Yoshihiro Takeyama
The multiple T-value, which is a variant of multiple zeta value of level two, is introduced by Kaneko and Tsumura. We show that the generating function of a weighted sum of the multiple T-values of fixed weight and depth is given in terms of the multiple T-values of depth one by solving a differential equation of Heun type.
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In this paper we prove a weighted sum formula for multiple harmonic sums modulo primes, thereby proving a weighted sum formula for finite multiple zeta values. Our proof utilizes difference equations for the generating series of multiple harmonic sums. We also conjecture several weighted sum formulas of similar flavor for finite multiple zeta values.
Weighted sum formulas for finite multiple mixed values
利用多重多对数迭代积分表达式，建立有限多重混合值的加权和公式，并推导出有限多重zeta值的二级变体间的关系。
We obtain a weighted sum formula of the zeta values at even arguments, and a weighted sum formula of the multiple zeta values with even arguments and its zeta-star analogue. The weight coefficients are given by (symmetric) polynomials of the arguments. These weighted sum formulas for the zeta values and for the multiple zeta values were conjectured by L. Guo, P. Lei and J. Zhao.
In this paper, we study the weighted sums of multiple t-values and of multiple t-star values at even arguments. Some general weighted sum formulas are given, where the weight coefficients are given by (symmetric) polynomials of the arguments.
A typical formula of multiple zeta values is the sum formula which expresses a Riemann zeta value as a sum of all multiple zeta values of fixed weight and depth. Recently weighted sum formulas, which are weighted analogues of the sum formula, have been studied by many people. In this paper, we give two formulas of weighted sums with two parameters of multiple zeta values. As applications of the formulas, we find some linear combinations of multiple zeta values which can be expressed as polynomials of usual zeta values with coeffcients in the rational polynomial ring generated by the two parame…
In this paper, we study the multiple $L$-values and the multiple zeta values of level $N$. We set up the algebraic framework for the double shuffle relations of the multiple zeta values of level $N$. Using the regularized double shuffle relations of multiple $L$-values, we give a sum formula and a weighted sum formula of multiple $L$-values. As applications, we give sum formulas and weighted sum formulas of double zeta values of level $2$ and $3$.
Hirose, Saito, and the author established the weighted sum formula for finite multiple zeta(-star) values. In this paper, we present its alternative proof. The proof is also valid for symmetric multiple zeta(-star) values.
In this paper, we investigate the asymptotics of a class of weighted sums over multiplicative functions and apply our results to deduce a stronger asymptotic form of Yitang Zhang's smoothened GPY sieve with coefficient expressions that are friendly to numerical computations.
We present a unified approach which gives completely elementary proofs of three weighted sum formulae for double zeta values. This approach also leads to new evaluations of sums relating to the harmonic numbers, the alternating double zeta values, and the Witten zeta function. We discuss a heuristic for finding or dismissing the existence of similar simple sums. We also produce some new sums from recursions involving the Riemann zeta and the Dirichlet beta functions.
We prove a sum formula with 4 parameters among finite alternating multiple zeta values which can be regarded as an alternating version of the result of Kamano on finite multiple zeta values.
Euler's sum formula and its multi-variable and weighted generalizations form a large class of the identities of multiple zeta values. In this paper we prove a family of identities involving Bernoulli numbers and apply them to obtain infinitely many weighted sum formulas for double zeta values and triple zeta values where the weight coefficients are given by symmetric polynomials. We give a general conjecture in arbitrary depth at the end of the paper.
