arXiv subjects

Habiba Kadiri

Publications and source records attributed to Habiba Kadiri.

At least 19 recordsLinked to original sources

arXiv2026

Explicit Exponential Sum Estimates and Approximate Functional Equations for the Zeta Function

We show an explicit version of the Van der Corput truncated Poisson summation formula (B-process). By using refined explicit exponential sum estimates, this improves the error term of previous explicit results by Patel and Yang (2024) and Arias de Reyna (2024). As an application, we obtain fine explicit estimates for the error terms in approximate functional equations for the Riemann zeta function, improving on previous explicit results of Simonič (2020) in certain ranges.

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arXiv2026

Bounds for Mertens Sums

In this article we provide new bounds for the Mertens sums and products including $\sum_{p \le x} p^{-1}$ and $\prod_{p \le x} (1-\frac{1}{p})$ which provide superior exponential and log type bounds for these sums in all ranges. These weighted prime number sums and products were extensively studied by Rosser and Schoenfeld (1962) and are employed in a wide range of applications in number theory, cryptography, and combinatorics. Extensive tables are provided in this article which will be useful for these types of applications. The main new ideas in this article are sharp bounds for weighted sums of zeros of zeros of the zeta function. We make use of a novel technique of Fiori-Kadiri-Swidinsky (2023) which relies on a recent explicit zero-density estimate for $N(σ,T)$ of Kadiri-Lumley-Ng (2018). The bounds and techniques in this article for weighted zeros sums will likely be useful in many other arithmetic applications. Our main theorem significantly improves the exponential decay result of Vanlalngaia (2017) and fills a gap in the literature by correcting work of Dusart (2018). The results are also presented in a way that are amenable to future improvements. In addition, we prove an exact ``Riemann-Guinand explicit formula" for the Mertens sum $\sum_{p \le x} p^{-1}$ that appears to be new.

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arXiv2025

An effective version of Chebotarev's density theorem

Chebotarev's density theorem asserts that the prime ideals are equidistributed among the conjugacy classes of the Galois group of any normal extension of number fields. An effective version of this theorem was first established by Lagarias and Odlyzko in 1977. In this article, we present an explicit refinement of their statement that applies to all non-rational fields, with every implicit constant expressed explicitly in terms of the field invariants. Additionally, we provide a sharper bound for extensions of sufficiently small degree. Our approach begins by proving an explicit formula for a smoothed prime ideal counting function. This relies on recent zero-free regions for Dedekind zeta functions, improved estimates on the number of low-lying zeros, and precise bounds for sums over the non-trivial zeros of the Dedekind $ζ$-function.

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arXiv2024

Comparative Prime Number Theory Problem List

This is a list of problems that were collected from participants at the Comparative Prime Number Theory Symposium held at UBC from June 17 to June 21, 2024. Its goal is to stimulate research and future collaborations in this growing field. This event was part of the PIMS (Pacific Institute of Mathematical Sciences) Collaborative Research Group L-functions in Analytic Number Theory: 2022- 2025.

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arXiv2024

Moments of $L$-functions Problem List

This is an ongoing list of problems that has resulted from the PIMS (Pacific Institute of Mathematical Sciences) Collaborative Research Group L-functions in Analytic Number Theory: 2022- 2025. The focus of this list is on Moments of $L$-functions and related topics.

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arXiv2023

Sharper bounds for the Chebyshev function $ψ(x)$

We improve the unconditional explicit bounds for the error term in the prime counting function $ψ(x)$. In particular, we prove that, for all $x>2$, we have \[ \left| ψ(x)-x \right| < 9.22106 \, x \, (\log x)^{3/2} \exp(-0.8476836\sqrt{\log x}), \] and that, for all $\log x \ge 3\,000$, \[ \left| ψ(x)-x \right| < 4.47\cdot 10^{-15} x. \] This compares to results of Platt \& Trudgian (2021) who obtained $4.51\cdot 10^{-13} x $. Our approach represents a significant refinement of ideas of Pintz which had been applied by Platt and Trudgian. Improvements are obtained by splitting the zeros into additional regions, carefully estimating all of the consequent terms, and a significant use of computational methods. Results concerning $π(x)$ will appear in a follow up work.

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arXiv2023

Sharper bounds for the error term in the Prime Number Theorem

We provide very effective methods to convert both asymptotic and explicit numeric bounds on the prime counting function $ψ(x)$ to bounds of the same type on both $θ(x)$ and $π(x)$. This follows up our previous work on $ψ(x)$ in \cite{FKS}, and prove that $ | π(x) - \mathrm{Li}(x) | \leq 9.2211\, x\sqrt{\log(x)} \exp \big( -0.8476 \sqrt{\log(x)} \big) $ for all $x\ge 2$. Additionally, we are able to obtain the best numeric bounds for $x$ on a very large interval (all $x$ up to $\exp(1.8\cdot10^9)$).

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arXiv2022

Primes in the Chebotarev density theorem for all number fields

We establish an explicit bound for the least prime occurring in the Chebotarev density theorem without any restriction. Let $L/K$ be any Galois extension of number fields such that $L\not=\mathbb{Q}$, and let $C$ be a conjugacy class in the Galois group of $L/K$. We show that there exists an unramified prime $\mathfrak{p}$ of $K$ such that $σ_{\mathfrak{p}}=C$ and $N \mathfrak{p} \le d_{L}^{B}$ with $B= 310$. This improves the value $B=12\,577$ as proven by Ahn and Kwon. In comparison to previous works on the subject, we modify the weights to detect the least prime, and we use a new version of Turán's power sum method which gives a stronger Deuring-Heilbronn (zero-repulsion) phenomenon. In addition, we refine the analysis of how the location of the potential exceptional zero for $ζ_L(s)$ affects the final result. We also use Fiori's numerical verification for $L$ up to a certain discriminant height. Finally, we provide a lower bound for the number of unramified primes $\mathfrak{p}$ of $K$ such that $σ_{\mathfrak{p}}=C$.

