arXiv subjects

Shehzad Hathi

Publications and source records attributed to Shehzad Hathi.

4 recordsLinked to original sources

arXiv2023

On the sum of a prime and a square-free number with divisibility conditions

Every integer greater than two can be expressed as the sum of a prime and a square-free number. Expanding on recent work, we provide explicit and asymptotic results when divisibility conditions are imposed on the square-free number. For example, we show for odd $k\leq 10^5$ and even $k\leq 2\cdot 10^5$ that any even integer $n\geq 40$ can be expressed as the sum of a prime and a squarefree number coprime to $k$. We also discuss applications to other Goldbach-like problems.

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arXiv2022

Representation of even integers as a sum of squares of primes and powers of two

In 1951, Linnik proved the existence of a constant $K$ such that every sufficiently large even number is the sum of two primes and at most $K$ powers of 2. Since then, this style of approximation has been considered for problems similar to the Goldbach conjecture. One such problem is the representation of a sufficiently large even number as a sum of four squares of primes and at most $k$ powers of two. In 2014, Zhao proved this to be true with $k = 46$. In this paper, we reduce this to $k = 31$.

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arXiv2022

Wolstenholme and Vandiver primes

A prime $p$ is a Wolstenholme prime if $\binom{2p}{p}\equiv2$ mod $p^4$, or, equivalently, if $p$ divides the numerator of the Bernoulli number $B_{p-3}$; a Vandiver prime $p$ is one that divides the Euler number $E_{p-3}$. Only two Wolstenholme primes and eight Vandiver primes are known. We increase the search range in the first case by a factor of $10$, and show that no additional Wolstenholme primes exist up to $10^{11}$, and in the second case by a factor of $20$, proving that no additional Vandiver primes occur up to this same bound. To facilitate this, we develop a number of new congruences for Bernoulli and Euler numbers mod $p$ that are favorable for computation, and we implement some highly parallel searches using GPUs.

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arXiv2021

Mertens' Third Theorem for Number Fields: A New Proof, Cram\'er's Inequality, Oscillations, and Bias

The first result of our article is another proof of Mertens' third theorem in the number field setting, which generalises a method of Hardy. The second result concerns the sign of the error term in Mertens' third theorem. Diamond and Pintz showed that the error term in the classical case changes sign infinitely often and in our article, we establish this result for number fields assuming a reasonable technical condition. In order to do so, we needed to prove Cram\'er's inequality for number fields, which is interesting in its own right. Lamzouri built upon Diamond and Pintz's work to prove the existence of the logarithmic density of the set of real numbers $x \ge 2$ such that the error term in Mertens' third theorem is positive, so the third result of our article generalises Lamzouri's results for number fields. We also include numerical investigations for the number fields $\mathbb{Q}(\sqrt{5})$ and $\mathbb{Q}(\sqrt{13})$, building upon similar work done by Rubinstein and Sarnak in the classical case.

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