Random process generated by the incomplete Gauss sums
In this paper we explore a random process generated by the incomplete Gauss sums and establish an analogue of weak invariance principle for these sums. We focus our attention exclusively on a generalization of the limit distribution of the long incomplete Gauss sums given by the family of periodic functions analyzed by the author and Marklof.
Random process generated by the incomplete Gauss sums
In this paper we explore a random process generated by the incomplete Gauss sums and establish an analogue of weak invariance principle for these sums. We focus our attention exclusively on a generalization of the limit distribution of the long incomplete Gauss sums given by the family of periodic functions analyzed by the author and Marklof.
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- Demirci Akarsu E, Marklof J. The value distribution of incomplete Gauss sums. Mathematika 2013; 59: 381–398.
- Demirci Akarsu E. Short incomplete Gauss sums and rational points on metaplectic horocycles. Int J Number Thr 2014; 10: 1553–1576.
- Evans R, Minei M, Yee B. Incomplete higher order Gauss sums. J Math Anal Appl 2003; 281: 454–476.
- Fiedler H, Jurkat WB, K ¨o rner O. Asymptotic expansions of finite theta series. Acta Arith 1977; 32: 129–146.
- Jurkat WB, van Horne JW. The proof of the central limit theorem for theta sums. Duke Math J 1981; 48: 873–885.
- Jurkat WB, van Horne JW. On the central limit theorem for theta series. Michigan Math J 1982; 29: 65–67.
- Jurkat WB, van Horne JW. The uniform central limit theorem for theta sums. Duke Math J 1983; 50: 649–666.
- Lehmer DH. Incomplete Gauss sums. Mathematika 1976; 23: 125–135.
- Marklof J. Limit theorems for Theta sums. Duke Math J 1999; 97: 127–153.
- Montgomery HL, Vaughan RC, Wooley TD. Some remarks on Gauss sums associated with kth powers. Math Proc Cambridge Philos Soc 1995; 118: 21–33.
- Oskolkov KI. On functional properties of incomplete Gaussian sums. Canadian J Math 1991; 43: 182–212.
- Paris RB. An asymptotic approximation for incomplete Gauss sums. J Comput Appl Math 2005; 180: 461–477.
- Paris RB. An asymptotic approximation for incomplete Gauss sums II. J Comput Appl Math 2008; 212: 16–30.