Shifted convolution sum with weighted average : $GL(3) \times GL(3)$ setup
研究针对一般 $GL(3) \ imes GL(3)$ 移位卷积和，运用圆法证明其平均及加权平均版本的非平凡估计。
Let $\gcd(j,k)$ be the greatest common divisor of the integers $j$ and $k$. In this paper, we give several interesting asymptotic formulas for weighted averages of the $\gcd$-sum function $f(\gcd(j,k)) $ and the function $\sum_{d|k, d^{s}|j}(f*μ)(d) $ for any positive integers $j$ and $k$, namely $$ \sum_{k\leq x}\frac{1}{k^{r+1}}\sum_{j=1}^{k}j^{r}f(\gcd(j,k)) \quad \text{and} \quad \sum_{k\leq x}\frac{1}{k^{s(r+1)}}\sum_{j=1}^{k^s}j^{r} \sum_{\substack{d|k d^{s}|j}}(f*μ)(d), $$ with any fixed integer $s> 1$ and any arithmetical function $f$. We also establish mean value formulas for the erro…
The constant $C_A(n)$ is defined to be the smallest natural number $k$ such that any sequence of $k$ elements in $\mathbb Z_n$ has a subsequence of consecutive terms whose $A$-weighted sum is zero, where the weight set $A\subseteq \mathbb Z_n\setminus \{0\}$. If $C_A(n)=k$, then a sequence in $\mathbb Z_n$ of length $k-1$ which has no $A$-weighted zero-sum subsequence of consecutive terms is called an $A$-extremal sequence. We characterize these sequences for some particular weight sets.
In this paper we use the generating functions and the double shuffle relations satisfied the multiple zeta values to derive some new families of identities.
Using combinatorial techniques, we prove that the weighted sum of the inverse number of automorphisms of all finite abelian $p$-groups $\sum_G |G|^{-u} |\text{Aut}(G)|^{-1}$ is equal to $\prod_{j=u+1}^\infty\left(1-1/p^j\right)^{-1}$, where $u$ is a non-negative integer. This result was originally obtained by H. Cohen and H. W. Lenstra, Jr. In this paper we give a new elementary proof of their result.
Recently, a new kind of multiple zeta value level two $T({\bf k})$ (which is called multiple $T$-values) was introduced and studied by Kaneko and Tsumura. In this paper, we define a kind of alternating version of multiple $T$-values, and study several duality formulas of weighted sum formulas about alternating multiple $T$-values by using the methods of iterated integral representations and series representations. Some special values of alternating multiple $T$-values can also be obtained.
Weighted sum formula for variants of half multiple zeta values
本文利用广义超几何函数${}_{5}F_{4}$的Dougall恒等式，证明了半多重zeta值及其变体的加权和公式，并讨论了插值细化对称多重zeta值的类似结果。
For a finite abelian group $(G,+)$, the constant $C(G)$ is defined to be the smallest natural number $k$ such that any sequence in $G$ having length $k$ will have a subsequence of consecutive terms whose sum is zero. For a subset $A\subseteq\mathbb Z_n$, the constant $C_A(n)$ is the smallest natural number $k$ such that any sequence in $G$ having length $k$ has an $A$-weighted zero-sum subsequence of consecutive terms. We determine the value of $C_A(n)$ for some particular weight-sets $A$.
The sum formula is a basic identity of multiple zeta values that expresses a Riemann zeta value as a homogeneous sum of multiple zeta values of a given dimension. This formula was already known to Euler in the dimension two case, conjectured in the early 1990s for higher dimensions and then proved by Granville and Zagier independently. Recently a weighted form of Euler's formula was obtained by Ohno and Zudilin. We generalize it to a weighted sum formula for multiple zeta values of all dimensions.
Let $a$ and $b$ be relatively prime positive integers. In this paper the weighted sum $\sum_{n\in{\rm NR}(a,b)}λ^{n-1}n^m$ is given explicitly or in terms of the Apostol-Bernoulli numbers, where $m$ is a nonnegative integer, and ${\rm NR}(a,b)$ denotes the set of positive integers nonrepresentable in terms of $a$ and $b$.
