We study the \lq \lq symmetry integral\rq \rq, \thinspace say $I_f$, of some arithmetic functions $f:\N \rightarrow \R$; we obtain from lower bounds of $I_f$ (for a large class of arithmetic functions $f$) lower bounds for the \lq \lq Selberg integral\rq \rq \thinspace of $f$, say $J_f$ (both these integrals give informations about $f$ in almost all the short intervals $[x-h,x+h]$, when $N\le x\le 2N$). In particular, when \thinspace $f=d_k$, the divisor function (having Dirichlet series \thinspace $ζ^k$, with \thinspace $ζ$ \thinspace the Riemann zeta function), where $k\ge 3$ is integer, we give lower bounds for the Selberg integrals, say \thinspace $J_k=J_{d_k}$, of the \thinspace $d_k$. We apply elementary methods (Cauchy inequality to get Large Sieve type bounds) in order to give $I_f$ lower bounds.
On the Correlations, Selberg Integral and Symmetry of Sieve Functions in Short Intervals
我们研究了算术（实）函数f=g*1，其中g“本质有界”且支撑在[1，Q]的整数上。特别地，我们通过f的“相关性”得到了几乎所有短区间[x-h，x+h]（N<x<2N）中f的“Selberg积分”和“对称积分”的非平凡界，这些短区间超过了“经典”水平，直到分布水平，比如lambda=log Q/log N < 2/3（对于足够大的h）。这一次我们不像文[C-S]那样应用大筛不等式。准确地说，我们的方法是完全基本的。
In this note, basing on a certain functional equation of the dilogarithm function, we establish nontrivial lower bounds for the $p$-adic valuation (where $p$ is a given prime number) of some type of rational numbers involving harmonic numbers. Then we use our estimate to derive the integrality of some sequences of rational numbers, which cannot be seen directly from their definitions.
We show that, based on Grabner's recent results on modular differential equations satisfied by quasimodular forms, there exist only finitely many normalized extremal quasimodular forms of depth $r$ that have all Fourier coefficients integral for each of $r=1,2,3,4$, and partly classifies them, where the classification is complete for $r=2,3,4$; in fact, we show that there exists no normalized extremal quasimodular forms of depth $4$ with all Fourier coefficients integral. Our result disproves a conjecture by Pellarin.
We obtain uniform lower bounds, true for all automorphic L-functions L(s) associated to cuspidal representations of GL(m,A) where A denotes the adeles of the rationals Q, of the integral on the vertical line (Re(s)=1/2) of the absolute value squared of L(s)/s; and also of L(s)/(s-s_0) when s_0 is a zero of the L-function on the critical line. Several variants are also obtained in small degrees m, for the vertical integrals at different abscissas in the critical strip. For the estimates required to prove convergence, we are led to generalise a result of Friedlander-Iwaniec (Can. J. Math. 57,200…
Mirror symmetry relates Gromov-Witten invariants of an elliptic curve with certain integrals over Feynman graphs. We prove a tropical generalization of mirror symmetry for elliptic curves, i.e., a statement relating certain labeled Gromov-Witten invariants of a tropical elliptic curve to more refined Feynman integrals. This result easily implies the tropical analogue of the mirror symmetry statement mentioned above and, using the necessary Correspondence Theorem, also the mirror symmetry statement itself. In this way, our tropical generalization leads to an alternative proof of mirror symmetry…
