Abstract. Motivated by classical identities of Euler, Schur, and Rogers and Ramanujan, Alder investigated q d (n) and Q d (n), the number of partitions of n into d-distinct parts and into parts which are ±1(mod d + 3), respectively. He conjectured that q d (n) ≥ Q d (n). Andrews and Yee proved the conjecture for d = 2 s − 1 and also for d ≥ 32. We complete the proof of Andrews's refinement of Alder's conjecture by determining effective asymptotic estimates for these partition functions (correcting and refining earlier work of Meinardus), thereby reducing the conjecture to a finite computation.
Bloch-Okounkov studied certain functions on partitions f called shifted symmetric polynomials. They showed that certain q-series arising from these functions (the so-called q-brackets f q ) are quasimodular forms. We revisit a family of such functions, denoted Q k , and study the p-adic properties of their q-brackets. To do this, we define regularized versions Q (p) k for primes p. We also use Jacobi forms to show that the Q (p) k q are quasimodular and find explicit expressions for them in terms of the Q k q .
Let f (z) = ∞ n=1 λ f (n)e 2πinz ∈ S new k (Γ 0 (N )) be a newform of even weight k ≥ 2 on Γ 0 (N ) without complex multiplication. Let P denote the set of all primes. We prove that the sequence {λ f (p)} p∈P does not satisfy Benford's Law in any integer base b ≥ 2. However, given a base b ≥ 2 and a string of digits S in base b, the set A λ f (b, S) := {p prime : the first digits of λ f (p) in base b are given by S} has logarithmic density equal to log b (1 + S −1 ). Thus {λ f (p)} p∈P follows Benford's Law with respect to logarithmic density. Both results rely on the now-proven Sato-Tate Conjecture.
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