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In 1999, Gadiyar and Padma discovered a simple heuristic to derive the generalized twin prime conjecture using an orthogonality principle for Ramanujan sums originally discovered by Carmichael. We derive a limit formula for higher convolutions of Ramanujan sums, generalizing an old result of Carmichael. We then apply this in conjunction with the general theory of arithmetical functions of several variables to give a heuristic derivation of the Hardy–Littlewood formula for the number of prime $k$-tuples less than $x$.

A zero-free region is a subset of the complex plane where the Riemann zeta-function does not vanish. Such regions have historically been used to further our understanding of prime-number distributions. In the classical approach, we first assume that a zero exists off the critical line, then arrive at an inequality involving its real and imaginary parts. One notable aspect of the classical argument is that it does not require any knowledge about the relationship between the zeroes. However, it is well known that the location of a hypothetical zero depends strongly on the behaviour of nearby zeroes—for instance, N. Levinson showed in 1969 that if zeroes of the zeta-function are well-spaced near the 1-line, then we can obtain a zero-free region stronger than any that are currently known. In this talk we will discuss some ideas on how one might incorporate information about distributions of hypothetical zeroes to improve existing zero-free regions.

In classical comparative prime number theory, it is customary to assume some kind of linear independence hypothesis about the zeros of the underlying L-functions. These hypotheses are completely out of reach of current methods. However, in the function field case, it is sometimes possible to prove them, or at least to show they hold generically. In this talk I will present recent results in comparative prime number theory over function fields that establish infinite families of “irreducible polynomial races” which we can study unconditionally. Some of those results are joint work with L. Devin, D. Keliher, and W. Li.

The Voronoi summation formulas for the divisor function and $r_2(n)$ are well known. Not only do these formulas have interesting structure, but they have also been used to improve the error term in the Dirichlet divisor problem and the Gauss circle problem, respectively. In this work with Atul Dixit, we derive Voronoi summation formulas for some other functions related to the generalized divisor function and the Liouville lambda function. We also derive beautiful analogues of Cohen’s identity and the Ramanujan–Guinand formula associated to these functions and show their equivalence.

Central to comparative number theory is the study of the difference $\Delta(t) = \pi(t) − li(t)$, where $\pi(t)$ is the prime counting function and $li(t)$ is the logarithmic integral. Prior to a celebrated 1914 paper of Littlewood, it was believed that $\Delta < 0$ for all $t > 2$. We now know however that $\Delta(t)$ changes sign infinitely often, with the first sign change occuring before 10320. Despite this, it still appears that $\Delta(t)$ is negative “on average”, in that integrating $\Delta (t)$ from $t = 2$ onwards yields a negative value. In this talk, we will explore this idea in detail, discussing links with the Riemann hypothesis and also extending such ideas to other differences involving arithmetic functions.

Based on work previously done by Gonek, Graham, and Lee, we show that the Riemann Hypothesis (RH) can be reformulated in terms of certain asymptotic estimates for twisted sums with k-fold convolution of von Mangoldt function and the generalized von Mangoldt function. For each $k \in\mathbb{N}$, we study two types of twisted sums:

1. $\sum_{n\leq x} \Lambda^k(n)n^{-iy}$, where $\Lambda^k(n) = \underbrace{\Lambda\star\cdots\Lambda}_\text{k copies}$
2. $\sum_{n\leq x} \Lambda_k(n)n^{-iy}$, where $\Lambda_k(n) :=\sum_{d|n}\mu(d)\left(\log{\frac{n}{d}}\right)^k$.

Where $\Lambda$ is the von Mangoldt function and $\mu$ is the Möbius function, and establish similar connections with RH.

Let $y\neq 0$ and $C>0$. Under the Riemann Hypothesis, there is a number $T_* > 0$ (depending on $y$ and $C$) such that for every $T>T_*$, both
$$
\zeta(\frac{1}{2}+i\gamma) = 0 \qquad \mbox{and} \qquad
\zeta(\frac{1}{2} + i(\gamma + y))\neq 0
$$
hold for at least one $\gamma$ in the interval $[T, T(1+\epsilon]$, where $\epsilon := T^{-C/\log\log T}$.

Let $E_1$ and $E_2$ be two non-CM elliptic curves defined over a number field $K$. By an isogeny theorem due to Kulkarni, Patankar, and Rajan, the two curves are geometrically isogenous if and only if the density of primes for which their Frobenius field coincide is positive. In this talk, we present a quantitative upper bounds of this criterion that improves the result of Baier–Patankar and Wong. The strategy relies on effective versions of the Chebotarev Density Theorem. This is joint work with Alina Cojocaru and Auden Hinz.

