arXiv subjects

Stelios Sachpazis

Publications and source records attributed to Stelios Sachpazis.

5 recordsLinked to original sources

arXiv2025

Primes in arithmetic progressions under the presence of Landau-Siegel zeroes

Let $x\geqslant 2$ and assume that $a$ and $q$ are coprime positive integers. As usual, $ψ(x;q,a):=\sum_{n\leqslant x,n\equiv a(\!\!\!\mod{\!\!q})}Λ(n)$, where $Λ$ is the von Mangoldt function. In 2003, Friedlander and Iwaniec assumed the existence of exceptional characters corresponding to "extreme" Landau-Siegel zeroes and established a meaningful asymptotic formula for $ψ(x;q,a)$ beyond the square-root barrier of the Generalized Riemann Hypothesis. In particular, their asymptotic yields non-trivial information for moduli $q\leqslant x^{1/2+1/231}$. In this paper, we considerably relax the extremity of the Landau-Siegel zero required in the work of Friedlander and Iwaniec and obtain a conditional asymptotic formula for $ψ(x;q,a)$ in a slightly wider range of $q$.

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arXiv2025

The Chowla conjecture and Landau-Siegel zeroes

Let $k\geq 2$ be an integer and let $λ$ be the Liouville function. Given $k$ non-negative distinct integers $h_1,\ldots,h_k$, the Chowla conjecture claims that $\sum_{n\leq x}λ(n+h_1)\cdots λ(n+h_k)=o(x)$ as $x\to\infty$. An unconditional answer to this conjecture is yet to be found, and in this paper, we take a conditional approach towards it. More precisely, we establish a non-trivial bound for the sums $\sum_{n\leq x}λ(n+h_1)\cdots λ(n+h_k)$ under the existence of a Landau-Siegel zero for $x$ in an interval that depends on the modulus of the character whose Dirichlet series corresponds to the Landau-Siegel zero. Our work constitutes an improvement over the previous related results of Germán and Kátai, Chinis, and Tao and Teräväinen.

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arXiv2024

A pretentious proof of Linnik's estimate for primes in arithmetic progressions

In the present paper, we adopt a pretentious approach and prove a strongly uniform estimate for the sums of the von Mangoldt function $Λ$ on arithmetic progressions. This estimate is analogous to an estimate that Linnik established in his attempt to prove his celebrated theorem concerning the size of the smallest prime number of an arithmetic progression. Our work builds on ideas coming from the pretentious large sieve of Granville, Harper and Soundararajan and it also borrows insights from the treatment of Koukoulopoulos on multiplicative functions with small averages.

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arXiv2023

On multiplicative functions with small partial sums

In analytic number theory, several results make use of information regarding the prime values of a multiplicative function in order to extract information about its averages. Examples of such results include Wirsing's theorem and the Landau-Selberg-Delange method. In this paper, we are interested in the opposite direction. In particular, we prove that when $f$ is a suitable divisor-bounded multiplicative function with small partial sums, then $f(p)\approx-p^{iγ_1}-\ldots-p^{iγ_m}$ on average, where the $γ_j$'s are the imaginary parts of the zeros of the Dirichet series of $f$ on the line $\Re(s)=1$. This extends a result of Koukoulopoulos and Soundararajan and it builds upon ideas coming from previous work of Koukoulopoulos for the case where $|f|\leqslant 1$.

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arXiv2021

Multiplicative Functions on Shifted Primes

Let $f$ be a positive multiplicative function and let $k\geq 2$ be an integer. We prove that if the prime values $f(p)$ converge to $1$ sufficiently slowly as $p\rightarrow +\infty$, in the sense that $\sum_{p}|f(p)-1|=\infty$, there exists a real number $c>0$ such that the $k$-tuples $(f(p+1),\ldots,f(p+k))$ are dense in the hypercube $[0,c]^k$ or in $[c,+\infty)^k$. In particular, the values $f(p+1),\ldots,f(p+k)$ can be put in any increasing order infinitely often. Our work generalises previous results of De Koninck and Luca.

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