Abstract
Let χ be a primitive multiplicative character modulo an integer m ≥ 1. Using some classical bounds of character sums, we estimate the average value of the character sums with subsequence sums \( T_m (\mathcal{S},\chi ) = \sum\nolimits_{\mathcal{I} \subseteq \{ 1, \ldots ,N\} } {\chi (\sum\nolimits_{i \in \mathcal{I}} {s_i } )} \) taken over all N-element sequences S = (s 1, …, s N) of integer elements in a given interval [K + 1, K + L]. In particular, we show that T m (S, χ) is small on average over all such sequences. We apply it to estimating the number of perfect squares in subsequence sums in almost all sequences.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. Gyarmati, A. Sárközy and C. L. Stewart, On sums which are powers, Acta Math. Hungar., 99 (2003), 1–24.
D. R. Heath-Brown, The square sieve and consecutive squarefree numbers, Math. Ann., 266 (1984), 251–259.
N. Hegyvári and A. Sárközy, On Hilbert cubes in certain sets, Ramanujan J., 3 (1999), 303–314.
H. Iwaniec and E. Kowalski, Analytic number theory, Amer. Math. Soc., Providence, RI, 2004.
P. Q. Nguyen, I. E. Shparlinski and J. Stern, Distribution of modular sums and the security of the server aided exponentiation, Proc. Workshop on Cryptography and Computational Number Theory, Singapore 1999, Birkhäuser, 2001, 331–342.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by András Sárközy
Rights and permissions
About this article
Cite this article
Balasuriya, S., Shparlinski, I.E. Character sums with subsequence sums. Period Math Hung 55, 215–221 (2007). https://doi.org/10.1007/s10998-007-4215-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10998-007-4215-4