Pure Mathematics


Exponential Sums In  Number Theory


András Sárközy






Differentiation and integration of the function e(a), the Parseval formula. Estimation of 1-e(a).  The Jensen-Ramanujan formula. Additive characters. Number of solutions of congruences of higher degree. Gaussian sums. Poisson’s summation formula (without proof), with application to compute Gaussian sums. Applications: 1. The number of representations of an element as the sum of two squares mod p. 2. The modular variant of an Erdős-Moser problem. Kloostermann sums. Multiplicative characters, their explicit form. Connection between additive and multiplicative characters. Complete and non-complete sums. Vinogradov’s method for estimation of non-complete sums. The Pólya-Vinogradov inequality. The problem of the smallest quadratic non-residue. Analytic (without proof), arithmetic and character form of the large sieve, with application. Converse of the large sieve, the Roth-inequality. Application for lower estimate of character sums. Rudin-Shapiro polynomials. Circle problem, the Erdős-Fuchs theorem. Uniform distribution, Weyl’s criterion, uniform distribution of the fractional part of the multiples of an irrational number. Double large sieve, with application. Van der Corput’s method, pairs of exponents.



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written/oral exam