arXiv subjects
Publications and source records attributed to T. S. Trudgian.
We consider Dirichlet $L$-functions $L(s, χ)$ where $χ$ is a non-principal quadratic character to the modulus $q$. We make explicit a result due to Pintz and Stephens by showing that $|L(1, χ)|\leq \frac{1}{2}\log q$ for all $q\geq 2\cdot 10^{23}$ and $|L(1, χ)|\leq \frac{9}{20}\log q$ for all $q\geq 5\cdot 10^{50}$.
This paper contains new explicit upper bounds for the number of zeroes of Dirichlet L-functions and Dedekind zeta-functions in rectangles.
Taking the easy option I prove that one can lower Zhang's bound on prime gaps from 70 million to 60 million. A poor man's improvement indeed!
This paper improves the bound on $|S(T)|$. The main result is to show that $|S(T)|\leq 0.111\log T + 0.275\log\log T + 2.450$, which is valid for all $T\geq e$.
This paper is concerned with the constancy in the sign of $L(X, α) = \sum_{1}^{X} \frac{λ(n)}{n^α}$, where $λ(n)$ the Liouville function. The non-positivity of $L(X, 0)$ is the Pólya conjecture, and the non-negativity of $L(X, 1)$ is the Turán conjecture --- both of which are false. By constructing an auxiliary function, evidence is provided that $L(X, \frac{1}{2})$ is the best contender for constancy in sign. The core of this paper is the conjecture that $L(X, \frac{1}{2}) \leq 0$ for all $X\geq 17$: this has been verified for $X\leq 300,001$.