The Riemann zeta function

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The Riemann Hypothesis asserts that all of the non-trivial zeros of the Riemann zeta function have real part equal to 1/2. Much progress has been made recently on the moments and extreme values of the Riemann zeta function on the line Res=1/2 (the critical line). Hardy and Littlewood established bounds on the fraction of the non-trivial zeroes satisfying the Riemann Hypothesis and found the asymptotics for the first moment of the zeta function on the critical line. The second moment was later found by Ingham. The Weyl-Hardy-Littlewood method gives bounds for sums of exponentials, which can be used to bound the extremal values of the zeta function in the critical strip using the Abel summation formula. The Keating-Snaith conjecture (later extended by Conrey, Farmer, Rubinstein and Snaith) gives precise asymptotics for all moments of the zeta function. This conjecture is motivated by random matrix theory, and specifically by the idea that on the critical line the zeta function behaves like the characteristic polynomial of certain random matrices.A key bound on the growth of the zeta function is the Lindelöf Hypothesis, which states that on the critical line, the zeta function grows slower than any positive power of the imaginary part; I hope to investigate the interplay between this and the Keating-Snaith conjecture, which states that the moments are a power of log. Since for large values of k, the 2k-th moment is dominated by its large values, we would expect that the Keating-Snaith conjecture implies something far stronger than the Lindelöf hypothesis.In between the maximum values and the moments of the zeta functions, there is the problem of determining the measure of an interval where the zeta function takes values above a certain threshhold. The work of Arguin-Bourgade-Radziwill gives decay rates for the measure of short intervals on the critical line where the zeta function is large, uniform for intervals in [T,2T], which are stronger than previous bounds and are consistent with the multifractality conjectured by Fyodorov and Keating. Many theorems about the Riemann zeta function can be extended to other L-functions; Dirichlet L-functions in particular have many similar properties. Having learnt the tools used in the work of Arguin, Bourgade and Radziwill in their bounds on the growth of the zeta function, I hope to apply them to other L-functions. This may be achievable, since if the terms relating to a period of a Dirichlet L-function with non-trivial character are grouped together, the results sum is absolutely convergent for positive real part, whereas for the Riemann zeta function, absolute convergence can only be provided for real part greater than 1. The research will contribute to the EPRSC areas of mathematical analysis, mathematical physics (Since the Keating-Snaith conjecture is motivated by considerations from Physics) and number theory.