We prove explicit estimates of $1/\zeta(s)$ of various orders, and use an improved version of the Perron formula to get explicit estimates for the Mertens function $M(x)$ of order $O(x)$, $O(x/\log^k x)$, and $O(x\log x exp(−\sqrt{\log{x}})$. These estimates are good for small, medium, and large ranges of $x$, respectively.

Let f be a self-dual Maass form for $SL(n, Z)$. We write $L_f (s)$ for the Godement–Jacquet L-function associated to $f$ and $L_{f\times f} (s)$ for the Rankin–Selberg L-function of $f$ with itself. The inverse of $L_{f\times f} (s)$ is defined by
$$
\frac{1}{L_{f\times f}(s)} := \sum_{m=1}^\infty \frac{c(m)}{m^s}, \mathfrak{R}(s) > 1.
$$
It is well known that the classical Mertens function $M(x) := \sum_{m\leq x} \mu(m)$ is related to
$$
\frac{1}{\zeta(s)} = \sum_{m=1}^\infty \frac{\mu(m)}{m^s}, \mathfrak{R}(s) > 1.
$$
We define the analogue of the Mertens function for $L_{f\times f} (s)$ as $\widetilde{M}(x) := \sum_{m\leq x} c(m)$ and obtain an upper bound for this analogue $\widetilde{M}(x)$, similar to what is known for the Mertens function $M(x)$. In particular, we prove that $\widetilde{M}(x) \ll_f x \exp(−A\sqrt{\log{x}}$ for sufficiently large $x$ and for some positive constant $A$. This is a joint work with my Ph.D. supervisor Prof. A. Sankaranarayanan.

One of the central problems in comparative prime number theory involves understanding primes in
arithmetic progressions. The distribution of primes in arithmetic progressions are sensitive to real zeros near $s = 1$ of L-functions associated to primitive real Dirichlet characters. The Generalized Riemann Hypothesis implies that such L-functions have no zeros near $s = 1$. In 1935, Siegel proved the strongest known upper bound for the largest such real zero, but his result is vastly inferior to what is known unconditionally for other L-functions. We exponentially improve Siegel’s bound under a mild hypothesis that permits real zeros to lie close to $s = 1$. Our hypothesis can be verified for almost all primitive real characters. Our work extends to other families of L-functions. This is joint work with Jesse Thorner and Alexandru Zaharescu.

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $C_l$ be the family of prime cyclic extensions of degree $l$ over $\mathbb{Q}$. Under GRH for elliptic L-functions, we give a lower bound for the probability for $K \in C_l$ such that the difference $r_K(E) − r_\mathbb{Q}(E)$ between analytic rank is less than a for $a \asymp l$. This result gives conjectural evidence that the Diophantine Stability problem suggested by Mazur and Rubin holds for most of $K \in C_l$.

The $\kappa$-factor function is a generalization of the irrational factor function introduced by Atanassov. We investigate the behavior of the sum of the $\kappa$-factor function over arithmetic progressions and study its bias over reduced residue classes. We also study this phenomenon for $\kappa$-factor functions defined over number fields and function fields.

Let $\pi(x; q, a)$ be the number of primes $p\leq x$ such that $p \equiv a (\mod q)$. The classical Shanks–Rényi prime number race problem asks, given positive integers $q \geq 3$ and $2 \leq r \leq \phi(q)$ and distinct reduced residue classes $a_1, a_2, . . . , a_r$ modulo $q$, whether there are infinitely many integers $n$ such that $\pi (n; q, a1) > \pi(n; q, a2) > \cdots > \pi(n; q, ar)$. In this talk, I will describe what is known on this problem when the number of competitors $r \geq 3$, and how this compares to the Chebyshev’s bias case which corresponds to $r = 2$.

Optimal transport operates on empirical distributions which may contain acquisition artifacts, such as outliers or noise, thereby hindering a robust calculation of the OT map. Additionally, it necessitates equal mass between the two distributions, which can be overly restrictive in certain machine learning or computer vision applications where distributions may have arbitrary masses, or when only a fraction of the total mass needs to be transported. Unbalanced Optimal Transport addresses the issue of rebalancing or removing some mass from the problem by relaxing the marginal conditions. Consequently, it is often considered to be more robust, to some extent, against these artifacts compared to its standard balanced counterpart. In this presentation, I will review several divergences for relaxing the marginals, ranging from vertical divergences like the Kullback-Leibler or the L2-norm, which allow for the removal of some mass, to horizontal ones, enabling a more robust formulation by redistributing the mass between the source and target distributions. Additionally, I will discuss efficient algorithms that do not necessitate additional regularization on the OT plan.

