Abstracts of upcoming talks at the MPIM. For an overview see also the calendar.

## A proof of the prime number theorem IV

The prime number theorem asserts that the $n$-th largest prime has approximate size $n \log n$.

We shall give the proof of Iwaniec in his recent AMS book on the Riemann zeta function.

These lectures are at the level of beginning graduate students.

## A proof of the prime number theorem III

The prime number theorem asserts that the $n$-th largest prime has approximate size $n \log n$.

We shall give the proof of Iwaniec in his recent AMS book on the Riemann zeta function.

These lectures are at the level of beginning graduate students.

## A proof of the prime number theorem II

The prime number theorem asserts that the $n$-th largest prime has approximate size $n \log n$.

We shall give the proof of Iwaniec in his recent AMS book on the Riemann zeta function.

These lectures are at the level of beginning graduate students.

## tba

## A proof of the prime number theorem I

We shall give the proof of Iwaniec in his recent AMS book on the Riemann zeta function.

These lectures are at the level of beginning graduate students.

## tba

## Summing $\mu(n)$: a better elementary algorithm

Joint with Lola Thompson.

Consider either of two related problems: determining the precise

number $\pi(x)$ of prime numbers $p\leq x$, and computing the Mertens

function $M(x) = \sum_{n\leq x} \mu(n)$, where $\mu$ is the Möbius function.

The two best algorithms known are the following:

## Locally harmonic Maass forms and central $L$-values

In this talk, we will discuss a relatively new modular-type object known as

a locally harmonic Maass form.

We will discuss recent joint work with Ehlen, Guerzhoy, and Kane with

applications to the theory of $L$-functions. In particular, we find

finite formulas for certain twisted central $L$-values of a family of

elliptic curves in terms of finite sums over canonical binary quadratic

forms. Applications to the congruent number problem will be given.

## On smooth square-free numbers in arithmetic progressions

A.~Booker and C.~Pomerance (2017) have shown that any residue class modulo a prime $p\ge 11$ can be represented by a positive $p$-smooth square-free integer $s = p^{O(\log p)}$ with all prime factors up to $p$ and conjectured that in fact one can find such $s$ with $s = p^{O(1)}$. Using bounds on double Kloosterman sums due to M.~Z.~Garaev (2010) we prove this conjecture in a stronger form $s \le p^{3/2 + o(1)}$. Furthermore, using some additional arguments we show that for almost all primes $p$ one can replace $3/2$ with $4/3$.

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