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 for these quantities on the critical line (and sharpens the error terms in such estimates). Our tools come not only from number theory, but also from Fourier analysis and approximation theory. An important element in our strategy is the ability to solve a Fourier optimization problem with constraints, namely, the problem of majorizing certain real-valued even functions by bandlimited functions, optimizing the L 1 pRq´error. Deriving explicit formulae for the Fourier transforms of such optimal approximations plays a crucial role in our approach.2010 Mathematics Subject Classification. 11M06, 11M26, 41A30.

Let $S(t) = {1}/{\pi} \arg \zeta \big(\hh + it \big)$ be the argument of the Riemann zeta-function at the point 1/2 + it. For n ⩾ 1 and t > 0 define its iterates $$\begin{equation*} S_n(t) = \int_0^t S_{n-1}(\tau) \,\d\tau\, + \delta_n\,, \end{equation*}$$ where δn is a specific constant depending on n and S0(t) ≔ S(t). In 1924, J. E. Littlewood proved, under the Riemann hypothesis (RH), that Sn(t) = O(log t/(log log t)n + 1). The order of magnitude of this estimate was never improved up to this date. The best bounds for S(t) and S1(t) are currently due to Carneiro, Chandee and Milinovich. In this paper we establish, under RH, an explicit form of this estimate $$\begin{equation*} -\left( C^-_n + o(1)\right) \frac{\log t}{(\log \log t)^{n+1}} \ \leq \ S_n(t) \ \leq \ \left( C^+_n + o(1)\right) \frac{\log t}{(\log \log t)^{n+1}}\,, \end{equation*}$$ for all n ⩾ 2, with the constants C±n decaying exponentially fast as n → ∞. This improves (for all n ⩾ 2) a result of Wakasa, who had previously obtained such bounds with constants tending to a stationary value when n → ∞. Our method uses special extremal functions of exponential type derived from the Gaussian subordination framework of Carneiro, Littmann and Vaaler for the cases when n is odd, and an optimized interpolation argument for the cases when n is even. In the final section we extend these results to a general class of L-functions.

In this paper we study the distribution of the non-trivial zeros of the zeta-function ζ(s) (and other L-functions) under Montgomery's pair correlation approach. We use semidefinite programming to improve the asymptotic bounds for N * (T ), N d (T ) and N (λ, T ).

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Let S(t) denote the argument of the Riemann zeta-function, defined asAssuming the Riemann hypothesis, we prove that S(t) = Ω± log t log log log t log log t .