A multi-variable version of the completed Riemann zeta function and other $L$-functions
A multi-variable version of the completed Riemann zeta function and other $L$-functions
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完整黎曼 zeta 函数和其他 $L$ 函数的多变量版本
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发表时间：
2019
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影响因子：
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通讯作者：
F. Brown
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文献类型：
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作者：
F. Brown
我们定义了一个广义的完成黎曼zeta函数在几个复杂的变量。它满足一个函数方程，洗牌产品的身份，并有简单的极点沿沿着许多超平面，与递归结构的剩余。两个变量的特殊情况可以写成一个真实的解析Eisenstein级数的部分Mellin变换，这使我们能够将它在正偶点对处的值与对应于高斯整数的CM椭圆曲线的周期（上同调的对称幂的简单扩展）联系起来。一般来说，这些函数的完全偶数值与我们称之为多重二次和的新量有关。
更一般地说，我们谨慎地定义多变量版本的动机$L$-功能，并询问是否有其特殊的价值观和一般的混合动机的时期之间的关系。我们表明，所有时期的混合泰特动机的整数，和所有时期的动机基本群体（或相对完成）的模群，确实是特殊的价值观的多动机$L$-值定义在这里。
We define a generalisation of the completed Riemann zeta function in several complex variables. It satisfies a functional equation, shuffle product identities, and has simple poles along finitely many hyperplanes, with a recursive structure on its residues. The special case of two variables can be written as a partial Mellin transform of a real analytic Eisenstein series, which enables us to relate its values at pairs of positive even points to periods of (simple extensions of symmetric powers of the cohomology of) the CM elliptic curve corresponding to the Gaussian integers. In general, the totally even values of these functions are related to new quantities which we call multiple quadratic sums.
More generally, we cautiously define multiple-variable versions of motivic $L$-functions and ask whether there is a relation between their special values and periods of general mixed motives. We show that all periods of mixed Tate motives over the integers, and all periods of motivic fundamental groups (or relative completions) of modular groups, are indeed special values of the multiple motivic $L$-values defined here.