The classical theory of hypergeometric functions, developed by generations of mathematicians including Gauss, Kummer, and Riemann, has been used substantially in the ensuing years within number theory, geometry, and the intersection thereof. In more recent decades, these classical ideas have been translated from the complex setting into the finite field and p
-adic settings as well.

In this talk, we will give a friendly introduction to hypergeometric functions, especially in the context of number theory.

Density functional theory (DFT) is one of the workhorses of quantum chemistry and material science. In principle, the joint probability of finding a specific electron configuration in a material is governed by a Schrödinger wave equation. But numerically computing this joint probability is computationally infeasible, due to the complexity scaling exponentially in the number of electrons. DFT aims to circumvent this difficulty by focusing on the marginal probability of one electron. In the last decade, a connection was found between DFT and a multi-marginal optimal transport problem with a repulsive cost. I will give a brief introduction to this topic, including some open problems, and recent progress.

Fix $N\geq 1$ and let $L_1, L_2, \ldots, L_N$ be Dirichlet L-functions with distinct, primitive and even Dirichlet characters. We assume that these functions satisfy the same functional equation. Let $F(s)∶= c_1L_1(s)+c_2L_2(s)+\ldots+c_NL_N(s)$ be a linear combination of these functions ($c_j \in\mathbb{R}^*$ are distinct). $F$ is known to have two kinds of zeros: trivial ones, and non-trivial zeros which are confined in a vertical strip. We denote the number of non-trivial zeros $\rho$ with $\mathfrak{I}(\rho)\leq T$ by $N(T)$, and we let $N_\theta(T)$ be the number of these zeros that are on the critical line. At the end of the 90's, Selberg proved that this linear combination had a positive proportion of zeros on the critical line, by showing that $\kappa F∶=\lim \inf T (N_\theta(2T)−N_\theta(T))/(N(2T)−N(T))\geq c/N^2$ for some $c>0$. Our goal is to provide an explicit value for $c$, and also to improve the lower bound above by showing that $\kappa_F \geq 2.16\times 10^{-6}/(N \log N)$, for any large enough $N$.

I will discuss the fourth moment of quadratic Dirichlet L-functions where we prove an asymptotic formula with four main terms unconditionally. Previously, the asymptotic formula was established with the leading main term under generalized Riemann hypothesis. This work is based on Li's recent work on the second moment of quadratic twists of modular L-functions. It is joint work with Joshua Stucky.

An orthogonal array with parameters \((N,n,q,t)\) (\(OA(N,n,q,t)\) for short) is an \(N\times n\) matrix with entries from the alphabet \(\{1,2,...,q\}\) such that in any of its \(t\) columns, all possible row vectors of length \(t\) occur equally often. Rao showed the following lower bound on \(N\) for \(OA(N,n,q,2e)\):
\[ N\geq \sum_{k=0}^e \binom{n}{k}(q-1)^k, \]
and an orthogonal array is said to be complete or tight if it achieves equality in this bound. It is known by Delsarte (1973) that for complete orthogonal arrays \(OA(N,n,q,2e)\), the number of Hamming distances between distinct two rows is \(e\). One of the classical problems is to classify complete orthogonal arrays.

We call an orthogonal array \(OA(N,n,q,2e-1)\) extremal if the number of Hamming distances between distinct two rows is \(e\). In this talk, we review the classification problem of complete orthogonal arrays with our contribution to the case \(t=4\) and show how to extend it to extremal orthogonal arrays. Moreover, we give a result for extremal orthogonal arrays which is a counterpart of a result in block designs by Ionin and Shrikhande in 1993.

In a linear population model that has a unique “largest” eigenvalue and is suitably irreducible, the corresponding left and right (Perron) eigenvectors determine the long-term relative prevalence and reproductive value of different types of individuals, as described by the Perron-Frobenius theorem and generalizations. It is therefore of interest to study how the Perron vectors depend on the generator of the model. Even when the generator is a finite-dimensional matrix, there are several approaches to the corresponding perturbation theory. We explore an approach that hinges on stochasticization (re-weighting the space of types to make the generator stochastic) and interprets formulas in terms of the corresponding Markov chain. The resulting expressions have a simple form that can also be obtained by differentiating the renewal-theoretic formula for the Perron vectors. The theory appears well-suited to the study of infection spread that persists in a population at a relatively low prevalence over an extended period of time, via a fast-slow decomposition with the fast/slow variables corresponding to infected/non-infected compartments, respectively. This is joint work with MSc student Tareque Hossain.