Weighted and Restricted Sum Formulas of Euler Sums
本文利用 Euler 和数的正则化双洗牌关系，统一证明了多重 zeta 值的求和公式及其变体，并在附录中以三个表格总结了包括新公式在内的多种结果。
In this paper, we establish the following two identities involving the Gamma function and Bernoulli polynomials, namely $$ \sum_{k\leq x}\frac{1}{k^s} \sum_{j=1}^{k^s}\logΓ\left(\frac{j}{k^s}\right) \sum_{\substack{d|k \\ d^{s}|j}}f*μ(d) \quad {\rm and } \quad \sum_{k\leq x}\frac{1}{k^s}\sum_{j=0}^{k^{s}-1} B_{m}\sum_{\substack{d|k \\ d^{s}|j}} f*μ(d) $$ with any fixed integer $s> 1$ and any arithmetical function $f$. We give asymptotic formulas for them with various multiplicative functions $f$. We also consider several formulas of the Dirichlet series associated with the above identities. Th…
A Weighted Sum Formula for Double Eisenstein Series
研究双艾森斯坦级数的加权和公式，通过作者与田坂证明的受限双洗牌关系得出双艾森斯坦级数恒等式，用组合方法及生成级数证明除数和恒等式，实现对相关恒等式的证明。
In this paper, we are going to perform the shuffle products of $Z_-(n) = \sum_{a+b=m} (-1)^{b} ζ(\{1\}^{a},b+2)$ and $Z_+^\star(n) = \sum_{c+d=n} ζ^{\star}(\{1\}^{c},d+2)$ with $m+n = p$. The resulted shuffle relation is a weighted sum formula given by \begin{equation*} \frac{(p+1)(p+2)}{2} ζ(p+4) =\sum_{m+n=p} \sum_{|\boldsymbolα|=p+3} ζ(α_{0}, α_{1}, \ldots, α_{m}, α_{m+1}+1) \sum_{a+b+c=m} \Bigl( W_{\boldsymbolα}(a,b,c) + W_{\boldsymbolα}(a,b,c=0) + W_{\boldsymbolα}(a=0,b,c) + W_{\boldsymbolα}(a=0,b=m,c=0) \Bigr), \end{equation*} where $W_{\boldsymbolα}(a,b,c) = 2^{σ(a+b+1)-σ(a)-(b+1)} (1-2…
Under the assumption of the Riemann Hypothesis, the Linear Independence Hypothesis, and a bound on negative discrete moments of the Riemann zeta function, we prove the existence of a limiting logarithmic distribution of the normalisation of the weighted sum of the Liouville function, $L_α(x) = \sum_{n \leq x}{λ(n) / n^α}$, for $0 \leq α< 1/2$. Using this, we conditionally show that these weighted sums have a negative bias, but that for each $0 \leq α< 1/2$, the set of all $x \geq 1$ for which $L_α(x)$ is positive has positive logarithmic density. For $α= 0$, this gives a conditional proof that…
A Positive Lower Bound for the Sum of Log-Reciprocal Twin Prime Products via Weighted Sieve
本文通过改进加权筛法，证明了孪生素数乘积对数倒数之和的正下界，从而推导出孪生素数对的无限性。
In this paper, we define some weighted sums of the alternating multiple $T$-values (AMTVs), and study several duality formulas for them by using the tools developed in our previous papers. Then we introduce the alternating version of the convoluted $T$-values and Kaneko-Tsumura $ψ$-function, which are proved to be closely related to the AMTVs. At the end of the paper, we study the $\Q$-vector space generated by the AMTVs of any fixed weight $w$ and provide some evidence for the conjecture that their dimensions $\{d_w\}_{w\ge 1}$ form the tribonacci sequence 1, 2, 4, 7, 13, ....
Let $G$ be a finite additive abelian group with exponent $d^kn, d,n>1,$ and $k$ a positive integer. For $S$ a sequence over $G$ and $A=\{1,2,\ldots,d^kn-1\}\setminus\{d^kn/d^i:i\in[1,k]\}, $ we investigate the lower bound of the number $N_{A,0}(S)$, which denotes the number of $A$-weighted zero-sum subsequences of $S.$ In particular, we prove that $N_{A,0}(S)\ge 2^{|S|-D_A(G)+1},$ where $D_A(G)$ is the $A$-weighted Davenport Constant. We also characterize the structures of the extremal sequences for which equality holds for some groups.
Double weighted sum involving $\mathrm{GL}(2)$ Fourier coefficients
本文对 $\mathrm{SL}(2,\mathbb{Z})$ 的 Hecke 全纯或 Hecke-Maass 尖形式的 Fourier 系数双线性和进行了非平凡估计，并推导了不同移位卷积和与求和函数的非平凡界。
In this paper, we give identities involving cyclic sums of regularized multiple zeta values of depth less than $5$. As a corollary, we present two kinds of extensions of Hoffman's theorem for symmetric sums of multiple zeta values for this case.
