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Neea Palojärvi

Publications and source records attributed to Neea Palojärvi.

11 recordsLinked to original sources
arXiv2026

The error term of the fourth moment of the Riemann zeta-function

We examine the size of $E_{2}(T)$, the error term in the asymptotic formula for $\int_{0}^{T} |ζ(1/2 + it)|^{4}\, dt$ where $ζ(s)$ is the Riemann zeta-function. We reduce the powers of $\log T$ in the known bounds for $E_{2}(T)$ and $\int_{0}^{T} E_{2}(t)^{2}\, dt$. As a consequence, we obtain small logarithmic improvements for $k$th moments where $8\leq k\leq 12$. In particular, we obtain $\int_0^T|ζ(1/2+it)|^{12}\,dt\ll T^2(\log T)^{15}$.

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arXiv2026

Explicit and Effective Estimates for the error term in the Generalised Divisor Problem

In this article, we obtain effective estimates for the error term $Δ_{k}(x)$ for all integers $k \geq2$, and completely explicit estimates for integers $k \in [3,9]$. The explicit results improve the powers of $x$ appearing in the known explicit bounds for $Δ_{k}(x)$, and the effective bounds provide a method to derive such bounds for all integers $k\geq 3$.

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arXiv2025

Conditional estimates for $L$-functions in the Selberg class II

Assuming the Generalized Riemann Hypothesis, we provide uniform upper and lower bounds with explicit main terms for $\log{\left|\cL(s)\right|}$ for $σ\in (1/2,1)$ and for functions in the Selberg class. In particular, we focus on the region $0\leqσ-1/2\ll 1/\log{\log{\left(\sq|t|^{\sdeg}\right)}}$. We also provide estimates under additional assumptions on the distribution of Dirichlet coefficients of $\cL(s)$ on prime numbers. Moreover, by assuming a polynomial Euler product representation for $\cL(s)$, we establish both uniform bounds and completely explicit estimates by also assuming the strong $λ$-conjecture. In addition to providing estimates for a large set of functions, our results improve the best known estimates for specific functions in the Selberg class including the lower bounds for the Riemann zeta function close to the critical line.

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arXiv2025

Explicit estimates for the Goldbach summatory function

In order to study the analytic properties of the Goldbach generating function we consider a smooth version, similar to the Chebyshev function for the Prime Number Theorem. In this paper, we obtain explicit numerical estimates for the average order of its summatory function both in the classical case and in arithmetic progressions. In addition, we derive new explicit estimates for sums over zeros and for the function $ψ(u,χ)$. Our results are general and describe how the explicit bounds depend on other known explicit estimates. These support the known asymptotic results under the (Generalised) Riemann Hypothesis involving error terms.

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arXiv2024

Conditional estimates for $L$-functions in the Selberg class

Assuming the Generalized Riemann Hypothesis, we provide uniform upper bounds with explicit main terms for moduli of $\left(\cL'/\cL\right)(s)$ and $\log{\cL(s)}$ for $1/2+δ\leqσ<1$, fixed $δ\in(0,1/2)$ and for functions in the Selberg class except for the identity function. We also provide estimates under additional assumptions on the distribution of Dirichlet coefficients of $\cL(s)$ on prime numbers. Moreover, by assuming a polynomial Euler product representation for $\cL(s)$, we establish uniform bounds for $|3/4-σ|\leq 1/4-1/\log{\log{\left(\sq|t|^{\sdeg}\right)}}$, $|1-σ|\leq 1/\log{\log{\left(\sq|t|^{\sdeg}\right)}}$ and $σ=1$, and completely explicit estimates by assuming also the strong $λ$-conjecture.

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arXiv2023

Explicit results for Euler's factorial series in arithmetic progressions under GRH

In this article, we study the Euler's factorial series $F_p(t)=\sum_{n=0}^\infty n!t^n$ in $p$-adic domain under the Generalized Riemann Hypothesis. First, we show that if we consider primes in $kφ(m)/(k+1)$ residue classes in the reduced residue system modulo $m$, then under certain explicit extra conditions we must have $λ_0+λ_1F_p(α_1)+\ldots+λ_kF_p(α_k) \neq 0$ for at least one such prime. We also prove an explicit $p$-adic lower bound for the previous linear form. Secondly, we consider the case where we take primes in arithmetic progressions from more than $kφ(m)/(k+1)$ residue classes. Then there is an infinite collection of intervals each containing at least one prime which is in those arithmetic progressions and for which we have $λ_0+λ_1F_p(α_1)+\ldots+λ_kF_p(α_k) \neq 0$. We also derive an explicit $p$-adic lower bound for the previous linear form.

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arXiv2022

A $p$-adic lower bound for a linear form in logarithms

Linear forms in logarithms have an important role in the theory of Diophantine equations. In this article, we prove explicit $p$-adic lower bounds for linear forms in $p$-adic logarithms of rational numbers using Padé approximations of the second kind.

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arXiv2020

Explicit zero-free regions and a $τ$-Li-type criterion

$τ$-Li coefficients describe if a function satisfies the Generalized Riemann Hypothesis or not. In this paper we prove that certain values of the $τ$-Li coefficients lead to existence or non-existence of certain zeros. The first main result gives explicit numbers $N_1$ and $N_2$ such that if all real parts of the $τ$-Li coefficients are non-negative for all indices between $N_1$ and $N_2$, then the function has non zeros outside a certain region. According to the second result, if some of the real parts of the $τ$-Li coefficients are negative for some index $n$ between numbers $n_1$ and $n_2$, then there is at least one zero outside a certain region.

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