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arXiv2021

Explicit zero density for the Riemann zeta function

Let $N(σ,T)$ denote the number of nontrivial zeros of the Riemann zeta function with real part greater than $σ$ and imaginary part between $0$ and $T$. We provide explicit upper bounds for $N(σ,T)$ commonly referred to as a zero density result. In 1937, Ingham showed the following asymptotic result $N(σ,T)=\mathcal{O} ( T^{\frac83(1-σ)} (\log T)^5 )$. Ramaré recently proved an explicit version of this estimate. We discuss a generalization of the method used in these two results which yields an explicit bound of a similar shape while also improving the constants.

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arXiv2021

Sharper Bounds for the Chebyshev function $θ(x)$

In this article, we provide explicit bounds for the prime counting function $θ(x)$ in all ranges of $x$. The bounds for the error term for $θ(x)- x$ are of the shape $εx$ and $\frac{c_k x}{(\log x)^k}$, for $k=1,\ldots,5$. Tables of values for $ε$ and $c_k$ are provided.

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arXiv2020

Short effective intervals containing primes in arithmetic progressions and the seven cubes problem

Let $q\ge 3$ be a non-exceptional modulus $q\ge3$, and let $a$ be a positive integer coprime with $q$. For any $ε>0$, there exists $α>0$ (computable), such that for all $x\ge α(\log q)^2$, the interval $\left[ e^x,e^{x+ε}\right]$ contains a prime $p$ in the arithmetic progression $a \bmod q$. This gives the bound for the least prime in this arithmetic progression: $P(a,q) \le e^{α(\log q)^2}$. For instance for all $q\ge 10^{30}$, $P(a,q) \le e^{4.401(\log q)^2}$. Finally, we apply this result to establish that every integer larger than $e^{71\,000}$ is a sum of seven cubes.

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arXiv2019

An explicit zero-free region for the Dirichlet L-functions

Let $L(s,χ)$ be the Dirichlet $L$-function associated to a non-principal primitive character $χ$ modulo $q$ with $3\le q \le 400\,000$. We prove a new explicit zero-free region for $L(s,χ)$: $L(s,χ)$ does not vanish in the region $\Re s \ge 1-\frac1{R \log \left(q\max(1,| \Im s|)\right) }$ with $R=5.60$. This improves a result of McCurley where $9.65$ was shown to be an admissible value for $R$.

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arXiv2019

The least prime ideal in the Chebotarev Density Theorem

In this article, we prove a new bound for the least prime ideal in the Chebotarev density theorem, which improves the main theorem of Zaman [Funct. Approx. Comment. Math. 57 (2017), no.1, 115-142] by a factor of $5/2$. Our main improvement comes from a new version of Turán's power sum method. The key new idea is to use Harnack's inequality for harmonic functions to derive a superior lower bound for the generalised Fejér kernel.

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arXiv2014

Short effective intervals containing primes

We prove that if $x$ is large enough, namely $x\ge x_0$, then there exists a prime between $x(1- Δ^{-1})$ and $x$, where $Δ$ is an effective constant computed in terms of $x_0$. This improves some previous results of Ramaré and Saouter and is used to verify the Ternary Goldbach Conjecture for the integers up to $7.86\cdot 10^{27}$.

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arXiv2014

A zero density result for the Riemann zeta function

In this article, we prove an explicit bound for $N(σ,T)$, the number of zeros of the Riemann zeta function satisfying $σ< \Re s <1 $ and $0 < \Im s < T$. This result provides a significant improvement over Rosser's bound for $N(T)$ when used for estimating prime counting functions. For instance this is applied to obtain new bounds for $ψ(x)$ (arXiv:1310.6374).

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arXiv2013

New bounds for $ψ(x)$

In this article we provide new explicit Chebyshev's bounds for the prime counting function $ψ(x)$. The proof relies on two new arguments: smoothing the prime counting function which allows to generalize the previous approaches, and a new explicit zero density estimate for the zeros of the Riemann zeta function.

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arXiv2012

Explicit zero density theorems for Dedekind zeta functions

This article studies the zeros of Dedekind zeta functions. In particular, we establish a smooth explicit formula for these zeros and we derive an effective version of the Deuring-Heilbronn phenomenon. In addition, we obtain an explicit bound for the number of zeros in a box.

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arXiv2011

Explicit zero-free regions for Dedekind Zeta functions

Let K be a number field, n_K its degree, and d_K the absolute value of its discriminant. We prove that, if d_K is sufficiently large, then the Dedekind zeta function associated to K has no zeros in the region: Re(s) > 1 - 1/(12.55 log d_K + 9.69 n_K log|Im s| + 3.03 n_K + 58.63) and |Im s| > 1. Moreover, it has at most one zero in the region: Re (s) > 1- 1/(12.74 log d_K) and |Im s| < 1. This zero if it exists is simple and is real. This argument also improves a result of Stark by a factor of 2: there is at most one zero in the region Re (s) > 1 - 1/(2 log d_K) and |Im s| < 1/(2 log d_K).

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