A proof of a weighted sum conjecture for finite multiple zeta values of level two
本文针对金子真等人提出的二级有限多重zeta值的加权和猜想，通过生成函数、线性递推关系，将问题归约为多项式恒等式，结合Racah型$_4F_3$超几何多项式求解二阶差分方程完成证明。
Multiple zeta values (MZVs) are real numbers which are defined by certain multiple series. Recently, many people have researched for relations among them and many relations are well known. In this paper, we get a new relation among them which is a generalization of a formula obtained by Eie, Liaw and Ong, and has four parameters. Moreover, by using similar method, we also obtain a relation which has six parameters.
Explicit estimates of the weighted sum $S(x)=\sum_{n \leq x} (-2)^{Ω(n)} \log\bigl(\frac{x}{n}\bigr).$
研究关于振荡算术函数\((-2)^{\Omega(n)}\)对数加权平均值\(S(x)\)的问题，运用复分析方法如对数加权佩龙公式，得到\(|S(x)| \leq 1.6x\)的新界。
We show that, modulo some odd prime p, the powers of two weighted sum of the first p-2 divided Bernoulli numbers equals the Agoh-Giuga quotient plus twice the number of permutations on p-2 letters with an even number of ascents and distinct from the identity. We provide a combinatorial characterization of Wieferich primes, as well as of primes p for which p^2 divides the Fermat quotient q_p(2).
Let $G$ be an additive finite abelian group and $Γ\subset \operatorname{End} (G)$ be a subset of the endomorphism group of $G$. A sequence $S = g_1 \cdot \ldots \cdot g_{\ell}$ over $G$ is a ($Γ$-)weighted zero-sum sequence if there are $γ_1, \ldots, γ_{\ell} \in Γ$ such that $γ_1 (g_1) + \ldots + γ_{\ell} (g_{\ell})=0$. We construct transfer homomorphisms from norm monoids (of Galois algebraic number fields with Galois group $Γ$) and from monoids of positive integers, represented by binary quadratic forms, to monoids of weighted zero-sum sequences. Then we study algebraic and arithmetic prope…
The set of minimal distances of the monoid of plus-minus weighted zero-sum sequences and applications to the characterization problem
本文延续正负加权零和序列幺半群的系统性研究，聚焦该类幺半群的重要算术不变量极小距离集合，并探讨其在刻画问题中的应用，研究受代数整数范数幺半群应用需求推动。
We give a short, self-contained proof of two key results from a paper of four of the authors. The first is a kind of weighted discrete Prékopa-Leindler inequality. This is then applied to show that if $A, B \subseteq \mathbb{Z}^d$ are finite sets and $U$ is a subset of a "quasicube" then $|A + B + U| \geq |A|^{1/2} |B|^{1/2} |U|$. This result is a key ingredient in forthcoming work of the fifth author and Pälvölgyi on the sum-product phenomenon.
Let $a_1,a_2,\dots,a_k$ be positive integers with $\gcd(a_1,a_2,\dots,a_k)=1$. Let ${\rm NR}={\rm NR}(a_1,a_2,\dots,a_k)$ denote the set of positive integers nonrepresentable in terms of $a_1,a_2,\dots,a_k$. The largest nonrepresentable integer $\max{\rm NR}$, the number of nonrepresentable positive integers $\sum_{n\in{\rm NR}}1$ and the sum of nonrepresentable positive integers $\sum_{n\in{\rm NR}}n$ have been widely studied for a long time as related to the famous Frobenius problem. In this paper by using Eulerian numbers, we give formulas for the weighted sum $\sum_{n\in{\rm NR}}λ^{n}n^μ$,…
Let $G$ be a finite abelian group of exponent $n$ and let $A$ be a non-empty subset of $[1,n-1]$. The Davenport constant of $G$ with weight $A$, denoted by $D_A(G)$, is defined to be the least positive integer $\ell$ such that any sequence over $G$ of length $\ell$ has a non-empty $A$-weighted zero-sum subsequence. Similarly, the combinatorial invariant $E_{A}(G)$ is defined to be the least positive integer $\ell$ such that any sequence over $G$ of length $\ell$ has an $A$-weighted zero-sum subsequence of length $|G|$. In this article, we determine the exact value of $D_A(\mathbb{Z}_n)$, for s…
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