A multiplicative miscellany
by
Hugh Montgomery
University of Michigan

1. Beurling primes. In joint work with Harold Diamond and Ulrike Vorhauer, we show that there is a system of Beurling primes such that the integer-counting function satisfies N(x) = \kappax + O(x^\theta) with \kappa > 0 and \theta < 1, but for which the associated zeta function has zeros within c/logt of the 1-line, and for which the error term in the prime number theorem is \Omega _+/- (xexp(-c\surd{ logx})). Thus Landau's proof of the prime number theorem by means of local lemmas is best possible.

2. The sum of the M function. In joint work with Helmut Maier, we show that if one assumes RH, then M(x) << x^1/2exp(c(logx)^39/61). The idea is to use Selberg's analysis of the distribution of (s) for s near the 1/2-line. 3. Primes in short intervals. The long-awaited paper with Soundararajan is at last ready for public inspection. By determining the asymptotics of a suitable singular series, we are able to generate heuristics that suggest that \int₀^X(\psi(x+h)-\psi(x)-h)^k dx = (\mu_k+o(1))X(hlogX/h)^k/2 when X^ h X^1-. Moreover, if the prime k-tuple conjecture holds with an error term O(x^1/2+), then the above is true uniformly for X^ h X^1/k-. 4. The mysterious constant B(). Let be a primitive Dirichlet character modulo q. In the Hadamard product for (s, ), one has a factor (A()+B()s). The B() is usually eliminated from consideration by differencing. However, we find that B(\chi) = \frac-12log\frac q\pi- \fracL'L(1, [`(\chi)])+\frac12C₀ + (1-\kappa)log2where if (-1) = 1, and if (-1) = -1

Date received: May 26, 2004