Erratum: Sums of Kloosterman Sums over Arithmetic Progressions (International Mathematics Research Notices DOI: 10.1093/imrn/rnr010)
International Mathematics Research Notices
The proof of Theorem 2 in  is erroneous, and the purpose of this note is to correct Theorem 2 and Corollary 3 in  and to correct a few more errors that are less serious. The error, as pointed out in a recent work by Steiner , lies in the treatment of the exceptional spectrum in the range x ≥ vmn > 0. The domain in question depends on x and that renders the proof invalid when x is large. Steiner  considered a more general sum with an extra additive twist e 2vcmna where a is any real number and with some additional restrictions on the parameters. His estimation of the contribution of the exceptional spectrum leads to a term of size O x2 (see [7, Corollary 2 and Theorem 3]). This is a strong bound in terms of x. However, in view of possible applications it may be of interest to give bounds that are strong in terms of the size of q as well. In this note we give such a bound for the contribution of the exceptional spectrum for the sum treated in . The bound we obtain is O (mn)θ+e(1 + q-1t(q)x 12 +e(mn)-1/4) in the range x vmn, where t(q) denotes the total number of divisors of q. Thus, by adding the above to the right-hand side of the equation in the statement of Theorem 2 in  and using the bound t(q) = O(qe), we obtain the theorem below, and by adding the corresponding term for θ = 7/64 (the Kim-Sarnak  bound) to the bound in [5, Corollary 3], we obtain the following corollary. We follow the same notation as in our article  throughout. (Formula Presented).
Ganguly, Satadal and Sengupta, Jyoti, "Erratum: Sums of Kloosterman Sums over Arithmetic Progressions (International Mathematics Research Notices DOI: 10.1093/imrn/rnr010)" (2021). Journal Articles. 1803.