Adelic formulas for gamma and beta functions in algebraic number fields (Q1377927)

In earlier works, the author proposed regularized adelic formulas for gamma and beta functions with unramified quasicharacters for all completions of an arbitrary field \(\mathbb{K}\) of algebraic numbers. The formula for gamma functions has the form \[ \Gamma^\sigma_\infty(\alpha) \Gamma^\tau_{-\infty} \text{reg}\prod^\infty_{p= 2} \prod^{m_p}_{j= 1} \Gamma_{q_{pj}}(\alpha)= | D|^{1/2- \alpha},\tag{1} \] where \(\sigma\) and \(\tau\) are the numbers of real and complex, respectively, roots of the minimal polynomial of degree \(n= \sigma+2\tau\) for an algebraic number \(\varepsilon\) generating the field \(\mathbb{K}= \mathbb{Q}(\varepsilon)\); \(D\) is the discriminant of the field \(\mathbb{K}\); \(m_p\) is the number of different prime divisors in the decomposition of the prime number \(p\); \(\Gamma_\infty\) and \(\Gamma_{-\infty}\) are the gamma functions for unramified quasicharacters of the fields \(\mathbb{Q}_\infty= \mathbb{R}\) and \(\mathbb{Q}_{-\infty}= \mathbb{C}\); the function \[ \Gamma_q(\alpha)= (1- q^{\alpha-1})/(1- q^{-\alpha}) \] is the gamma function of the local \(p\)-field with the modulus \(q= p^f\). Formulas similar to (1) also hold for beta functions. For unramified quasicharacters, he obtained regularized adelic formulas for gamma and beta functions in more specific forms for the field \(\mathbb{Q}\) of rational numbers, quadratic fields \(\mathbb{Q}(\sqrt d)\), \(m\)-circular fields, and some cubic fields. Now, the author generalizes the obtained results to ramified quasicharacters. The corresponding adelic formulas are given for gamma functions and for beta functions. The adelic formulas for beta functions are applied to relate the Veneziano and Virasoro-Shapiro amplitudes to string amplitudes (for open and closed strings, respectively). For beta functions with ramified quasicharacters, he introduces new amplitudes similar to the Veneziano and Virasoro-Shapiro amplitudes.