Welcome! It’s great to have you here. This page provides information about the seminars regularly organized by Bilecik Algebra & Number Theory Team (BANT). These seminars cover various important disciplines of mathematics, especially Number Theory.
Everyone is invited to attend! This academic year, all seminars will be held via Zoom. Zoom links will be shared one day before each seminar. To register, please feel free to contact us at [ilker.inam at gmail.com].
We look forward to the opportunity to meet face to face and turn coffee meetings into theorems as soon as possible.
Kind regards, BANT Team
In these seminars, we examine Algebra and number theory topics, and our goals are to increase our mathematical maturity and say hello to new bridges in mathematics.
This academic year, we will use the book Apostol, T. M. (1998). Introduction to Analytic Number Theory. Springer Science & Business Media in these seminars.
| Date | Speaker | Title |
|---|---|---|
| 03.10.2024 | İlker İnam | Divisibility, Greatest common divisor, Prime numbers |
| 10.10.2024 | Zeynep Demirkol Özkaya | The fundamental theorem of arithmetic, The series of reciprocals of the primes, The Euclidean algorithm, The greatest common divisor of more than two numbers |
| 17.10.2024 | Zekiye Pınar Cihan | The Mobius function μ(n), The Euler totient function φ(n), A relation connecting μ(n) and φ(n), A product formula for φ(n) |
| 24.10.2024 | İlker İnam | The Dirichlet product of arithmetical functions, Dirichlet inverses and the Mobius inversion formula, The Mangoldt function Λ(n), Multiplicative functions |
| 31.10.2024 | Mine Ateş | Multiplicative functions and Dirichlet multiplication, The inverse of a completely multiplicative function, Liouville's function λ(n), The divisor functions σ_z(n), Generalized convolutions, Formal power series |
| 07.11.2024 | Zekiye Pınar Cihan | The Bell series of an arithmetical function, Bell series and Dirichlet multiplication, Derivatives of arithmetical functions, The Selberg identity |
| 14.11.2024 | Şevval Dündar | The big oh notation, Asymptotic equality of functions, Euler's summation formula, Some elementary asymptotic formulas |
| 21.11.2024 | Zeynep Demirkol Özkaya | The average order of d(n), The average order of the divisor functions σ_z(n), The average order of φ(n), An application to the distribution of lattice points visible from the origin |
| 28.11.2024 | Mine Ateş | The average order of μ(n) and of Λ(n), The partial sums of a Dirichlet product, Applications to μ(n) and Λ(n), Another identity for the partial sums of a Dirichlet product |
| 05.12.2024 | Sashadhar Dutta | Chebyshev's functions θ(x) and ψ(x), Relations connecting θ(x) and π(n), Some equivalent forms of the prime number theorem |
| 12.12.2024 | İlker İnam | Some equivalent forms of the prime number theorem, Inequalities for π(n) and p_n, Shapiro's Tauberian theorem, Applications of Shapiro's theorem, An asymptotic formula for the partial sum ∑ 1/p, The partial sums of the Möbius function, A brief sketch of an elementary proof of the prime number theorem, Selberg's asymptotic formula |
| 19.12.2024 | Zeynep Demirkol Özkaya | Definition and basic properties of congruences, Residue classes and complete residue systems, Linear congruences, Reduced residue systems and the Euler-Fermat theorem |
| 26.12.2024 | BANT Team | Time Out |
| 02.01.2025 | BANT Team | Holiday |
| 09.01.2025 | Mine Ateş | Polynomial congruences modulo p, Lagrange's theorem, Applications of Lagrange's theorem, Simultaneous linear congruences, The Chinese remainder theorem, Applications of the Chinese remainder theorem, Polynomial congruences with prime power moduli |
| 16.01.2025 | Zekiye Pınar Cihan | The principle of cross-classification, A decomposition property of reduced residue systems |
| 23.01.2025 | BANT Team | Winter Break |
| 30.01.2025 | BANT Team | Winter Break |
