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Vorrapan Chandee

Publications and source records attributed to Vorrapan Chandee.

25 records · Page 2Linked to original sources
arXiv2013

Bounding $S(t)$ and $S_1(t)$ on the Riemann hypothesis

Let $πS(t)$ denote the argument of the Riemann zeta-function, $ζ(s)$, at the point $s=\frac{1}{2}+it$. Assuming the Riemann hypothesis, we present two proofs of the bound $$ |S(t)| \leq \left(\tfrac{1}{4} + o(1) \right)\tfrac{\log t}{\log \log t} $$ for large $t$. This improves a result of Goldston and Gonek by a factor of 2. The first method consists in bounding the auxiliary function $S_1(t) = \int_0^{t} S(u) du$ using extremal functions constructed by Carneiro, Littmann and Vaaler. We then relate the size of $S(t)$ to the size of the functions $S_1(t\pm h)-S_1(t)$ when $h\asymp 1/\log\log t$. The alternative approach bounds $S(t)$ directly, relying on the solution of the Beurling-Selberg extremal problem for the odd function $f(x) = \arctan\left(\tfrac{1}{x}\right) - \tfrac{x}{1 + x^2}$. This draws upon recent work by Carneiro and Littmann.

math.NT↗
arXiv2013

Simple zeros of primitive Dirichlet $L$-functions and the asymptotic large sieve

Assuming the Generalized Riemann Hypothesis (GRH), we show using the asymptotic large sieve that 91% of the zeros of primitive Dirichlet $L$-functions are simple. This improves on earlier work of Özlük which gives a proportion of at most 86%. We further compute an $q$-analogue of the Pair Correlation Function $F(α)$ averaged over all primitive Dirichlet $L$-functions in the range $|α| < 2$ . Previously such a result was available only when the average included all the characters $χ$.

math.NT↗
arXiv2010

Bounding $ζ(s)$ in the critical strip

Assuming the Riemann Hypothesis, we make use of the recently discovered \cite{CLV} extremal majorants and minorants of prescribed exponential type for the function $\log\left(\tfrac{4 + x^2}{(α-1/2)^2 + x^2}\right)$ to find upper and lower bounds with explicit constants for $\log|ζ(α+ it)|$ in the critical strip, extending the work of Chandee and Soundararajan \cite{CS}.

math.NT↗
arXiv2009

On the correlation of shifted values of the Riemann zeta function

In 2007, assuming the Riemann Hypothesis (RH), Soundararajan \cite{Moment} proved that $\int_{0}^T |ζ(1/2 + it)|^{2k} dt \ll_{k, ε} T(\log T)^{k^2 + ε}$ for every $k$ positive real number and every $ε> 0.$ In this paper we generalized his methods to find upper bounds for shifted moments. We also obtained their lower bounds and conjectured asymptotic formulas based on Random matrix model, which is analogous to Keating and Snaith's work. These upper and lower bounds suggest that the correlation of $|ζ(\h + it + iα_1)|$ and $|ζ(\h + it + iα_2)|$ transition at $|α_1 - α_2| \approx \frac{1}{\log T}$. In particular these distribution appear independent when $|α_1 - α_2|$ is much larger than $\frac{1}{\log T}.$

math.NT↗
arXiv2009

Bounding |ζ(1/2 + it)| on the Riemann hypothesis

In 1924 Littlewood showed that, assuming the Riemann Hypothesis, for large t there is a constant C such that |ζ(1/2+it)| \ll \exp(C\log t/\log \log t). In this note we show how the problem of bounding |ζ(1/2+it)| may be framed in terms of minorizing the function \log ((4+x^2)/x^2) by functions whose Fourier transforms are supported in a given interval, and drawing upon recent work of Carneiro and Vaaler we find the optimal such minorant. Thus we establish that any C> (\log 2)/2 is permissible in Littlewood's result.

math.NT↗
arXiv2009

Explicit Upper Bounds for L-functions on the critical line

We find an explicit upper bound for general $L$-functions on the critical line, assuming the Generalized Riemann Hypothesis, and give as illustrative examples its application to some families of $L$-functions and Dedekind zeta functions. Further, this upper bound is used to obtain lower bounds beyond which all eligible integers are represented by Ramanujan's ternary form and Kaplansky's ternary forms. This improves on previous work of Ono and Soundararajan on Ramanujan's form and Reinke on Kaplansky's form with a substantially easier proof.

math.NT↗