Jianqiang Zhao

In this paper, we study an Apéry-type series involving the central binomial coefficients ${\sum }_{n_1>\dots >n_d>0}{\displaystyle \frac{1}{4^{n_1}}}\left({\displaystyle \genfrac{}{}{0pt}{}{2n_1}{n_1}}\right)n_1^{-s_1}\dots n_d^{-s_d}$ and its variations where the summation indices may have mixed parities and some or all “>” are replaced by “≥”, as long as the series are defined. These sums have naturally appeared in the calculation of massive Feynman integrals by the work of Jegerlehner, Kalmykov, and Veretin. We show that all these sums can be expressed as $\mathbb{Q}$-linear combinations of the real and/or imaginary parts of the colored multiple zeta values at level four, i.e., special values of multiple polylogarithms at fourth roots of unity. For example, our main theorem shows that when $n_1^{-s_1}$ is replaced by ${\left(2n_1\right)}^{-s_1}$ and other $n_j^{-s_j}$’s are replaced by either ${\left(2n_j\right)}^{-s_j}$ or ${(2n_j+1)}^{-s_j}$, then all the colored multiple zeta values can be chosen to have the same weight $s_1+\dots +s_d$, but the weights of these values are only bounded by $s_1+\dots +s_d$ for general variant Apéry-type series of mixed parities. We also show that the corresponding series where $\left({\displaystyle \genfrac{}{}{0pt}{}{2n_1}{n_1}}\right)/4^{n_1}$ is replaced by ${\left({\displaystyle \genfrac{}{}{0pt}{}{2n_1}{n_1}}\right)}^2/{16}^{n_1}$ can be expressed in a similar way except for a possible extra factor of $1/\pi$, with the weight of the colored multiple zeta values similarly bounded.

In this paper, we propose a class of elementary plane geometry problems closely related to the title of this paper. Here, a circle is the one-dimensional curve bounding a disk. For any non-negative integer, a circle is called n-enclosing if it contains exactly n lattice points on the $xy$-plane in its interior. In this paper, we are mainly interested in when the largest n-enclosing circle exists and what the largest radius is. We study the small integer cases by hand and extend to all $n<1100$ with the aid of a computer. We find that frequently such a circle does not exist, e.g., when $n=5,6$. We then show a few general results on these circles including some regularities among their radii and an easy criterion to determine exactly when the largest n-enclosing circles exist. Further, from numerical evidence, we conjecture that the set of integers whose largest enclosing circles exist is infinite, and so is its complementary in the set of non-negative integers. Throughout this paper, we present more mysteries/problems/conjectures than answers/solutions/theorems. In particular, we list many conjectures and some unsolved problems including possible higher-dimensional generalizations at the end of the last two sections.

Many variations of the multiple zeta values have been found to play important roles in different branches of mathematics and theoretical physics in recent years, such as the cyclotomic/color version, which appears prominently in the computation of Feynman integrals. In this paper, we introduce a trigonometric variant of the Kaneko–Tsumura $\psi$-function (called the Kaneko–Tsumura $\tilde{\psi }$-function) and discover some nice properties similar to those for ordinary Kaneko–Tsumura $\psi$-values using the method of iterated integrals, which was first studied systematically by K.T. Chen in the 1960s. In particular, we establish some duality formulas involving the Kaneko–Tsumura $\tilde{\psi }$-function and alternating multiple T-values by adapting Yamamoto’s graphical representation method for computing special types of iterated integrals.