{2}\right)_k4^k}{k!^3}\equiv p-6p^3\left(\frac{-1}{p}\right)+2p^3\left(\frac{-1}{p}\right)E_{p-3}\pmod{p^4},\\ \sum_{k=0}^{p-1}(3k-1)\frac{\left(-\frac{1}{2}\right)_k^2\left(\frac{1}{2}\right)_k4^k}{k!^3}\equiv p-2p^3\pmod{p^4}, \end{gather*} where $p>3$ is a prime, $E_{p-3}$ is the $(p-3)$-th Euler number and $\left(-\right)$ is the Legendre symbol. The first congruence modulo $p^3$ was conjectured by Guo and Schlosser recently.
We cover some useful techniques in computational aspects of analytic number theory, with specific emphasis on ideas relevant to the evaluation of L-functions. These techniques overlap considerably with basic methods from analytic number theory. On the elementary side, summation by parts, Euler Maclaurin summation, and Mobius inversion play a prominent role. In the slightly less elementary sphere, we find tools from analysis, such as Poisson summation, generating function methods, Cauchy's residue theorem, asymptotic methods, and the fast Fourier transform. We then describe conjectures and expe…
Jacob's ladders, Hardy-Littlewood integral (1918) and new asymptotic functional equations for Euler's Gamma function together with the tenth equivalent of the Fermat-Wiles theorem
The main goal of this paper is to give a modular type representation for the infinite product $(1-x)(1-xq)(1-xq^2)(1-xq^3)...$. It is shown that this representation essentially contains the well-known modular formulae either for Dedekind's eta function, Jacobi theta function or for certain Lambert series. Thus a new and unified approach is outlined for the study of elliptic and modular functions and related series.
We extend the classical Euler-Maclaurin expansion to sums over multidimensional lattices that involve functions with algebraic singularities. This offers a tool for the precise quantification of the effect of microscopic discreteness on macroscopic properties of a system. First, the Euler-Maclaurin summation formula is generalised to lattices in higher dimensions, assuming a sufficiently regular summand function. We then develop this new expansion further and construct the singular Euler-Maclaurin (SEM) expansion in higher dimensions, an extension of our previous work in one dimension, which r…
