Signs of Fourier coefficients of cusp form at sum of two squares

Sign changes of cusp form coefficients on indices that are sums of two squares

“…We note that we use the notation L(s, f ⊗ g) to refer to the Rankin-Selberg L-function, including the associated Dirichlet L-function or zeta function. This is notationally different than in [BP20]. We note that certain equations in [BP20] have ζ(2s) instead of L(2s, χ −4 ) in Rankin-Selberg L-functions, but these do not affect their results.…”

“…This is notationally different than in [BP20]. We note that certain equations in [BP20] have ζ(2s) instead of L(2s, χ −4 ) in Rankin-Selberg L-functions, but these do not affect their results.…”

“…To prove our results, we generalize [MRM14] and refine the original arguments of [BP20]. If we wanted to use the axiomatized criteria for sign changes from [MRM14], we would study the partial sums…”

“…Proof of Lemma 5. We apply an explicit truncated Perron's formula, similar to the analysis in [BP20]. We use the presentation of Perron's formula in Theorem 5.1, Theorem 5.2, and Corollary 5.3 of [MV07].…”

“…Landau [Lan08] showed that the number of integers up to X that can be written as a sum of two squares is of the order X/ √ log X, and thus this is a density 0 subsequence. In [BP20], Banerjee and Pandey show that this sequence has at least X 1 8 −ǫ many sign changes for n ∈ [X, 2X] for X ≫ 1.…”