discipline 
Pure
Mathematics

subject 
Exponential Sums In Number Theory

lecturers 
András Sárközy 
credits 
2 
period 

curriculum 
Differentiation and integration of the
function e(a), the Parseval formula. Estimation of
1e(a). The JensenRamanujan 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 ErdősMoser problem. Kloostermann sums. Multiplicative characters, their
explicit form. Connection between additive and multiplicative characters.
Complete and noncomplete sums. Vinogradov’s method
for estimation of noncomplete sums. The PólyaVinogradov inequality. The problem of the
smallest quadratic nonresidue. Analytic (without proof), arithmetic and
character form of the large sieve, with application. Converse of the large
sieve, the Rothinequality. Application for lower estimate of character sums.
RudinShapiro polynomials. Circle problem, the ErdősFuchs 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. 
literature 

form of tuition 
Lecture 
mode of assessment 
written/oral
exam 