Department of Mathematics
Royal Holloway University Of London

Number Theory Seminar

Previous Number theory activities 2004/2005

Graduate lectures:
Glyn Harman: Prime detecting sieves

Christian Elsholtz: Combinatorial number theory
Start: Both lectures will start Wednesday 6th October 2-5 pm, in 229.
But note that after a few weeks there will be a change of the time, due to the UCAS interviews.

For some talks in the Tuesday seminar see here
5th October 2004
Yoichi Motohashi, Nihon University (Japan),
"Sieve Methods --- Looking back on the stormy years".

12th October 2004
Thomas Prellberg, Queen Mary (London),
"Monotonicity of partition functions".

19th October 2004
Gihan Marasingha, Oxford, "On Pairs of Quadratic Forms".

26th October 2004
Nigel Watt, RHUL, "Fourier coefficients of modular forms and eigenvalues of a Hecke operator".

16th November 2004
Glyn Harman (RHUL),
"There are infinitely many more Carmichael numbers".

23th November 2004
Lillian Pierce (Princeton),
"A Bound for the 3-Part of Class Numbers of Quadratic Fields via the Square Sieve".

30th November 2004
Richard Pinch, title to be announced.

OLD Number theory activities 2003/2004

Here are some number theory seminar talks.
Some of them are in fact part of the Tuesday Pure mathematics seminar, but listed for completeness.
Monday seminars take place at 4pm in C229.
Tuesday seminars take place at 4pm in C219.

Tuesday 30th September Jan-Christoph Schlage-Puchta Freiburg
Modular subgroup arithmetic for surface groups
Monday 6th October Christian Elsholtz
Sums of unit fractions
This is a survey on the solutions of the diophantine equation \frac{m}{n}= \frac{1}{x_1} + \cdots +\frac{1}{x_k}. We use methods from elementary, combinatorial and analytic number theory and some basic group theory. We develop a method that could be useful for other diophantine equations.
Tuesday 21st October Christian Elsholtz
Additive decompositions of the set of primes
This is a survey on Ostmann's problem. Ostmann asked whether there exist two sets A and B (with at least two elements each) so that their sumset A+B equals the set of primes, for sufficiently large primes. Using a new version of the large sieve method I can show, that such sets A and B would need to have counting functions of size N^(1/2 +o(1)), whereas previously only a lower bound of N^(o(1)) and an upper bound of N^(1+o(1)) was known. This implies, for example, that the set of primes cannot be decomposed into three such sets.
Monday 27th October James McKee
A killer exponent for maximal torsion cosets
Results of Laurent, Bombieri and Zannier, and Schmidt state that for any variety V defined over a number field, the union of all torsion cosets contained in V is in fact contained in a union of finitely many maximal torsion cosets. The first part of this talk will attempt to explain what all these words mean. The finiteness of the number of maximal torsion cosets in V immediately implies the existence of a single "killer" exponent for all these cosets. The second part of the talk will describe an effective version of this result, giving an explicit bound for the size of the killer exponent. This is joint work with Chris Smyth (Edinburgh), motivated by the problem of showing that there exist Salem numbers of every trace.
Monday November 10th Igor Shparlinski
Quadratic residues and non-residues in the sequence n!
Tuesday 18th November James McKee
Salem numbers via interlacing
Monday 24th November Igor Shparlinski
Pseudorandom Points on Elliptic Curves
Period, Distribution and other properties of sequences of the form
P_n = P_{n-1} + G (= n G) and
P_n = e P_{n-1} = (e^n G) where P_0 = O on elliptic curves. Survey with some sketches of proofs.
Saturday 29th November SECANTS at Royal Holloway.
Monday 8th December Igor Shparlinski
Arithmetic Functions on Sparse Integers Abstract: We evaluate the average value \sum_{s \in S, s \le x} \phi(s)/s
taken over various sets S of integers with restricted g-ary expansions. In particular we settle an open question of Erdos, Mauduit and Sarkozy, and improve another result of Mauduit and Sarkozy.

Further Number theory talks in the Tuesday seminar 2003/2004 were given by
by Nelson Stephens, Somos sequences
Edlyn Teske (Waterloo), Cryptographic applications of Weil decent
Michael Scott (Dublin City University), IBE as fast as RSA?
Nigel Smart (Bristol), Revisiting the relationship between DLP and DHP for elliptic curves
Alan Lauder (Oxford) Factoring sparse polynomials


Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX
Tel/Fax: +44 (0)1784 443093/430766