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Problems in Number Theory related to Mathematical Physics
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2008 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

This thesis consists of an introduction and four papers. All four papers are devoted to problems in Number Theory. In Paper I, a special class of local ζ-functions is studied. The main theorem states that the functions have all zeros on the line Re(s)=1/2.This is a natural generalization of the result of Bump and Ng stating that the zeros of the Mellin transform of Hermite functions have Re(s)=1/2.In Paper II and Paper III we study eigenfunctions of desymmetrized quantized cat maps.If N denotes the inverse of Planck's constant, we show that the behavior of the eigenfunctions is very dependent on the arithmetic properties of N. If N is a square, then there are normalized eigenfunctions with supremum norm equal to $N^{1/4}$, but if N is a prime, the supremum norm of all eigenfunctions is uniformly bounded. We prove the sharp estimate $\|\psi\|_\infty=O(N^{1/4})$ for all normalized eigenfunctions and all $N$ outside of a small exceptional set. For normalized eigenfunctions of the cat map (not necessarily desymmetrized), we also prove an entropy estimate and show that our functions satisfy equality in this estimate.We call a special class of eigenfunctions newforms and for most of these we are able to calculate their supremum norm explicitly.For a given $N=p^k$, with k>1, the newforms can be divided in two parts (leaving out a small number of them in some cases), the first half all have supremum norm about $2/\sqrt{1\pm 1/p}$ and the supremum norm of the newforms in the second half have at most three different values, all of the order $N^{1/6}$. The only dependence of A is that the normalization factor is different if A has eigenvectors modulo p or not. We also calculate the joint value distribution of the absolute value of n different newforms.In Paper IV we prove a generalization of Mertens' theorem to Beurling primes, namely that

\lim_{n \to \infty}\frac{1}{\ln n}\prod_{p \leq n} \left(1-p^{-1}\right)^{-1}=Ae^{\gamma}$\lim_{n \to \infty}\frac{1}{\ln n}\prod_{p \leq n} \left(1-p^{-1}\right)^{-1}=Ae^{\gamma},$where γ is Euler's constant and Ax is the asymptotic number of generalized integers less than x. Thus the limit $M=\lim_{n\to\infty}\left(\sum_{p\le n}p^{-1}-\ln(\ln n)\right)$exists. We also show that this limit coincides with $\lim_{\alpha\to 0^+} \left(\sum_p p^{-1}(\ln p)^{-\alpha}-1/\alpha\right)$ ; for ordinary primes this claim is called Meissel's theorem. Finally we will discuss a problem posed by Beurling, namely how small |N(x)-[x] | can be made for a Beurling prime number system Q≠P, where P is the rational primes. We prove that for each c>0 there exists a Q such that |N(x)-[x] |

##### Place, publisher, year, edition, pages
Stockholm: KTH , 2008. , viii, 28 p.
##### Series
Trita-MAT. MA, ISSN 1401-2278 ; 08:12
##### Keyword [en]
number theory, mathetical physics, local zeta functions, cat maps, Beurling primes
Mathematics
##### Identifiers
ISBN: 978-91-7415-177-0OAI: oai:DiVA.org:kth-9514DiVA: diva2:117335
##### Public defence
2008-12-08, F3, KTH, Lindstedtsvägen 26, Stockholm, 13:00 (English)
##### Note
QC 20100902Available from: 2008-11-21 Created: 2008-11-11 Last updated: 2010-09-02Bibliographically approved
##### List of papers
1. Local Riemann hypothesis for complex numbers
Open this publication in new window or tab >>Local Riemann hypothesis for complex numbers
2009 (English)In: International Journal of Number Theory, ISSN 1793-0421, Vol. 5, no 7, 1221-1230 p.Article in journal (Refereed) Published
##### Abstract [en]

In this paper, a special class of local zeta-functions is studied. The main theorem states that the functions have all zeros on the line R(s) = 1/2. This is a natural generalization of the result of Bump and Ng stating that the zeros of the Mellin transform of Hermite functions have R(s) = 1/2.

