Primes, Patterns & Propagation
Schedule

[Talk] Moments of averages of Ramanujan sums 

Prof. M. Ram Murty

Abstract: We will discuss higher moments of averages of Ramanujan sums. Chan and Kumchev studied the first and second moment using classical methods. A similar attempt was made (unsuccessfully) by Robles and Roy. The study of higher moments requires the theory of arithmetical functions of several variables and their associated Dirichlet series. We will apply a multi-dimensional Tauberian theorem to derive the correct asymptotic formulas for all the higher moments. This is a report of joint work with Shivani Goel.

[Talk] TBA

Dr. Sudhir Pujahari

Abstract: TBA

[Talk] Moments and nonvanishing for quadratic twists 

Prof. Ritabrata Munshi

Abstract: TBA

[Talk*] Joint moments

Prof. Jon Keating

Abstract: I will discuss the evaluation of the joint moments of the characteristic polynomials of random unitary matrices and their derivatives, and the joint moments of the Riemann zeta-function and its derivatives on the critical line, in both cases emphasizing the connections to integrable odes.

[Talk*] The Sign of Linear Periods

Prof. U.K. Anandabardhanan

Abstract: Distinguished representations are the central objects of study in the relative Langlands programme. The linear periods in the title refer to the study of distinction for the symmetric space (GL(n, F ), GL(p, F ) × GL(q, F)), with p + q = n. The pioneering work here is the 1996 paper of Jacquet and Rallis. We discuss this theme in this talk.

[Talk] Catalan's Constant

Prof. V Kumar Murty

Abstract: Catalan’s constant is the value L(2, χ) where χ is the nontrivial character mod 4. It is unknown whether this number is irrational. We discuss the construction of a mixed motive that has Catalan’s constant and π as periods. We use this description to produce many rational linear forms in 1 and L(2, χ) and study what implication this has for irrationality. This is joint work with Payman Eskandari and Yusuke Nemoto.

[Talk] On the Brauer-Siegel and Stark's Conjecture on almost S_n Fields

Dr. Anup Dixit

Abstract: The classical Brauer-Siegel conjecture predicts how the class number times the regulator behaves on a family of number fields. This conjecture holds true under GRH and is known for several families, such as Galois extensions, almost normal extensions and solvable extensions. A related conjecture is due to Stark which asserts that there are only finitely many CM fields with a bounded class number. In this talk, we introduce a new family of ”almost Sn-fields” and address both conjectures for this family.

[Talk] Fundamental Fourier coefficients of Siegel modular forms: level aspect

Dr. Soumya Das

Abstract: Arithmetically distinguished Fourier coefficients of automorphic forms often imply decisive results on the associated L-functions or their special value results.

In this talk we will focus on this problem for Siegel modular forms (SMF) of any level , and any degree. As an application this completes the proof of the special value results (as part of the Deligne’s conjectures) of the spinor L-functions for holomorphic SMF on the congruence subgroups of Sp(6), based on the recent work of Eischen, Rosso, Shah.

[Talk*] The distribution of values of L-functions in the critical strip and applications

Prof. Youness Lamzouri

Abstract: In this talk, I will survey the history of the value distribution theory of L-functions in the critical strip, focusing in particular on what is known at the right of the critical line. I will then give two main applications of this theory: counting zeros of linear combinations of L-functions near the critical line, and counting sign changes of quadratic character sums and real zeros of Fekete polynomials.

[Talk] TBA

Dr. Soumyarup Banerjee

Abstract: TBA

[Talk] On real zeroes of the first derivative of quadratic Dirichlet L-functions

Dr. Kunjakanan Nath

Abstract: One of the central topics in number theory is the study of L-functions and the distribution of their zeros. For example, the celebrated Prime Number Theorem is equivalent to the fact that the Riemann zeta function ζ(s) does not vanish on the line Re(s) = 1. In this talk, we will focus on quadratic Dirichlet L-functions: in particular, the real zeros of the derivative of quadratic Dirichlet L-functions L′(s, χd), where d ranges over fundamental discriminants. In 1990, Baker and Montgomery conjectured that there are ≍ log log |d| real zeros of L′ (s, χd ) in the interval [1/2, 1] for almost all fundamental discriminants d. We will highlight some recent exciting progress that comes close to proving this conjecture. This is based on joint work with Youness Lamzouri.  

