arXiv · 0707.1761
We give a more strong heuristic justification of our conjecture on the excess of the odious primes.
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Vladimir Shevelev. 2007-10-12. On Excess of the Odious Primes. https://arxiv.org/abs/0707.1761
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The lattice point problems of the $p$-circle (for example, the astroid), which a generalized circle for positive real numbers $p$, have been solved for approximately $p$ more than 3, based on the series representation of the error term using the generalized Bessel functions by E. Krätzel and the results of G. Kuba. On the other hand, for the cases $0<p<2$, the method via this series representation cannot make progress. Therefore, in such cases, it is necessary to consider another method. In this paper, we prove that certain functions closely related to the problems can be displayed as series by newly generalized Bessel functions based on the property $p$-radial, generalization of spherical symmetry, and highlight the possibility that attempts to solve the problems via this display are suitable especially for the cases $0<p\leq1$. This study is based on the harmonic-analytic method by S. Kuratsubo and E. Nakai, using certain functions generalizing the error term of the circle problem by variables and series representation of the functions by the Bessel functions.
We investigate the mean value of the first moment of primitive quartic $L$-functions over $\mathbb{F}_q(T)$ in the non-Kummer setting. Specifically, we study the sum \begin{equation*} \sum_{\substack{χ primitive\ quartic\\ χ^2 primitive\\ genus(χ)=g}}L_q(\frac{1}{2}, χ), \end{equation*} where $L_q(s,χ)$ denotes the $L$-function associated with primitive quartic character $χ$. Using double Dirichlet series, we derive an error term of size $q^{(\frac{3}{5}+\varepsilon)g}$.
We consider a set of generators for the space of Eisenstein series of varying even weight $k$ for any fixed congruence group $Γ$ and study the set of all of their zeros taken for all $Γ(1)$-conjugates of $Γ$ in the standard fundamental domain for $Γ(1)$. We describe (a) an upper bound $κ_Γ+ O(1/k)$ for their imaginary part; (b) a finite configuration of geodesics segments to which all zeros converge in Hausdorff distance as $k \rightarrow \infty$; (c) a finite set containing all algebraic zeros for all weights. The bound in (a) depends on the (non-)vanishing of a new generalization of Ramanujan sums. The proof of (b) originates in a method used to study phase transitions in statistical physics. The proof of (c) relies on the theory of complex multiplication. The results can be made quantitative for specific groups. For $Γ=Γ(N)$ with $4 \nmid N$, $κ_Γ=1$ and the zeros tend to the unit circle, whereas if $4 \mid N$, $κ_Γ=2$ and the limit configuration includes parts of vertical geodesics and circles of radius $2$. In both cases, for sufficiently large $k$, the only algebraic zeros are at $\mathrm{i}$ and $\exp(2π\mathrm{i}/3)$, and, for $4 \mid N$, at $2\mathrm{i}$. For $Γ(N)$ with $N$ odd, we use finer estimates to prove a trichotomy for the exact `convergence speed' of the zeros to the unit circle, as well as angular equidistribution of the zeros as $k \rightarrow + \infty$, implying weak convergence of the normalized zero counting measure on $Y(1)$ to the image of the argument measure on the lower arc of the fundamental domain.