arXiv subjects

Micah B. Milinovich

Publications and source records attributed to Micah B. Milinovich.

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.

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arXiv2013

Simple zeros of modular L-functions

Assuming the generalized Riemann hypothesis, we prove quantitative estimates for the number of simple zeros on the critical line for the L-functions attached to classical holomorphic newforms.

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arXiv2013

A hybrid Euler-Hadamard product and moments of ζ'(ρ)

Keating and Snaith modeled the Riemann zeta-function ζ(s) by characteristic polynomials of random NxN unitary matrices, and used this to conjecture the asymptotic main term for the 2k-th moment of ζ(1/2+it) when k>-1/2. However, an arithmetical factor, widely believed to be part of the leading term coefficient, had to be inserted in an ad hoc manner. Gonek, Hughes and Keating later developed a hybrid formula for ζ(s) that combines a truncation of its Euler product with a product over its zeros. Using it, they recovered the moment conjecture of Keating and Snaith in a way that naturally includes the arithmetical factor. Here we use the hybrid formula to recover a conjecture of Hughes, Keating and O'Connell concerning discrete moments of the derivative of the Riemann zeta-function averaged over the zeros of ζ(s), incorporating the arithmetical factor in a natural way.

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arXiv2011

Moments of the Riemann zeta-function at its relative extrema on the critical line

Assuming the Riemann hypothesis, we obtain upper and lower bounds for moments of the Riemann zeta-function averaged over the extreme values between its zeros on the critical line. Our bounds are very nearly the same order of magnitude. The proof requires upper and lower bounds for continuous moments of derivatives of Riemann zeta-function on the critical line.

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arXiv2011

A note on a conjecture of Gonek

We derive a lower bound for a second moment of the reciprocal of the derivative of the Riemann zeta-function averaged over the zeros of the zeta-function that is half the size of the conjectured value. Our result is conditional upon the assumption of the Riemann Hypothesis and the conjecture that the zeros of the zeta-function are simple.

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arXiv2008

Upper bounds for the moments of zeta prime rho

Assuming the Riemann Hypothesis, we obtain an upper bound for the 2k-th moment of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of $ζ(s)$ for every positive integer k. Our bounds are nearly as sharp as the conjectured asymptotic formulae for these moments. The proof is based upon a recent method of K. Soundararajan that provides analogous bounds for continuous moments of the Riemann zeta-function as well as for moments L-functions at the central point, averaged over families.

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arXiv2007

Lower bounds for moments of zeta prime rho

Assuming the Riemann Hypothesis, we establish lower bounds for moments of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of $ζ(s)$. Our proof is based upon a recent method of Rudnick and Soundararajan that provides analogous bounds for moments of $L$-functions at the central point, averaged over families.

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