[Talk] Partition statistics, their modularity, and asymptotics: a walk in Ramanujan’s garden

Dr. Koustav Banerjee

Abstract: In this talk, I will present a few examples of partition statistics and discuss their modularity through two q-series identities of Ramanujan. In the end, I will also discuss briefly on asymptotics of these sequences.

[Talk*] On exceptions of unimodular quadratic forms

Prof. M Manickam

Abstract: In this work we derive an explicit bound for growth of arbitrary cusp form of even weight for the full modular group. Then using theory of modular forms we get a tight  natural number N which depends on the weight and  it has a polynomial growth in weight. Finally we derive that if n > N any such unimodular quadratic form represents 2n. 

[Talk] Numbers, primes and equidistribution

Prof. Anirban Mukhopadhyay

Abstract: We will be discussing the distribution of the fractional parts of some sequences of real numbers. This includes integral and prime multiples of irrational numbers, and also some similar generalised polynomial sequences.

[Talk*] The large sieve with square moduli, revisited

Prof. Stephen Baier

Abstract: We start with giving a review of the classical large sieve and its applications in analytic number theory. Then we turn our attention to variants of the large sieve, in particular, the large sieve with square moduli. We discuss results on this variant and recent conditional improvements using additive energies of modular square roots.

[Talk*] Integro-differential equations for glassy type memory

Prof. Paola Loreti

Abstract: The study of glassy materials fits naturally within the framework of viscoelasticity theory and integro-differential equations. In the talk we investigate a class of kernels that arise in viscoelastic materials and analyze regular kernels expressed as sums of exponential functions, as commonly used in linear vis- coelasticity, see [1]. We also discuss (using multiplier techniques) the decay estimates for a class of second- order integro-differential evolution equations for glassy type memory, see [2]. Finally, we talk about the observability of the wave equation with memory, as the Burgers model, using a spectral approach, see [3].

[1] Paola Loreti; Daniela Sforza, Viscoelastic aspects of glass relaxation models, Phys. A 526, 2019, 120768.

[2]  Paola Loreti; Daniela Sforza, Energy decay for evolution equations with glassy type memory, Applied Mathematics Letters, 2025, 109834.

[3]  Paola Loreti; Daniela Sforza, Controllability for the Burgers model, J. Math. Anal. 2. 531, 2024, no. 2, part 2, 127836.

[Talk] Surfaces Associated to Zeores of Automorphic L-functions

Dr. Debmalya Basak

Abstract: Assuming the Riemann Hypothesis, Montgomery established results concerning pair correla- tion of zeros of the Riemann zeta function. Rudnick and Sarnak extended these results to automorphic L-functions and all level correlations. We show that automorphic L-functions exhibit additional geometric structures related to the correlation of their zeros. This is joint work with Cruz Castillo and Alexandru Zaharescu.

[Talk] Omega Bound for The Divisor Problems

Dr. Kamalakshya Mahatab

Abstract: We will discuss application of the resonance method to exponential sums with positive coeffi- cients. We will apply this method to obtain omega bounds for the classical divisor problem, Piltz divisor problem over number fields, circle problem and other lattice points problems. We will also discuss limita- tions of this method and possible extensions.

[Talk*] On convolution sums of divisor functions

Dr. Biswajyoti Saha

Abstract: The study of convolution sums of the divisor functions has a towering history, starting with the fundamental work Ingham, where he established asymptotic estimates for the shifted and additive convolution sum of the divisor function. However, the study of the triple convolution of the divisor function turns out to be rather challenging. Browning provided a conjectural statement in this context. In this talk, we will discuss some facets of these studies

[Talk*] Simultaneous nonvanishing of central L-values with large squarefree level

Prof. B. Ramakrishnan

Abstract: For a given normalised newform f of weight 2k and squarefree level N, we establish a lower bound (with respect to N ) for the number of normalised newforms g of the same weight and level as f such that the twisted L-values of f and g, both twisted by a quadratic character, do not vanish. This is joint work with E. M. Sandeep.

