New series for some special values of $L$-functions
For number theorists, this provides many new conjectures that may deepen understanding of special values of L-functions, but the results are purely conjectural without proofs.
The paper presents 48 new conjectural series involving harmonic numbers for special values of L-functions and related constants, such as ζ(5) and ζ(3)^2, based on transformations of congruences and computational evidence.
Dirichlet's $L$-functions are natural extensions of the Riemann zeta function. In this paper we first give a brief survey of Apéry-like series for some special values of the zeta function and certain $L$-functions. Then, we establish two theorems on transformations of certain kinds of congruences. Motivated by the results and based on our computation, we pose 48 new conjectural series (most of which involve harmonic numbers) for such special values and related constants. For example, we conjecture that \begin{align*}\sum_{k=1}^\infty\frac1{k^4\binom{2k}k}\bigg(\frac1k+\sum_{j=k}^{2k}\frac1j\bigg)=&\frac{11}9ζ(5), \\\sum_{k=1}^\infty\frac{(-1)^{k-1}}{k^3\binom{2k}k}\bigg(\frac1{5k^3}+\sum_{j=1}^{k}\frac1{j^3}\bigg)=&\frac{2}5ζ(3)^2, \end{align*} and $$\sum_{k=1}^\infty\frac{48^k}{k(2k-1)\binom{4k}{2k}\binom{2k}k}=\frac{15}2\sum_{k=1}^\infty\frac{(\frac k3)}{k^2},$$ where $(\frac k3)$ denotes the Legendre symbol.