Moments of L-functions, correlation sums, and primes

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An important field of research in analytic number theory is the study of moments of L-functions. At the beginning of the twentieth century, researchers including Bohr, Landau, Hardy, and Littlewood realized that bounds for L-functions had important number theoretic applications. Hardy and Littlewood studied the 2k-th moments of the Riemann zeta function on the critical line, denoted I_k(T) where k is positive. In 1918 Hardy-Littlewood asymptotically evaluated the second moment and in 1926 Ingham asymptotically evaluated the fourth moment. I_k(T) has not been evaluated for any other value of k. In 1998 Keating and Snaith, using a random matrix model, gave a precise conjecture for the asymptotic size of I_k(T). In 1998 Conrey and Gonek linked I_k(T) to correlation sums of divisor functions in the cases k=3 and k=4. In 2006 Conrey et al. conjectured the full main term asymptotic formula for I_k(T). In 2016, I gave a proof in that an asymptotic formula for certain correlation sums of ternary divisor functions implies the full main term asymptotic for the sixth moment of the zeta function. The main theme of this proposal is to study the connection between mean values of L-functions and correlation sums of arithmetic functions. In particular, I shall focus on correlations sums of higher divisor functions. I aim to establish an asymptotic formula for the eighth moment of the Riemann zeta function based on my recent NSERC funded work on the sixth moment of the zeta function and also establish asymptotics for related moments of the zeta function. The moments of I_k(T) can be modelled by certain mean values of long Dirichlet polynomials with higher divisor coefficients. In collaboration with Alia Hamieh we aim to prove a more precise asymptotic for these mean values thus verifying a 1998 conjecture of Conrey-Gonek and special cases of a recent conjecture of Conrey-Keating. Using techniques from my work on the sixth moment of the zeta function, I will also study other families of mean values of L-functions. This includes the discrete mean values of the derivative of the zeta function and mean values of quadratic Dirichlet L-functions. A final element of the proposal is to explore the size of prime counting functions and related error terms. Montgomery has made a deep conjecture regarding the size of the error term in the prime number theorem. I aim to prove this assuming the Linear Independence conjecture. Also I shall study the size of the sum of the Mobius function, M(x), attempting to exhibit large values. A number of these objectives will be suitable for undergraduate and graduate students.