proved for k = 1 or 2, but the case k = 3 seems much harder and is still an open problem. There is a much more precise conjecture about the asymptotic behavior of the integral: it is believed that for some constants ck,j&hairsp;. This has been proved by Littlewood for k = 1 and by for k = 2 (extending a result of who found the leading term). suggested the value for the leading coefficient when k is 6, and used random matrix theory to suggest some conjectures for the values of the coefficients for higher k. The leading coefficients are conjectured to be the product of an elementary factor, a certain product over primes, and the number of n × n Young tableaux given by the sequence 1, 1, 2, 42, 24024, 701149020, ... . Other consequences Denoting by pn the n-th prime number, a result by Albert Ingham shows that the Lindelöf hypothesis implies that, for any ε > 0, if n is sufficiently large. However, this result is much worse than that of the large prime gap conjecture. L-functions The Riemann zeta function belongs to a more general family of functions called L-functions. In 2010, new methods to obtain sub-convexity estimates for L-functions in the PGL(2) case were given by Joseph Bernstein and Andre Reznikov and in the GL(1) and GL(2) case by Akshay Venkatesh and Philippe Michel and in 2021 for the GL(n) case by Paul Nelson. Notes and references 2001 pbk reprint Conjectures Zeta and L-functions Unsolved problems in number theory Analytic number theory