| 07.02.2025 | BANT Team | Winter Break |
| 13.02.2025 | Sashadhar Dutta | Examples of groups and subgroups, Elementary properties of groups, Construction of subgroups, Characters of finite abelian groups |
| 20.02.2025 | Murat Özyurt | The character group, The orthogonality relations for characters, Dirichlet characters, Sums involving Dirichlet characters, The nonvanishing of L(1, x) for real nonprincipal x |
| 27.02.2025 | İlker İnam | Chapter 7, whole |
| 05.03.2025 | Zeynep Demirkol Özkaya | Functions periodic modulo k. Existence of finite Fourier series for periodic arithmetical functions. Ramanujan's sum and generalizations. Multiplicative properties of the sums sk(n). |
| 13.03.2025 | Mine Ateş | Gauss sums associated with Dirichlet characters. Dirichlet characters with nonvanishing Gauss sums. Induced moduli and primitive characters. |
| 20.03.2025 | Sashadhar Dutta | Further properties of induced moduli. The conductor of a character. Primitive characters and separable Gauss sums. The finite Fourier series of the Dirichlet characters. Pólya’s inequality for the partial sums of primitive characters. Exercises for Chapter 8. |
| 27.03.2025 | İlker İnam | Quadratic residues. Legendre’s symbol and its properties. Evaluation of (-1|p) and (2|p). Gauss' lemma. The quadratic reciprocity law. |
| 03.04.2025 | Zekiye Pınar Cihan | Applications of the reciprocity law. The Jacobi symbol. Applications to Diophantine equations. Gauss sums and the quadratic reciprocity law. The reciprocity law for quadratic Gauss sums. Another proof of the quadratic reciprocity law. Exercises for Chapter 9. |
| 10.04.2025 | Zeynep Demirkol Özkaya | The exponent of a number mod m. Primitive roots. Primitive roots and reduced residue systems. The nonexistence of primitive roots mod 2^α for α ≥ 3. The existence of primitive roots mod p for odd primes p. Primitive roots and quadratic residues. The existence of primitive roots mod p^x. The existence of primitive roots mod 2p^x. |
| 17.04.2025 | Murat Özyurt | The nonexistence of primitive roots in the remaining cases. The number of primitive roots mod m. The index calculus. Primitive roots and Dirichlet characters. Real-valued Dirichlet characters mod p^x. Primitive Dirichlet characters mod p^x. Exercises for Chapter 10. |
| 24.04.2025 | İlker İnam | The half-plane of absolute convergence of a Dirichlet series. The function defined by a Dirichlet series. Multiplication of Dirichlet series. Euler products. The half-plane of convergence of a Dirichlet series. Analytic properties of Dirichlet series. |
| 01.05.2025 | BANT Team | Holiday |
| 08.05.2025 | Mine Ateş | Dirichlet series with nonnegative coefficients. Dirichlet series expressed as exponentials of Dirichlet series. Mean value formulas for Dirichlet series. An integral formula for the coefficients of a Dirichlet series. An integral formula for the partial sums of a Dirichlet series. Exercises for Chapter 11. |
| 15.05.2025 | Sashadhar Dutta | Properties of the gamma function. Integral representation for the Hurwitz zeta function. A contour integral representation for the Hurwitz zeta function. The analytic continuation of the Hurwitz zeta function. Analytic continuation of ζ(s) and L(s, χ). Hurwitz’s formula for ζ(s, a). The functional equation for the Riemann zeta function. |
| 22.05.2025 | Zekiye Pınar Cihan | A functional equation for the Hurwitz zeta function. The functional equation for L-functions. Evaluation of ζ(−n, a). Properties of Bernoulli numbers and Bernoulli polynomials. Formulas for L(0, χ). Approximation of ζ(s, a) by finite sums. Inequalities for |ζ(s, a)|. Inequalities for |ζ(s)| and |L(s, χ)|. Exercises for Chapter 12. |
| 29.05.2025 | İlker İnam | Chapter 13 whole |
This academic year, we will use the book D. S Malik, John M. Mordeson and M. K. Sen, Fundamentals of Abstract Algebra. in these seminars.
