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Alisa Sedunova

Publications and source records attributed to Alisa Sedunova.

13 recordsLinked to original sources
arXiv2024

Relative class numbers and Euler-Kronecker constants of maximal real cyclotomic subfields

The Euler--Kronecker constant of a number field $K$ is the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function $ζ_K(s)$ at $s=1$. We study the distribution of the Euler--Kronecker constant $γ_q^+$ of the maximal real subfield of $\mathbb Q(ζ_q)$ as $q$ ranges over the primes. Further, we consider the distribution of $γ_q^+-γ_q$, with $γ_q$ the Euler--Kronecker constant of $\mathbb Q(ζ_q)$ and show how it is connected with Kummer's conjecture, which predicts the asymptotic growth of the relative class number of $\mathbb Q(ζ_q)$. We improve, for example, the known results on the bounds on average for the Kummer ratio and we prove analogous sharp bounds for $γ_q^+-γ_q$. The methods employed are partly inspired by those used by Granville (1990) and Croot and Granville (2002) to investigate Kummer's conjecture. We supplement our theoretical findings with numerical illustrations to reinforce our conclusions.

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arXiv2024

The Kummer ratio of the relative class number for prime cyclotomic fields

Kummer's conjecture predicts the asymptotic growth of the relative class number of prime cyclotomic fields. We substantially improve the known bounds of Kummer's ratio under three scenarios: no Siegel zero, presence of Siegel zero and assuming the Riemann Hypothesis for the Dirichlet $L$-series attached to odd characters only. The numerical work in this paper extends and improves on our earlier preprint (arXiv:1908.01152) and demonstrates our theoretical results.

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arXiv2023

The multiplication table constant and sums of two squares

We will show that the number of integers $\leq x$ that can be written as the square of an integer plus the square of a prime equals $\fracπ{2} \cdot \frac {x}{\log x}$ minus a secondary term of size $x/(\log x)^{ 1+δ+o(1)}$, where $δ:= 1 - \frac{1+\log\log 2}{\log 2} = 0.0860713320\dots$ is the multiplication table constant. Detailed heuristics suggest that this secondary term is asymptotic to $\frac{1 }{\sqrt{\log\log x}} \cdot \frac x{(\log x)^{ 1+δ}} $ times a bounded, positive, $1$-periodic, non-constant function of $\frac{\log\log x}{\log 2}$.

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arXiv2022

A higher order Levin-Feinleib theorem

When restricted to some non-negative multiplicative function, say f, bounded on primes and that vanishes on non square-free integers, our result provides us with an asymptotic for $\sum_{n \le X}f(n)/n$ with error term $O((\log X)^{κ-h-1+\varepsilon})$ (for any positive $\varepsilon>0$) as soon as we have $\sum_{p\le Q}f(p)(\log p)/p=κ\log Q+η+O(1/(\log2Q)^h)$ for a non-negative $κ$ and some non-negative integer $h$. The method generalizes the 1967-approach of Levin and Fainleib and uses a differential equation.

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arXiv2021

Intersections of binary quadratic forms in primes and the paucity phenomenon

The number of solutions to $a^2+b^2=c^2+d^2 \le x$ in integers is a well-known result, while if one restricts all the variables to primes Erdos showed that only the diagonal solutions, namely, the ones with $\{a,b\}=\{c,d\}$ contribute to the main term, hence there is a paucity of the off-diagonal solutions. Daniel considered the case of $a,c$ being prime and proved that the main term has both the diagonal and the non-diagonal contributions. Here we investigate the remaining cases, namely when only $c$ is a prime and when both c,d are primes and, finally, when $b,c,d$ are primes by combining techniques of Daniel, Hooley and Plaksin.

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arXiv2019

Computation of the Kummer ratio of the class number for prime cyclotomic fields

Let $ζ_q$ be a primitive $q^{\text{th}}$ root of unity with $q$ an arbitrary odd prime. The ratio of Kummer's first factor of the class number of the cyclotomic number field $\mathbb{Q}(ζ_q)$ and its expected order of magnitude (a simple function of $q$) is called the Kummer ratio and denoted by $r(q)$. It is known that typically $r(q)$ is close to 1, but nevertheless it is believed that it is unbounded, but only large on a very thin sequence of primes $q$. We propose an algorithm to compute $r(q)$ requiring the evaluation of $O(q\log q)$ products and $O(q)$ logarithms. Using it we obtain a new record maximum for $r(q)$, namely $r(6766811) =1.709379\dotsc$ (the old record being $r(5231)=1.556562\dotsc$). The program used and the results described here, are collected at the following address \url{http://www.math.unipd.it/~languasc/rq-comput.html}. This is a (preliminary) report about the computational part of a joint project with Pieter Moree, Sumaia Saad Eddin, and Alisa Sedunova.

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arXiv2019

Jordan totient quotients

The Jordan totient $J_k(n)$ can be defined by $J_k(n)=n^k\prod_{p\mid n}(1-p^{-k})$. In this paper, we study the average behavior of fractions $P/Q$ of two products $P$ and $Q$ of Jordan totients, which we call Jordan totient quotients. To this end, we describe two general and ready-to-use methods that allow one to deal with a larger class of totient functions. The first one is elementary and the second one uses an advanced method due to Balakrishnan and Pétermann. As an application, we determine the average behavior of the Jordan totient quotient, the $k^{th}$ normalized derivative of the $n^{th}$ cyclotomic polynomial $Φ_n(z)$ at $z=1$, the second normalized derivative of the $n^{th}$ cyclotomic polynomial $Φ_n(z)$ at $z=-1$, and the average order of the Schwarzian derivative of $Φ_n(z)$ at $z=1$.

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arXiv2018

Constrained ternary integers

An integer $n$ is said to be ternary if it is composed of three distinct odd primes. In this paper, we asymptotically count the number of ternary integers $n \leq x$ with the constituent primes satisfying various constraints. We apply our results to the study of the simplest class of (inverse) cyclotomic polynomials that can have coefficients that are greater than 1 in absolute value, namely to the $n^{th}$ (inverse) cyclotomic polynomials with ternary $n$. We show, for example, that the corrected Sister Beiter conjecture is true for a fraction $\ge 0.925$ of ternary integers.

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arXiv2017

A logarithmic improvement in the Bombieri-Vinogradov theorem

In this paper we improve the best known to date result of Dress-Iwaniec-Tenenbaum, getting (log x)^2 instead of (log x)^(5/2). We use a weighted form of Vaughan's identity, allowing a smooth truncation inside the procedure, and an estimate due to Barban-Vehov and Graham related to Selberg's sieve. We give effective and non-effective versions of the result.

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arXiv2017

Bounds for the integral points on elliptic curves over function fields

In this paper we give an upper bound for the number of integral points on an elliptic curve E over F_q[T] in terms of its conductor N and q. We proceed by applying the lower bounds for the canonical height that are analogous to those given by Silverman and extend the technique developed by Helfgott-Venkatesh to express the number of integral points on E in terms of its algebraic rank. We also use the sphere packing results to optimize the size of an implied constant. In the end we use partial Birch Swinnerton-Dyer conjecture that is known to be true over function fields to bound the algebraic rank by the analytic one and apply the explicit formula for the analytic rank of E.

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arXiv2016

A variant of Bombieri-Vinogradov theorem with explicit constants

In this paper we improve the result of Akbary and Hambrook by a factor of log x by obtaining a better version of Vaughan's inequality and using the explicit variant of an inequality connected to the Möbius function, derived by Helfgott in his work on ternary Goldbach conjecture.

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