Claire Frechette (Boston College)

When studying a multiplicative function (such as the coefficients of a modular form), it is often fruitful to simultaneously look at the associated Dirichlet series or the partial sums of the function. For Dirichlet characters, this story has been extensively studied, with recent improvements by Granville and Soundararajan that relate the asymptotics of the partial sum to the location of zeros of the L-function. Inspired by their techniques, we investigate the asymptotics of partial sums for cusp forms, showing that zeros of the associated L-function determine when the sum of coefficients up to x grows like xlog x. In a similar vein, it is natural to ask when partial sums of individual functions affect the partial sum of the product: in this case, we can extend from just modular forms to any multiplicative function bounded by the divisor function and show that lower bounds on partial sums of divisor-bounded functions result in lower bounds on the partial sums associated to their products.