arXiv subjects

Elchin Hasanalizade

Publications and source records attributed to Elchin Hasanalizade.

7 recordsLinked to original sources

arXiv2026

Explicit Burgess inequalities for cubefree moduli

Burgess proved that for $χ_q$ a primitive Dirichlet character modulo $q$ with $q$ cubefree, $\sum_{M< n\le M+N}χ_q(n)= O\left(N^{1-\frac{1}{r}}q^{\frac{r+1}{4r^2}+ε}\right)$ for all integers $r\ge1.$ More recently, explicit versions with prime moduli $q$ were computed by Booker, McGown, Treviño, and Francis, with applications to finding the least $k$-th power residue, and bounding the size of Dirichlet $L$-functions just to name a few. Jain-Sharma, Khale, and Liu proved an explicit estimate for $r=2.$ We improve their explicit constant for $r = 2$ and compute an explicit Burgess bound for cubefree $q$ for $r\ge 3$.

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arXiv2022

On near-perfect numbers of special forms

In this paper we discuss near-perfect numbers of various forms. In particular, we study the existence of near-perfect numbers in the Fibonacci and Lucas sequences, near-perfect values taken by integer polynomials and repdigit near-perfect numbers.

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arXiv2022

Remark on a paper by Wu

In this short note, we give an affirmative answer to Wu's conjecture on practical numbers, which was posed in [X.-H. Wu, {\it Special forms and the distribution of practical numbers}, Acta Math. Hungar., {\bf 160}(2020), 405-411].

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arXiv2022

On a conjecture of Deaconescu

In 2000 Deaconescu raised a question whether there exists a composite $n$ for which $S_2(n)|ϕ(n)-1$, where $ϕ(n)$ is Euler's function and $S_2(n)$ is Schemmel's totient function. In this paper we prove that any such $n$ is odd, squarefree and has at least seven distinct prime factors. We also prove that any such $n$ with exactly $K$ distinct prime divisors is necessarily less than $2^{2^{K+1}}$.

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arXiv2022

Sums of Fibonacci numbers close to a power of 2

In this paper, we find all sums of two Fibonacci numbers which are close to a power of 2. As a corollary, we also determine all Lucas numbers close to a power of 2. The main tools used in this work are lower bounds for linear forms in logarithms due to Matveev and Dujella-Pethö version of the Baker-Davenport reduction method in diophantine approximation. This paper continues and extends the previous work of Chern and Cui.

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arXiv2021

Counting zeros of the Riemann zeta function

In this article, we show that $$ \left| N (T) - \frac{T}{ 2 π} \log \left( \frac{T}{2πe}\right) \right| \le 0.1038 \log T + 0.2573 \log\log T + 9.3675 $$ where $N(T)$ denotes the number of non-trivial zeros $ρ$, with $0<\Im(ρ) \le T$, of the Riemann zeta function. This improves the previous result of Trudgian for sufficiently large $T$. The improvement comes from the use of various subconvexity bounds and ideas from the work of Bennett $et$ $al.$ on counting zeros of Dirichlet $L$-functions.

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arXiv2021

Counting zeros of Dedekind zeta functions

Given a number field $K$ of degree $n_K$ and with absolute discriminant $d_K$, we obtain an explicit bound for the number $N_K(T)$ of non-trivial zeros (counted with multiplicity), with height at most $T$, of the Dedekind zeta function $ζ_K(s)$ of $K$. More precisely, we show that for $T \geq 1$, $$ \Big| N_K (T) - \frac{T}π \log \Big( d_K \Big( \frac{T}{2πe}\Big)^{n_K}\Big)\Big| \le 0.228 (\log d_K + n_K \log T) + 23.108 n_K + 4.520, $$ which improves previous results of Kadiri and Ng, and Trudgian. The improvement is based on ideas from the recent work of Bennett $et$ $al.$ on counting zeros of Dirichlet $L$-functions.

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