DIOPHANTINE PROPERTIES FOR q-ANALOGUES OF DIRICHLET'S BETA FUNCTION AT POSITIVE INTEGERS

作者：Jouhet Frederic^*; Mosaki Elie

来源：Transactions of the American Mathematical Society, 2011, 363(3): 1533-1554.

DOI：10.1090/S0002-9947-2010-05236-5

• q-analogues of the values of Dirichlet's beta function at integers
• modular forms
• irrationality
• basic hypergeometric series

In this paper, we define q-analogues of Dirichlet's beta function at positive integers, which can be written as beta(q)(s) = Sigma(k >= 1) Sigma(d vertical bar k) chi(k/d)d(s-1)q(k) for s is an element of N*, where q is a complex number such that vertical bar q vertical bar < 1 and chi is the nontrivial Dirichlet character modulo 4. For odd s, these expressions are connected with the automorphic world, in particular with Eisenstein series of level 4. From this, we derive through Nesterenko's work the transcendance of the numbers beta(q)(2s + 1) for q algebraic such that 0 < vertical bar q vertical bar < 1. Our main result concerns the nature of the numbers,beta(q)(2s): we give a lower bound for the dimension of the vector space over Q spanned by 1,beta(q)(2), beta(q)(4), ..., beta(q)(A), where 1/q is an element of Z \ {1; 1} and A is an even integer. As consequences for 1/q is an element of Z \ {-1;1}, on the one hand there is an infinity of irrational numbers among beta(q)(2), beta(q)(4),..., and on the other hand at least one of the numbers beta(q)(2),beta(q)(4), ..., beta(q)(20) is irrational.

• 出版日期2011-3