| Date | Speaker | Title |
|---|---|---|
| 12/10/2023 | İlker İnam | Set, Relations, and Integers |
| 19/10/2023 | Mine Ateş | Introduction to Groups |
| 26/10/2023 | Murat Özyurt | Permutation Groups |
| 02/11/2023 | Zeynep Demirkol Özkaya | Subgroups and Normal Subgroups |
| 09/11/2023 | Zeynep Demirkol Özkaya | Subgroups and Normal Subgroups |
| 16/11/2023 | BANT Team | Time Out |
| 23/11/2023 | Zeynep Demirkol Özkaya | Subgroups and Normal Subgroups |
| 30/11/2023 | Elif Ilgaz Çağlayan | Homomorphisms and Isomorphisms of Groups |
| 07/12/2023 | Elif Ilgaz Çağlayan | Homomorphisms and Isomorphisms of Groups |
| 14/12/2023 | İlker İnam | Direct Product of Groups |
| 21/12/2023 | Zeynep Demirkol Özkaya | Sylow Theorems |
| 28/12/2023 | Murat Özyurt | Solvable and Nilpotent Groups |
| 04/01/2024 | Pınar Cihan | Finitely Generated Abelian Groups |
| 11/01/2024 | Mine Ateş | Introduction to Rings |
| 18/01/2024 | Mine Ateş | Some Important Rings |
| 25/01/2024 | BANT Team | Winter Break |
| 01/02/2024 | BANT Team | Winter Break |
| 21/02/2024 | BANT Team | Winter Break |
| 28/02/2024 | Pınar Cihan | Subrings, Ideals, and Homomorphisms |
| 06/03/2024 | Elif Ilgaz Çağlayan | Ring Embedding |
| 13/03/2024 | İlker İnam | Direct Sum of Rings |
| 20/03/2024 | Zeynep Demirkol Özkaya | Polynomial Rings |
| 27/03/2024 | Zeynep Demirkol Özkaya | Euclidean Domains |
| 03/04/2024 | İlker İnam | Unique Factorization Domains |
| 10/04/2024 | BANT Team | Holiday |
| 17/04/2024 | İlker İnam | Unique Factorization Domains |
| 24/04/2024 | Zeynep Demirkol Özkaya | Maximal, Prime, and Maximal Ideals |
| 01/05/2024 | BANT Team | Holiday |
| 08/05/2024 | Mine Ateş | Noetherian and Artinian Rings |
| 15/05/2024 | Elif Ilgaz Çağlayan | Modules and Vector Spaces |
| 22/05/2024 | Murat Özyurt | Rings of Matrices |
| 29/05/2024 | Zeynep Demirkol Özkaya | Field Extension |
| 05/06/2024 | Elif Ilgaz Çağlayan | Field Extension |
Carl Friedrich Gauss’s statement:
“Mathematics is the queen of sciences, the queen of Mathematics is Number Theory”.
To participate in the seminar, please kindly fill out the form:
Seminar Form
| Date | Time | Speaker | Affiliation | Title | Abstract |
|---|---|---|---|---|---|
| 15/10/2024 | 10:00 İstanbul / 11:00 Berlin / 18:00 Sydney / 12:00 Londra | Igor Shparlinski | University of New South Wales | Moments and non-vanishing of L-functions over thin subgroups | Moments and non-vanishing of L-functions over thin subgroups. We obtain an asymptotic formula for all moments of Dirichlet L-functions L(1,χ) modulo p when averaged over a subgroup of characters χ of size (p-1)/d with φ(d)=o(log p). Assuming the infinitude of Mersenne primes, the range of our result is optimal and improves and generalises the previous result of S. Louboutin and M. Munsch (2022) for second moments. We also give an asymptotic formula for the second moment of L(1/2,χ) over subgroups of characters of similar size. This leads to non-vanishing results which in some cases improve those of E. Fouvry, E. Kowalski and P. Michel (2023). Additionally, we prove that, in both cases, we can take much smaller subgroups for almost all primes p. Our method relies on pointwise and average estimates on small solutions of linear congruences which in turn leads us to use and modify some results of J. Bourgain, S. V. Konyagin and I. E. Shparlinski (2008) on product sets of Farey fractions. Joint work with Marc Munsch. |
| 12/11/2024 | 19:00 Istanbul / 17:00 Berlin / 16:00 London | Florian Luca | Stellenbosch University | On a question of Douglass and Ono | It is known that the partition function p(n) obeys Benford’s law in any integer base b ≥ 2. A similar result was obtained by Douglass and Ono for the plane partition function PL(n) in a recent paper. In their paper, Douglass and Ono asked for an explicit version of this result. In particular, given an integer base b ≥ 2 and a string f of digits in base b, they asked for an explicit value N(b, f) such that there exists n ≤ N(b, f) with the property that PL(n) starts with the string f when written in base b. In my talk, I will present an explicit value for N(b, f) both for the partition function p(n) as well as for the plane partition function PL(n). |
