A simple note on the number-theoretic function $$\Omega (n)$$ and Dirichlet L-functions

A Note on the Fourth Moment of Dirichlet L-functions

For χ a Dirichlet character (mod q), the moments of L(s, χ) have many applications, for example to the distribution of primes in the arithmetic progressions to modulus q. The asymptotic formula of the fourth power moment in the q-aspect has been obtained by Heath-Brown [1], for q prime, and more recently by Soundararajan [5] for general q. Following Soundararajan’s work, Young [7] pushed the re...

A NOTE ON SIMPLE a-POINTS OF L-FUNCTIONS

We prove, subject to certain hypotheses, that a positive proportion of the a-points of the Riemann zeta-function and Dirichlet L-functions with primitive characters are simple and discuss corresponding results for other functions in the Selberg class. We also prove an unconditional result of this type for the a-points in fixed strips to the right of the line s = 1/2.

Prime number races and zeros of Dirichlet L-functions 09rit148

This Research in Teams meeting focused on the finer behaviour of the function π(x; q, a), which denotes the number of prime numbers of the form qn+ a that are less than or equal to x. Dirichlet’s famous theorem on primes in arithmetic progressions asserts that that there are infinitely many primes of the form qn + a when a is a reduced residue modulo q (that is, when a and q are relatively prim...

On the Order of Dirichlet L-functions

A Note on the Implementation of the Number Theoretic Transform

The Number Theoretic Transform (NTT) is a time critical function required by many post-quantum cryptographic protocols based on lattices. For example it is commonly used in the context of the Ring Learning With Errors problem (RLWE), which is a popular basis for post-quantum key exchange, digital signature, and encryption. Here we apply a simple methodology to convert the NTT and its inverse fr...