0 |ζ( 1 2 + it)| dt. For positive real numbers k, it is believed that Mk(T ) ∼ CkT (logT ) 2 for a positive constant Ck. A precise value for Ck was conjectured by Keating and Snaith [9] based on considerations from random matrix theory. Subsequently, an alternative approach, based on multiple Dirichlet series and producing the same conjecture, was given by Diaconu, Goldfeld and Hoffstein [4]. R...

c © W W L Chen, 1990, 2013. This chapter originates from material used by the author at Imperial College London between 1981 and 1990. It is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage an...

The Riemann Hypothesis, which specifies the location of zeros of the Riemann zeta function, and thus describes the behavior of primes, is one of the most famous unsolved problems in mathematics, and extensive efforts have been made over more than a century to check it numerically for large sets of cases. Recently a new algorithm, invented by the speaker and A. Scho . . nhage, has been implement...

We expose techniques that permit to approximate the Riemann Zeta function in different context. The motivation does not restrict only to the numerical verification of the Riemann hypothesis (see Numerical computations about the zeros of the zeta function) : in [5] for example, Lagarias and Odlyzko obtained the best asymptotic known method to compute π(x), the number of primes less than x, thank...

The “hybrid” moments Z 2T T |ζ( 1 2 + it)| „ Z t+G t−G |ζ( 1 2 + ix)| dx m dt of the Riemann zeta-function ζ(s) on the critical line Re s = 1 2 are studied. The expected upper bound for the above expression is Oε(T G). This is shown to be true for certain specific values of k, l,m ∈ N, and the explicitly determined range of G = G(T ; k, l,m). The application to a mean square bound for the Melli...

Assuming the Riemann Hypothesis, Montgomery and Taylor showed that at least 67.25% of the zeros of the Riemann zeta-function are simple. Using Montgomery and Taylor's argument together with an elementary combinatorial argument, we prove that assuming the Riemann Hypothesis at least 67.275% of the zeros are simple.

Let P be the set of all primes and ψ(n) = n ∏ n∈P,p|n(1 + 1/p) be the Dedekind psi function. We prove that, under the assumption that the n-th prime pn is always larger than the first Chebyshev function θ(pn), the Riemann hypothesis is satisfied if and only if f(n) = ψ(n)/n − e log logn < 0 for all integers n > n0 = 30 (D), where γ ≈ 0.577 is Euler’s constant. This inequality is equivalent to R...

According to my first teacher Gustave Choquet one does, by openly facing a well known unsolved problem, run the risk of being remembered more by one’s failure than anything else. After reaching a certain age, I realized that waiting “safely” until one reaches the end-point of one’s life is an equally selfdefeating alternative. In this paper I shall first look back at my early work on the classi...