| 17/12/2024 | 16:00 Istanbul / 22:00 Tokyo / 13:00 London / 14:00 Berlin | Ade Irma Suriajaya | Kyushu University | Pair correlation of zeros of the Riemann zeta-function, zero density and simple zeros | Assuming the Riemann Hypothesis (RH), Montgomery (1973) proved a theorem concerning the pair correlation of nontrivial zeros of the Riemann zeta-function. One consequence of this theorem was that, under RH, at least 2/3 of the zeros are simple. We show that this theorem of Montgomery holds unconditionally. In an earlier paper, as an application, under a much weaker hypothesis than RH that all the zeros lie within a narrow vertical box centered on the critical line, we showed that at least 61.7% of zeros of the Riemann zeta-function are simple. We can further weaken the hypothesis using a density hypothesis. Recently we were able to improve a little bit on this proportion, and furthermore found a connection to finding proportion of zeros on the critical line. Inspired by a recent preprint of J. Maynard and K. Pratt, we can additionally weaken our assumption by copying the box finitely many times. This is joint work with Siegfred Alan C. Baluyot, Daniel Alan Goldston, and Caroline L. Turnage-Butterbaugh. |
| 14/01/2025 | 16:00 Istanbul / 13:00 London / 14:00 Berlin / 22:00 Tokyo | Cem Yalçın Yıldırım | Boğaziçi University | Some analogues of pair correlation of zeta zeros | An alternative way to carry out Montgomery’s original calculation of the pair correlation of zeta zeros allows us to obtain some other analogous results. In particular, we can approach the problems of correlating zeta zeros and relative maxima on the critical line, and the pair correlation of these maxima. |
| 25/02/2025 | 16:00 Istanbul / 13:00 London / 14:00 Berlin / 22:00 Tokyo | A. Muhammed Uludağ | Galatasaray University | Dyer’s outer automorphism of PGL(2,Z) and the codenominator | The codenominator is a function F that extends the Fibonacci sequence to the index set of positive rational numbers. Many known Fibonacci identities carry over to the codenominator. One can express Dyer’s outer automorphism of the extended modular group PGL(2, Z) in terms of F. This automorphism can be viewed as an automorphism group of the trivalent tree. The real-covariant modular function Jimm J on the real line is defined via the codenominator. J relates the Stern-Brocot tree to the Bird tree. Jimm induces an involution of the moduli space of rank-2 pseudolattices and is related to the arithmetic of real quadratic irrationals. |
| 25/03/2025 | 16:00 Istanbul / 13:00 London / 14:00 Berlin / 22:00 Tokyo | Özlem İmamoğlu | ETH Zurich | A Lyapunov exponent attached to modular functions | In joint work with Paloma Bengoechea and Sebastian Herrero we recently defined a Lyapunov exponent attached to modular functions and proved its properties. Our work was motivated by the work of Spalding and Veselov as well as conjectures of Kaneko on the values of modular functions at real quadratic irrationalities. In this talk I will first explain the results of Spalding and Veselov and the conjectures of Kaneko. I will then talk about the new results. |
| 22/04/2025 | TBA | TBA | TBA | TBA | TBA |
| 27/05/2025 | TBA | John Voight | University of Sydney | TBA | TBA |
| 10/06/2025 | TBA | Ian Kiming | University of Copenhagen | TBA | TBA |
| Date | Time | Speaker | Affiliation | Title | Abstract |
|---|---|---|---|---|---|