To each square-free monic polynomial $D$ in a fixed polynomial ring $\mathbb{F}_q[t]$, we can associate a real quadratic character $\chi_D$, and then a Dirichlet $L$-function $L(s,\chi_D)$. We compute the limiting distribution of the family of values $L'(1,\chi_D)/L(1,\chi_D)$ as $D$ runs through the square-free monic polynomials of $\mathbb{F}_q[t]$ and establish that this distribution has a smooth density function. Time permitting, we discuss connections of this result with Euler-Kronecker constants and ideal class groups of quadratic extensions. This is joint work with Amir Akbary.

The class number one problem is one of the central subjects in algebraic number theory that turns back to the time of Gauss. This problem has led to the classical embedding problem which asks whether or not any number field $K$ can be embedded in a finite extension $L$ with class number one. Although Golod and Shafarevich gave a counterexample for the classical embedding problem, yet one may ask about the embedding in 'Polya fields', a special generalization of class number one number fields. The latter is the 'new embedding problem' investigated by Leriche in 2014. In this talk, I briefly review some well-known results in the literature on the embedding problems. Then, I will present the 'relativized' version of the new embedding problem studied in a joint work with Ali Rajaei.

Let \(p\) be an odd prime. We study Mazur's conjecture on the growth of the Mordell--Weil ranks of an elliptic curve \(E/\mathbb{Q}\) over an imaginary quadratic field in which \(p\) splits and \(E\) has good reduction at \(p\). In particular, we obtain criteria that may be checked through explicit calculation, thus allowing for the verification of Mazur's conjecture in specific examples. This is joint work with Rylan Gajek-Leonard, Jeffrey Hatley, and Antonio Lei.

The distribution of values of Dirichlet L-functions \(L(s, \chi)\) for variable \(χ\) has been studied extensively and has a vast literature. Moments of higher derivatives has been studied as well, by Soundarajan, Sono, Heath-Brown etc. However, the study of the same for the logarithmic derivative \(L'(s, \chi)/ L(s, \chi)\) is much more recent and was initiated by Ihara, Murty etc. In this talk we will discuss higher derivatives of the logarithmic derivative and present some new results related to their distribution and moments at s=1.

In this talk, we will discuss an estimate for a discrete mean value of the Riemann zeta function and its derivatives multiplied by Dirichlet polynomials. Assuming the Riemann Hypothesis, we obtain a lower bound for the 2kth moment of all the derivatives of the Riemann zeta function evaluated at its nontrivial zeros. This is based on a joint work with Kübra Benli and Nathan Ng.

Let $s(n)$ denote the sum of proper divisors of a positive integer $n$. In 1992, Erdös, Granville, Pomerance, and Spiro conjectured that if \(\square\) is a set of integers with asymptotic density zero then the preimage set \(s^{−1}(\square)\) also has asymptotic density zero. In this talk, we will discuss the verification of this conjecture when \(\square\) is the set of integers with missing digits (also known as ellipsephic integers) by giving a quantitative estimate on the size of the set \(s^{-1}(\square)\). This talk is based on the joint work with Giulia Cesana, C\'{e}cile Dartyge, Charlotte Dombrowsky and Lola Thompson.

Evolutionary game theory is a discipline devoted to studying populations of individuals that are subject to evolutionary pressures, and whose success generally depends on the composition of the population. In biological contexts, individuals could be molecules, simple organisms or animals, and evolutionary pressures often take the form of natural selection and mutations. In socioeconomic contexts, individuals could be humans, firms or other institutions, and evolutionary pressures often derive from competition for scarce resources and experimentation.

In this talk I will give a very basic introduction to agent-based evolutionary game theory, a bottom-up approach to modelling and analyzing these systems. The defining feature of this modelling approach is that the individual units of the system and their interactions are explicitly and individually represented in the model. The models thus defined can be usefully formalized as stochastic processes, whose dynamics can be explored using computer simulation and approximated using various mathematical theories.

Exponential sums play a role in many different problems in number theory. For instance, Gauss sums are at the heart of some early proofs of the quadratic reciprocity law, while Kloosterman sums are involved in the study of modular and automorphic forms. Another example of application of exponential sums is the circle method, an analytic approach to problems involving the enumeration of integer solutions to certain equations. In many cases, obtaining upper bounds on the modulus of these sums allow us to draw conclusions, but once the modulus has been bounded, it is natural to ask the question of the distribution of exponential sums in the region of the complex plane in which they live. After a brief overview of the motivations mentioned above, I will present some results obtained with Emmanuel Kowalski on the equidistribution of exponential sums indexed by the roots modulo p of a polynomial with integer coefficients.