We obtain asymptotic formulas for the \( 2k\)th moments of partially smoothed divisor sums of the M\"obius function. When \( 2k\) is small compared with \( A\), the level of smoothing, then the main contribution to the moments come from integers with only large prime factors, as one would hope for in sieve weights. However if \( 2k\) is any larger, compared with \( A\), then the main contribution to the moments come from integers with quite a few prime factors, which is not the intention when designing sieve weights. The threshold for ``small'' occurs when \( A=\frac 1{2k} \binom{2k}{k}-1\). One can ask analogous questions for polynomials over finite fields and for permutations, and in these cases the moments behave rather differently, with even less cancellation in the divisor sums. We give, we hope, a plausible explanation for this phenomenon, by studying the analogous sums for Dirichlet characters, and obtaining each type of behaviour depending on whether or not the character is ``exceptional''.

We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length \( y\) around \( x\), where \( y\ll (\log x)^2\). In particular we conjecture that the maximum grows surprisingly slowly as \( y\) ranges from \( \log x\) to \( (\log x)^2\). We will show that our conjectures are somewhat supported by available data, though not so well that there may not be room for some modification.

A celebration of the life of one of the most colorful figures in contemporary mathematics. Richard Guy, popular author, columnist, educator, researcher, and avid mountain climber, passed away in 2020 at the age of 103.

We show that if an exponential sum with multiplicative coefficients is large then the associated multiplicative function is ``pretentious''. This leads to applications in the circle method, and a natural interpretation of the local-global principle.