Strong $q$-analogues for values of the Dirichlet beta function

The essentials

• Ankush Goswami and Timothy Huber present an infinite class of relations between modular forms that generalize evaluations of the Dirichlet beta function at odd positive integers.
• The authors are motivated by a base case appearing in Ramanujan's Notebooks and a parallel construction for the Riemann zeta function.
• The identities presented in this work are shown to be strong $q$-analogues, reducing to classical beta evaluations as $q\to 1^{-}$ and having explicit evaluations at CM points for $|q|<1$.
• This expansion of the classical beta evaluations provides new insights into the behavior of modular forms.
• Inequalities of Deligne determine asymptotic formulas for the Fourier coefficients of the associated modular forms.
• These formulas give us a deeper understanding into the underlying structure and properties of these forms.
• Goswami and Huber's work contributes to our knowledge about modular forms and their connections to other areas in mathematics such as number theory and analysis.
• Their findings may have implications for future research in these fields.

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