| 19/03/2024 | 16:00 Istanbul / 14:00 Berlin / 13:00 London / 22:00 Seoul / 08:00 New York | Kaisa Matomäki | University of Turku | Primes in arithmetic progressions and short intervals without L-functions | I will discuss my joint work with Jori Merikoski and Joni Teräväinen where we develop a sieve that can detect primes in multiplicatively structured sets under certain conditions. In particular, I will discuss the following two applications: a new L-function free approach to Linnik's problem of bounding the least prime p such that p ≡ a (mod q) (obtaining the bound p << q350) and a new L-function free proof that the interval (x−x39/40, x] contains primes for every large x. |
| 02/04/2024 | 16.00 Istanbul / 15:00 Berlin / 14:00 London / 23:00 Seoul / 07:00 New York | Kağan Kurşungöz | Sabancı University | A Decomposition of Cylindric Partitions and Cylindric Partitions into Distinct Parts | After relevant definitions, some motivation, and some results from the literature, we will show that cylindrical decompositions correspond exactly to pairs of an ordinary decomposition and a colored decomposition into different parts. According to the remaining time, we will explain how to obtain the generator functions of cylindrical decompositions in different sections and give examples. This study is a joint work with Halime Ömrüuzun Seyrek (https://arxiv.org/abs/2308.14514). |
| 16/04/2024 | 16:00 Istanbul / 15:00 Berlin / 14:00 London / 22:00 Seoul / 06:00 New York | Gabor Wiese | University of Luxembourg | Splitting fields of Xn-X-1 and modular forms | In his article 'On a theorem of Jordan', Serre considered the family of polynomials fn(X) = Xn-X-1 and the counting function of the number of roots of fn over the finite field Fp, seen as function in p. He explicitly showed the 'modularity' of this function for n=3,4. In this talk, I report on joint work with Alfio Fabio La Rosa and Chandrashekhar Khare, in which we treat the case n=5 in several different ways. |
| 30/04/2024 | 16:00 Istanbul / 15:00 Berlin / 14:00 London / 22:00 Seoul / 06:00 New York | Ken Ono | University of Virginia | The partition function modulo 2 and 4 | The Ramanujan congruences for the partition function have an extraordinary legacy in mathematics. These days research abounds with new congruences for various sorts of restricted partition functions. Unfortunately, very little is known about p(n) modulo powers of 2. In this talk, the speaker will discuss new and old results about the partition function modulo 2 and 4, and will offer a few precise open questions with the idea of catalyzing work in the area. |
| 14/05/2024 | 18:00 Istanbul / 17:00 Berlin / 16:00 London / 00:00 Seoul / 08:00 New York | Kenneth Ribet | University of California, Berkeley | Cyclotomic points on abelian varieties | Roughly 30 years ago, I proved: Suppose that A is an abelian variety over a number field K. Then A has only a finite number of torsion points defined over the maximal cyclotomic extension of K. After explaining the ingredients of the proof, I will highlight some questions suggested by this theorem. One natural project is to compute the group of cyclotomic torsion points in some specific examples. If A is J0(N), where N is a prime number, then the group of torsion points on A over the maximal cyclotomic extension of Q is the kernel of the Eisenstein ideal on A. |
| 28/05/2024 | 16:00 Istanbul / 15:00 Berlin / 14:00 London / 22:00 Seoul / 06:00 New York | Kathrin Bringmann | University of Cologne | Modular-Type Objects and Asymptotics of Their Coefficients | In my talk I will report on asymptotics for Fourier coefficients of modular forms and related objects. |
| Name & Surname |
|---|
| Mine Ateş |
| Zekiye Pınar Cihan |
| Elif Ilgaz Çağlayan |
| Şevval Dündar |
| İlker İnam |
| Bahar Kuloğlu |
| Zeynep Demirkol Özkaya |
| Murat Özyurt |
| Sashadhar Dutta |
Edited by Goca Patron