Who is the "young student" André Weil is referring to in his letter from the prison?

This must have been Heinrich Kornblum (1890-1914).

_{[note by E. Landau in German, my translation]}

$^1$ The author, born in Wohlau on August 23, 1890, had before the war independently made the discovery that Dirichlet's classic proof of the theorem of prime numbers in an arithmetic progression (along with the later elementary reasons for the non-vanishing of the known series) had an analogue in the theory of prime functions in residue classes with a double module ($p,M$). His doctoral dissertation on this self-chosen topic was already essentially finished when, as a war volunteer, he fell in October 1914 at Роёl-Сареllе. Only recently I received from his estate the manuscript (known to me since 1914). I hereby publish the most beautiful and interesting parts. The Kornblum approach is characterized by high elegance and shows that science has lost in him a very promising researcher.

That was most certainly Heinrich Kornblum. His paper titled 'Über die Primfunktionen in einer arithmetischen Progression' was published in Math. Z. 5 (1919) pp 100–111 (EuDML), see the zbMath revew. In the paper he establishes the analogue of Dirichlet's theorem on primes in arithmetic progressions, in the polynomial setting (with natural density).

His history and contribution are mentioned several times in Roquette's historical account of the Riemann Hypothesis in positive characteristic.