[Talk] A revisit to Ramanujan-Serre derivative map on quasimodular forms and some applications

Dr. Brundaban Sahu

Abstract: We revisit the Ramanujan-Serre derivative map on the space of modular forms, give a re-interpretation as a differential operator on the space of quasimodular forms. We study various algebraic properties of the differential operator, give applications in the direction of the study of the Chazy equation, Niebur’s identity, van der Pol’s identity and evaluation of convolution sums of divisor functions. This is a joint work with Raveena Ganash.

[Talk*] Voronoï Summation Formula for the Generalized Divisor Function σ_z^(k) (n).

Dr. Atul Dixit

Abstract:  For fixed z ∈ C and k ∈ N, let σ_z^(k) (n) denote the sum of z-th powers of those divisors d of n whose k-th powers also divide n. This arithmetic function is a simultaneous generalization of the well- known divisor function σ_z^ (n) as well as of the divisor function d^(k)(n), first studied by Wigert. We obtain the Voronoï summation formula for σ_z^(k) (n). An important thing to note here is that this arithmetic function does not fall under the purview of the general setting of the Hecke functional function with multiple gamma factors studied by Chandrasekharan and Narasimhan for which the such a summation formula is already known. The kernel of the integral transform occurring in this formula is a Meijer G-function not studied previously. By deriving a grand generalization of a result of G. H. Hardy, we show that the Meijer G-function can also be represented as a conditionally convergent integral of a real variable. The proof of this fact em- ploys techniques from differential equations, and also demonstrates that the differential equation of this particular Meijer G -function simplifies considerably. The latter is shown using properties of combinatorial objects such as Stirling numbers and elementary symmetric polynomials. This talk is based on joint work with Bibekananda Maji and Akshaa Vatwani, and with an appendix to it written jointly with Shashank Chorge and Aviral Srivastava.

[Talk*] Lattice Counting with Local Hecke Series

Prof. Gautami Bhowmik

Abstract: We count maximal and polarized lattices over p-adic fields. The tool used is the local Hecke series for a reductive group developped by Andrianov and Hina-Sugano. The Euler products give related zeta functions of classical groups, which were often studied with p-adic cone integrals.

We recover the known zeta functions and mention some new ones, especially for non-split forms. This is joint work with Masao Tsuzuki.

[Talk*] Monogenity: A "Prayag" of Algebraic Number Theory with Differential Equations

Dr. Tapas Chatterjee

Abstract: A number field K is called monogenic if it has a power basis, i.e., there exists an algebraic integer α such that {1,α,...,α^(n−1)} is a basis of K over Q. Let f(x) = x^n +a x^3 +b x+c be the minimal polynomial of an algebraic integer θ over Q with certain conditions on a, b, c, and n. Let K = Q(θ) be a number field and O_K the ring of integers of K. In this talk, we characterize all prime divisors of the discriminant of f(x) which do not divide the index of Z[θ] in O_K. As an interesting corollary, we establish necessary and sufficient conditions for Z[θ] to be integrally closed, which implies that the number field K = Q(θ) is monogenic. In addition, we investigate the types of solutions to certain differential equations associated with the polynomial f(x) with the help of monogenity of the splitting field of f(x). This is a joint work with Karishan Kumar, recently accepted for publication in the journal Research in Number Theory.

If time permits, we discuss a similar problem for the polynomial f (x) = x^n + a x^(n−1) + b x + c.

[Talk] A pearl of Number Theory: some old and new applications

Prof. Sukumar Das Adhikari

Abstract: After stating the classical van der Waerden’s theorem, we go through some old and new applications of the theorem. We shall also see some open questions in Ramsey Theory.

[Talk*] TBA

Prof. Anish Ghosh

Abstract: TBA

[Public Lecture] The Art of Research

Prof. M Ram Murty

[Public Lecture] Reflections on the Computer and the Human Brain

Prof. V Kumar Murty