##### Keyword
Local Tate zeta-function, Weil representation, Mellin transform, orthogonal polynomials, Riemann hypothesis
Mathematics
##### Identifiers
urn:nbn:se:kth:diva-24328 (URN)10.1142/S1793042109002651 (DOI)000272094400006 ()
##### Note
QC 20100901Available from: 2010-09-01 Created: 2010-09-01 Last updated: 2010-09-02Bibliographically approved
2. Large Supremum Norms and Small Shannon Entropy for Hecke Eigenfunctions of Quantized Cat Maps
Open this publication in new window or tab >>Large Supremum Norms and Small Shannon Entropy for Hecke Eigenfunctions of Quantized Cat Maps
2009 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 286, no 3, 1051-1072 p.Article in journal (Refereed) Published
##### Abstract [en]

This paper concerns the behavior of eigenfunctions of quantized cat maps and in particular their supremum norm. We observe that for composite integer values of N, the inverse of Planck's constant, some of the desymmetrized eigenfunctions have very small support and hence very large supremum norm. We also prove an entropy estimate and show that our functions satisfy equality in this estimate. In the case when N is a prime power with even exponent we calculate the supremum norm for a large proportion of all desymmetrized eigenfunctions and we find that for a given N there is essentially at most four different values these assume.

Mathematics
##### Identifiers
urn:nbn:se:kth:diva-24329 (URN)10.1007/s00220-008-0627-x (DOI)000263059600008 ()2-s2.0-59449109020 (ScopusID)
##### Note
QC 20100901Available from: 2010-09-01 Created: 2010-09-01 Last updated: 2010-09-02Bibliographically approved
3. Hecke Eigenfunctions of Quantized Cat Maps Modulo Prime Powers
Open this publication in new window or tab >>Hecke Eigenfunctions of Quantized Cat Maps Modulo Prime Powers
2009 (English)In: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, Vol. 10, no 6, 1111-1139 p.Article in journal (Refereed) Published
##### Abstract [en]

This paper continues the work done in Olofsson [Commun Math Phys 286(3):1051-1072, 2009] about the supremum norm of eigenfunctions of desymmetrized quantized cat maps. N will denote the inverse of Planck's constant and we will see that the arithmetic properties of N play an important role. We prove the sharp estimate parallel to psi parallel to(infinity) - O(N-1/4) for all normalized eigenfunctions and all N outside of a small exceptional set. We are also able to calculate the value of the supremum norms for most of the so called newforms. For a given N = p(n), with n > 2, the newforms can be divided in two parts (leaving out a small number of them in some cases), the first half all have supremum norm about 2/root 1 +/- 1/p and the supremum norm of the newforms in the second half have at most three different values, all of the order N-1/6. The only dependence of A is that the normalization factor is different if A has eigenvectors modulo p or not. We also calculate the joint value distribution of the absolute value of n different newforms.

Mathematics
##### Identifiers
urn:nbn:se:kth:diva-24330 (URN)10.1007/s00023-009-0011-1 (DOI)000271402700004 ()
##### Note
QC 20100901Available from: 2010-09-01 Created: 2010-09-01 Last updated: 2010-09-02Bibliographically approved
4. Properties of the Beurling generalized primes
Open this publication in new window or tab >>Properties of the Beurling generalized primes
2011 (English)In: Journal of Number Theory, ISSN 0022-314X, E-ISSN 1096-1658, Vol. 131, no 1, 45-58 p.Article in journal (Refereed) Published
##### Abstract [en]

In this paper, we prove a generalization of Mertens' theorem to Beurling primes, namely that lim(x ->infinity) 1/Inx Pi(p <= x)(1 - p(-1))(-1) = Ae(gamma), where gamma is Euler's constant and Ax is the asymptotic number of generalized integers less than x. Thus the limit M = lim(x ->infinity) (Sigma(p <= x) p(-1) - In(Inx)) exists. We also show that this limit coincides with lim(alpha -> 0+)(Sigma(p) p(-1)(In p)(-alpha) - 1/alpha); for ordinary primes this claim is called Meissel's theorem. Finally, we will discuss a problem posed by Beurling, namely how small vertical bar N(x)-[4]vertical bar can be made for a Beurling prime number system Q not equal P. where P is the rational primes. We prove that for each c > 0 there exists a Q such that vertical bar N(x) - [x]vertical bar < cInx and conjecture that this is the best possible bound.

Mathematics
##### Identifiers
urn:nbn:se:kth:diva-24337 (URN)10.1016/j.jnt.2010.06.014 (DOI)000283401800004 ()
##### Funder
Knut and Alice Wallenberg Foundation
##### Note
Updated from submitted to published. QC 20120327Available from: 2010-09-02 Created: 2010-09-02 Last updated: 2012-03-27Bibliographically approved

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