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Register a new userExplicit Evaluations of Sums of Sequence Tails
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In this paper, we use Abels summation formula to evaluate several quadratic and cubic sums of the form: [{F_N}left( {A,B;x} right) := sumlimits_{n = 1}^N {left( {A - {A_n}} right)left( {B - {B_n}} right){x^n}} ,;x in [ - 1,1]] and [Fleft( {A,B,zeta (r)} right): = sumlimits_{n = 1}^infty {left( {A - {A_n}} right)left( {B - {B_n}} right)left( {zeta left( r right) - {zeta_n}left( r right)} right)} ,] where the sequences $A_n,B_n$ are defined by the finite sums ${A_n} := sumlimits_{k = 1}^n {{a_k}} , {B_n} := sumlimits_{k = 1}^n {{b_k}} ( {a_k},{b_k} =o(n^{-p}),{mathop{Re} olimits} left( p right) > 1 $) and $A = mathop {lim }limits_{n to infty } {A_n},B = mathop {lim }limits_{n to infty } {B_n},Fleft( {A,B;x} right) = mathop {lim }limits_{n to infty } {F_n}left( {A,B;x} right)$. Namely, the sequences $A_n$ and $B_n$ are the partial sums of the convergent series $A$ and $B$, respectively. We give an explicit formula of ${F_n}left( {A,B;x} right)$ by using the method of Abels summation formula. Then we use apply it to obtain a family of identities relating harmonic numbers to multiple zeta values. Furthermore, we also evaluate several other series involving multiple zeta star values. Some interesting (known or new) consequences and illustrative examples are considered.
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We study several variants of Euler sums by using the methods of contour integration and residue theorem. These variants exhibit nice properties such as closed forms, reduction, etc., like classical Euler sums. In addition, we also define a variant of multiple zeta values of level 2, and give some identities on relations between these variants of Euler sums and the variant of multiple zeta values.
In recent work, Sun constructed two $q$-series, and he showed that their limits as $qrightarrow1$ give new derivations of the Riemann-zeta values $zeta(2)=pi^2/6$ and $zeta(4)=pi^4/90$. Goswami extended these series to an infinite family of $q$-series, which he analogously used to obtain new derivations of the evaluations of $zeta(2k)inmathbb{Q}cdotpi^{2k}$ for every positive integer $k$. Since it is well known that $Gammaleft(frac{1}{2}right)=sqrt{pi}$, it is natural to seek further specializations of these series which involve special values of the $Gamma$-function. Thanks to the theory of complex multiplication, we show that the values of these series at all CM points $tau$, where $q:=e^{2pi itau}$, are algebraic multiples of specific ratios of $Gamma$-values. In particular, classical formulas of Ramanujan allow us to explicitly evaluate these series as algebraic multiples of powers of $Gammaleft(frac{1}{4}right)^4/pi^3$ when $q=e^{-pi}$, $e^{-2pi}$.
We define a new kind of classical digamma function, and establish its some fundamental identities. Then we apply the formulas obtained, and extend tools developed by Flajolet and Salvy to study more general Euler type sums. The main results of Flajolet and Salvys paper cite{FS1998} are the immediate corollaries of main results in this paper. Furthermore, we provide some parameterized extensions of Ramanujan-type identities that involve hyperbolic series. Some interesting new consequences and illustrative examples are considered.
We develop a new method for studying sums of Kloosterman sums related to the spectral exponential sum. As a corollary, we obtain a new proof of the estimate of Soundararajan and Young for the error term in the prime geodesic theorem.
The paper compares the asymptotic of the expressions $frac {1} {x} sumlimits_{n leq x} {f(n)}$ and $sumlimits_{n leq x} {frac {f(n)} {n}}$, $frac {1} {x} sumlimits_{p leq x} {f(p)}$ and $sumlimits_{p leq x} {frac {f(p)} {p}}$. The asymptotic of sums $sumlimits_{n leq x} {frac {f(n)} {n}}$ and $sumlimits_{p leq x} {frac {f(p)} {p}}$ ($n,p$ - respectively, positive and prime numbers) are determined if the asymptotic of sums are known, respectively: $sumlimits_{n leq x} {f(n)}$,$sumlimits_{p leq x} {f(p)}$.
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