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Table Of Integrals Series And Products

DORIN GHISA Riemann Zeta function is one of the most studied transcendental functions, having in view its many applications in number theory,algebra, complex analysis, statistics, as well as in physics. Another reason why this function has drawn so much attention is the celebrated Riemann conjecture regarding its non trivial zeros, which resisted proof or disproof until now. There has been lately a lot of progress in the sense of proving its truthfulness.

/pdf/the-riemann-zeta-function-3l59kypwcz.pdf

The Riemann zeta function

Independent and stationary sequences of random variables

Limit Theorems of Probability Theory.

We introduce a general result relating "short averages" of a multiplicative function to "long averages" which are well understood. This result has several consequences. First, for the M\"obius function we show that there are cancellations in the sum of $\mu(n)$ in almost all intervals of the form $[x, x + \psi(x)]$ with $\psi(x) \rightarrow \infty$ arbitrarily slowly. This goes beyond what was previously known conditionally on the Density Hypothesis or the stronger Riemann Hypothesis. Second, we settle the long-standing conjecture on the existence of $x^{\epsilon}$-smooth numbers in intervals of the form $[x, x + c(\varepsilon) \sqrt{x}]$, recovering unconditionally a conditional (on the Riemann Hypothesis) result of Soundararajan. Third, we show that the mean-value of $\lambda(n)\lambda(n+1)$, with $\lambda(n)$ Liouville's function, is non-trivially bounded in absolute value by $1 - \delta$ for some $\delta > 0$. This settles an old folklore conjecture and constitutes progress towards Chowla's conjecture. Fourth, we show that a (general) real-valued multiplicative function $f$ has a positive proportion of sign changes if and only if $f$ is negative on at least one integer and non-zero on a positive proportion of the integers. This improves on many previous works, and is new already in the case of the M\"obius function. We also obtain some additional results on smooth numbers in almost all intervals, and sign changes of multiplicative functions in all intervals of square-root length.

/pdf/multiplicative-functions-in-short-intervals-33nzwrqbg6.pdf

Multiplicative functions in short intervals

Let $\lambda$ denote the Liouville function. A well known conjecture of Chowla asserts that for any distinct natural numbers $h_1,\dots,h_k$, one has $\sum_{1 \leq n \leq X} \lambda(n+h_1) \dotsm \lambda(n+h_k) = o(X)$ as $X \to \infty$. This conjecture remains unproven for any $h_1,\dots,h_k$ with $k \geq 2$. In this paper, using the recent results of the first two authors on mean values of multiplicative functions in short intervals, combined with an argument of Katai and Bourgain-Sarnak-Ziegler, we establish an averaged version of this conjecture, namely $$\sum_{h_1,\dots,h_k \leq H} \left|\sum_{1 \leq n \leq X} \lambda(n+h_1) \dotsm \lambda(n+h_k)\right| = o(H^kX)$$ as $X \to \infty$ whenever $H = H(X) \leq X$ goes to infinity as $X \to \infty$, and $k$ is fixed. Related to this, we give the exponential sum estimate $$ \int_0^X \left|\sum_{x \leq n \leq x+H} \lambda(n) e(\alpha n)\right| dx = o( HX )$$ as $X \to \infty$ uniformly for all $\alpha \in \mathbb{R}$, with $H$ as before. Our arguments in fact give quantitative bounds on the decay rate (roughly on the order of $\frac{\log\log H}{\log H}$), and extend to more general bounded multiplicative functions than the Liouville function, yielding an averaged form of a (corrected) conjecture of Elliott.

An averaged form of Chowla's conjecture

Let λ denote the Liouville function. A well-known conjecture of Chowla asserts that, for any distinct natural numbers h1,…,hk, one has ∑ 1≤n≤Xλ(n + h1)⋯λ(n + hk) = o(X) as X →∞. This conjecture remains unproven for any h1,…,hk with k ≥ 2. Using the recent results of Matomaki and Radziwill on mean values of multiplicative functions in short intervals, combined with an argument of Katai and Bourgain, Sarnak, and Ziegler, we establish an averaged version of this conjecture, namely ∑ h1,…,hk≤H|∑ 1≤n≤Xλ(n + h1)⋯λ(n + hk)| = o(HkX) as X →∞, whenever H = H(X) ≤ X goes to infinity as X →∞ and k is fixed. Related to this, we give the exponential sum estimate ∫ 0X|∑ x≤n≤x+Hλ(n)e(αn)|dx = o(H“X) as X →∞ uniformly for all α ∈ ℝ, with H as before. Our arguments in fact give quantitative bounds on the decay rate (roughly on the order of loglogH∕logH) and extend to more general bounded multiplicative functions than the Liouville function, yielding an averaged form of a (corrected) conjecture of Elliott.

/pdf/an-averaged-form-of-chowla-s-conjecture-13bkcld9io.pdf

An averaged form of Chowla’s conjecture

We prove the leading order of a conjecture by Fyodorov, Hiary and Keating, about the maximum of the Riemann zeta function on random intervals along the critical line. More precisely, as T→∞ for a set of t∈[T,2T] of measure (1−o(1))T, we have max|t−u|≤1log∣∣ζ(12+iu)∣∣=(1+o(1))loglogT.

https://par.nsf.gov/servlets/purl/10100827

Maximum of the Riemann zeta function on a short interval of the critical line

We show that if one can compute a little more than a particular moment for some family of L-functions, then one has upper bounds of the conjectured order of magnitude for all smaller (positive, real) moments and a one-sided central limit theorem holds. We illustrate our method for the family of quadratic twists of an elliptic curve, obtaining sharp upper bounds for all moments below the first. We also establish a one sided central limit theorem supporting a conjecture of Keating and Snaith. Our work leads to a conjecture on the distribution of the order of the Tate-Shafarevich group for rank zero quadratic twists of an elliptic curve, and establishes the upper bound part of this conjecture (assuming the Birch-Swinnerton-Dyer conjecture).

Maksym Radziwiłł

Papers

Multiplicative functions in short intervals

An averaged form of Chowla's conjecture

An averaged form of Chowla’s conjecture

Maximum of the Riemann zeta function on a short interval of the critical line

Moments and distribution of central $$L$$ L -values of quadratic twists of elliptic curves