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Valentin Blomer

Publications and source records attributed to Valentin Blomer.

76 records · Page 5Linked to original sources
arXiv2009

Subconvexity for a double Dirichlet series

For Dirichlet series roughly of the type $Z(s, w) = sum_d L(s, chi_d) d^{-w}$ the subconvexity bound $Z(s, w) \ll (sw(s+w))^{1/6+\varepsilon}$ is proved on the critical lines $\Re s = \Re w = 1/2$. The convexity bound would replace 1/6 with 1/4. In addition, a mean square bound is proved that is consistent with the Lindelöf hypothesis. An interesting specialization is $s=1/2$ in which case the above result give a subconvex bound for a Dirichlet series without an Euler product.

math.NT↗
arXiv2009

Bounding sup-norms of cusp forms of large level

Let f be an $L^2$-normalized weight zero Hecke-Maass cusp form of square-free level N, character $χ$ and Laplacian eigenvalue $λ\geq 1/4$. It is shown that $\| f \|_{\infty} \ll_λ N^{-1/37}$, from which the hybrid bound $\|f \|_{\infty} \ll λ^{1/4} (Nλ)^{-δ}$ (for some $δ> 0$) is derived. The first bound holds also for $f = y^{k/2}F$ where F is a holomorphic cusp form of weight k with the implied constant now depending on k.

math.NT↗
arXiv2008

Sums of Hecke eigenvalues over quadratic polynomials

Let f(z) = sum_n a(n) n^{(k-1)/2} e(nz) be a cusp form for Gamma_0(N), character chi and weight k geq 4. Let q(x) = x^2 + sx + t be a polynomial with integral coefficients. It is shown that sum_{n \leq X} a(q(n)) = cX + O(X^{6/7+eps}) for some constant c depending on f and q. The constant vanishes in many cases, for example if k is even. On the way a Kuznetsov formula for half-integral weight and entries having different sign is derived.

math.NT↗