arXiv subjects

Guilherme França

Publications and source records attributed to Guilherme França.

22 records · Page 2Linked to original sources

arXiv2015

On the validity of the Euler product inside the critical strip

The Euler product formula relates Dirichlet $L(s,χ)$ functions to an infinite product over primes, and is known to be valid for $\Re (s) >1$, where it converges absolutely. We provide arguments that the formula is actually valid for $\Re (s) > 1/2$ in a specific sense. Namely, the logarithm of the Euler product, although formally divergent, is meaningful because it is Cesàro summable, and its Cesàro average converges to $\log L (s,χ)$. Our argument relies on the prime number theorem, an Abel transform, and a central limit theorem for the Random Walk of the Primes, the series $\sum_{n=1}^N \cos\left(t\log p_n\right)$, and its generalization to other Dirichlet $L$-functions. The significance of ${\Re(s) > 1/2}$ arises from the $\sqrt{N}$ growth of this series, since it satisfies a central limit theorem. $L$-functions based on principal Dirichlet characters, such as the Riemann $ζ$-function, are exceptional due to the pole at $s=1$, and require $\Im (s) \neq 0$ and a truncation of the Euler product. Compelling numerical evidence of this surprising result is presented, and some of its consequences are discussed.

math.NT↗

arXiv2015

Transcendental equations satisfied by the individual zeros of Riemann $ζ$, Dirichlet and modular $L$-functions

We consider the non-trivial zeros of the Riemann $ζ$-function and two classes of $L$-functions; Dirichlet $L$-functions and those based on level one modular forms. We show that there are an infinite number of zeros on the critical line in one-to-one correspondence with the zeros of the cosine function, and thus enumerated by an integer $n$. From this it follows that the ordinate of the $n$-th zero satisfies a transcendental equation that depends only on $n$. Under weak assumptions, we show that the number of solutions of this equation already saturates the counting formula on the entire critical strip. We compute numerical solutions of these transcendental equations and also its asymptotic limit of large ordinate. The starting point is an explicit formula, yielding an approximate solution for the ordinates of the zeros in terms of the Lambert $W$-function. Our approach is a novel and simple method, that takes into account $\arg L$, to numerically compute non-trivial zeros of $L$-functions. The method is surprisingly accurate, fast and easy to implement. Employing these numerical solutions, in particular for the $ζ$-function, we verify that the leading order asymptotic expansion is accurate enough to numerically support Montgomery's and Odlyzko's pair correlation conjectures, and also to reconstruct the prime number counting function. Furthermore, the numerical solutions of the exact transcendental equation can determine the ordinates of the zeros to any desired accuracy. We also study in detail Dirichlet $L$-functions and the $L$-function for the modular form based on the Ramanujan $τ$-function, which is closely related to the bosonic string partition function.

math.NT↗

arXiv2014

Statistical and other properties of Riemann zeros based on an explicit equation for the $n$-th zero on the critical line

We show that there are an infinite number of Riemann zeros on the critical line, enumerated by the positive integers $n=1,2,\dotsc$, whose ordinates can be obtained as the solution of a new transcendental equation that depends only on $n$. Under weak assumptions, we show that the number of such zeros already saturates the counting formula for the numbers of zeros on the entire critical strip. These results thus constitute a concrete proposal toward verifying the Riemann hypothesis. We perform numerical analyses of the exact equation, and its asymptotic limit of large ordinate. The starting point is an explicit analytical formula for an approximate solution to the exact equation in terms of the Lambert $W$ function. In this way, we neither have to use Gram points or deal with violations of Gram's law. Our numerical approach thus constitutes a novel method to compute the zeros. Employing these numerical solutions, we verify that solutions of the asymptotic version are accurate enough to confirm Montgomery's and Odlyzko's pair correlation conjectures and also to reconstruct the prime number counting function.

math.NT↗

arXiv2014

On the zeros of L-functions

We generalize our recent construction of the zeros of the Riemann $ζ$-function to two infinite classes of $L$-functions, Dirichlet $L$-functions and those based on level one modular forms. More specifically, we show that there are an infinite number of zeros on the critical line which are in one-to-one correspondence with the zeros of the cosine function, and thus enumerated by an integer $n$. We obtain an exact equation on the critical line that determines the $n$-th zero of these $L$-functions. We show that the counting formula on the critical line derived from such an equation agrees with the known counting formula on the entire critical strip. We provide numerical evidence supporting our statements, by computing numerical solutions of this equation, yielding $L$-zeros to high accuracy. We study in detail the $L$-function for the modular form based on the Ramanujan $τ$-function, which is closely related to the bosonic string partition function. The same analysis for a more general class of $L$-functions is also considered.

math.NT↗