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<p>Ramanujan’s 1920 last letter to Hardy contains seventeen examples of mock theta functions which he organized into three “orders.” The most famous of these is the third-order function <i>f</i>(<i>q</i>) which has received the most attention of any individual mock theta function in the intervening century. In 1964, Andrews—improving on a result of Dragonette—gave an asymptotic formula for the coefficients of <i>f</i>(<i>q</i>) and conjectured an exact formula for the coefficients: a conditionally convergent series that closely resembles the Hardy–Ramanujan–Rademacher formula for the partition function. To prove that the conjectured series converges, it is necessary to carefully measure the cancellation among the Kloosterman sums appearing in the formula. Andrews’ conjecture was proved four decades later by Bringmann and Ono. Here we prove exact formulas for all seventeen of the mock theta functions appearing in Ramanujan’s last letter. Along the way, we prove a general theorem bounding sums of Kloosterman sums for the Weil representation attached to a lattice of odd rank.</p>

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Transcendental formulas for the coefficients of Ramanujan’s mock theta functions

  • 期刊论文

  • 发表日期：

    2026年3月5日

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  • Nickolas Andersen,
  • Gradin Anderson

摘要

Ramanujan’s 1920 last letter to Hardy contains seventeen examples of mock theta functions which he organized into three “orders.” The most famous of these is the third-order function f(q) which has received the most attention of any individual mock theta function in the intervening century. In 1964, Andrews—improving on a result of Dragonette—gave an asymptotic formula for the coefficients of f(q) and conjectured an exact formula for the coefficients: a conditionally convergent series that closely resembles the Hardy–Ramanujan–Rademacher formula for the partition function. To prove that the conjectured series converges, it is necessary to carefully measure the cancellation among the Kloosterman sums appearing in the formula. Andrews’ conjecture was proved four decades later by Bringmann and Ono. Here we prove exact formulas for all seventeen of the mock theta functions appearing in Ramanujan’s last letter. Along the way, we prove a general theorem bounding sums of Kloosterman sums for the Weil representation attached to a lattice of odd rank.