On short exponential sums related to holomorphic cusp forms
by
Anne-Maria Ernvall-Hytönen
Department of Mathematics, University of Turku

Holomorphic cusp forms with respect to the full modular group can be represented as Fourier series

∞ å n=1  a(n)n(k-1)/2e(tn),

where k is the weight of the form. We consider normalized and truncated sums

å M ≤ n ≤ M+D  a(n) e(na),

where a ∈ R. Wilton proved the estimate ∑_{1 ≤ n ≤ M}a(n)e(na) << MlogM from which Jutila was able to remove the logarithm. Using Rankin's result ∑_{1 ≤ n ≤ M}|a(n)|²=AM+O(M^3/5), where A is a constant, we see that Jutila's estimate is best possible. However, one knows a lot less about short sums ∑_{M ≤ n ≤ M+D}a(n)e(an), where D ≤ M. We show that when D >> M^3/4, no better general upper bound is possible than O(M^1/2), which is obtained from Jutila's estimate using triangle inequality. Further, we will show that for smaller D, we always obtain a better estimate than M^1/2. Part of the results is joint work with K. Karppinen.

Date received: March 20, 2008