Related papers: Explicit bounds for the Riemann zeta-function on t…
We optimized the implicit constant for the refined upper bounds for moments of the Riemann zeta-function proved by Harper. We also computed the implicit constant for the upper bounds for moments of the Riemann zeta-function proved by…
In this paper, we provide explicit upper and lower bounds for the argument of the Riemann zeta-function and its antiderivatives in the critical strip under the assumption of the Riemann hypothesis. This extends the previously known bounds…
We study lower bounds for the Riemann zeta function $\zeta(s)$ along vertical arithmetic progressions in the right-half of the critical strip. We show that the lower bounds obtained in the discrete case coincide, up to the constants in the…
We show that the higher derivatives of the Riemann zeta function may be expressed in terms of integrals involving the digamma function. Related integrals for the Stieltjes constants are also shown. We also present a formula for the…
In this paper, we derive new lower bounds for the normalized distances between consecutive maxima of the Riemann zeta-function on the critical line subject to the truth of the Riemann hypothesis. The method of our proofs relies on a Sobolev…
We investigate the distribution of the Riemann zeta-function on the line $\Re(s)=\sigma$. For $\tfrac 12 < \sigma \le 1$ we obtain an upper bound on the discrepancy between the distribution of $\zeta(s)$ and that of its random model,…
The derivative of the Riemann zeta function was computed numerically on several large sets of zeros at large heights. Comparisons to known and conjectured asymptotics are presented.
Assuming the Generalized Riemann Hypothesis, we provide uniform upper and lower bounds with explicit main terms for $\log{\left|\cL(s)\right|}$ for $\sigma \in (1/2,1)$ and for functions in the Selberg class. In particular, we focus on the…
We present drawings on the complex plane of the lines Im(zeta(s))=0 and Re(zeta(s))=0. This allow to illustrate many properties of the zeta function of Riemann. This is an expository paper. It does not pretend to prove any new result about…
We present highlights of computations of the Riemann zeta function around large values and high zeros. The main new ingredient in these computations is an implementation of the second author's fast algorithm for numerically evaluating…
In this paper, we obtain a series of new conditional lower bounds for the modulus and the argument of the Riemann zeta function on very short segments of the critical line, based on the Riemann hypothesis. In particular, the conditional…
We show the estimates \inf_T \int_T^{T+\delta} |\zeta(1+it)|^{-1} dt =e^{-\gamma}/4 \delta^2+ O(\delta^4) and \inf_T \int_T^{T+\delta} |\zeta(1+it)| dt =e^{-\gamma} \pi^2/24 \delta^2+ O(\delta^4) as well as corresponding results for…
Several identities for the Riemann zeta-function $\zeta(s)$ are proved. For example, if $s = \sigma + it$ and $\sigma > 0$, then $$ \int_{-\infty}^\infty |{(1-2^{1-s})\zeta(s)\over s}|^2dt = {\pi\over\sigma}(1 -…
We study the conditional upper bounds and extreme values of derivatives of the Riemann zeta function and Dirichlet $L$-functions near the 1-line. Let $\ell$ be a fixed natural number. We show that, if $|\sigma-1|\ll1/\log_2t$, then…
The Riemann zeta function on the critical line can be computed using a straightforward application of the Riemann-Siegel formula, Sch\"onhage's method, or Heath-Brown's method. The complexities of these methods have exponents 1/2, 3/8…
Assume the Riemann hypothesis. On the right-hand side of the critical strip, we obtain an asymptotic formula for the discrete mean square of the Riemann zeta-function over imaginary parts of its zeros.
Assuming the Generalised Riemann Hypothesis, we prove a sharp upper bound on moments of shifted Dirichlet $L$-functions. We use this to obtain conditional upper bounds on high moments of theta functions. Both of these results strengthen…
Riemann numerically approximated at least three zeta zeros. According to Edwards, Riemann even took steps to verify that the lowest zero he computed was indeed the first zeta zero. This approach to verification is developed, improved, and…
In previous work, the first author obtained conjecturally sharp upper bounds for the joint moments of the $(2k-2h)^{\text{th}}$ power of the Riemann zeta function with the $2h^{\text{th}}$ power of its derivative on the critical line in the…
New expansions for some functions related to the Zeta function in terms of the Pochhammer's polynomials are given (coefficients b(k), d(k), d_(k) and d__(k). In some formal limit our expansion b(k) obtained via the alternating series gives…