Two open points in Ramanujan's Lost Notebook, Part IV

Entry 12.2.3: Andrews and Berndt expect L(s,χ)L(s−r,χ) = Σ χ(n)σ_r(n)n^−s to hold for Re s > sup{0, Re r}. For r = 0 this fails: the series Σ χ(n)d(n)n^−s has abscissa of convergence in [1/4, 1/3]. Entry 19.2.2 fails also when its divergent series is read by Abel summation. self-test …

1. Entry 12.2.3 (r = 0): the partial sums A(x) = Σ_n≤x χ(n)d(n)

x (1 – 10¹²)

s for the partial sums Σ_n≤x χ(n)d(n)n^−s, x ≤ 2·10⁵: 0.20

A(x) is computed exactly by the hyperbola method A(x) = 2Σ_a≤√x χ(a)S(x/a) − S(√x)², S(m) = Σ_b≤m χ(b). Theorem: the series diverges for Re s < 1/4 and converges for Re s > 1/3.

2. Entry 19.2.2 in the Abel reading

α/2π

s (< −2, not an odd integer)

S(α) = ½Σ_d d^scot(dα/2) is the Abel value of Σσ_s(n)sin(nα); the sum is truncated at d ≤ 2·10⁵ and the tail bound uses ‖dξ‖ > 1/(4d) (partial quotients ≤ 2). Entry 19.2.2 would require the two sides to be equal.

The note, the verification script (10 checks) and the figure code accompany this page. Everything runs locally in your browser.

Two open points in Ramanujan's Lost Notebook, Part IV

1. Entry 12.2.3 (r = 0): the partial sums A(x) = Σn≤x χ(n)d(n)

2. Entry 19.2.2 in the Abel reading

1. Entry 12.2.3 (r = 0): the partial sums A(x) = Σ_n≤x χ(n)d(n)