Some bounds and limits in the theory of Riemann’s zeta function

For any real $a>0$ we determine the supremum of the real $\sigma$ such that $\zeta (\sigma +it)=a$ for some real $t$. For $0<a<1$, $a=1$, and $a>1$ the results turn out to be quite different.We also determine the supremum $E$ of the real parts of the ‘turning points’, that is points $\sigma +it$ where a curve $\mathrm{Im}\zeta (\sigma +it)=0$ has a vertical tangent. This supremum $E$ (also considered by Titchmarsh) coincides with the supremum of the real $\sigma$ such that ${\zeta }^{\prime }(\sigma +it)=0$ for some real $t$.We find a surprising connection between the three indicated problems: $\zeta \left(s\right)=1$, ${\zeta }^{\prime }\left(s\right)=0$ and turning points of $\zeta \left(s\right)$. The almost extremal values for these three problems appear to be located at approximately the same height.     

A proof of Terras’ conjecture on the radius of convergence of the Ihara zeta function

We prove the conjecture by Audrey Terras (2011) on the inequalities among the spectral radius ${\rho }_X$ of a finite graph $X$, the radius of convergence $R$ of its Ihara zeta function $Z_X\left(u\right)$, and the average degree $\overline{d_X}$ of $X$. Relating to Terras’ conjecture, we propose a new conjecture between $R$ and certain Rayleigh quotients.     

On fluctuations of Riemann’s zeta zeros

It is shown that the normalized fluctuations of Riemann’s zeta zeros around their predicted locations follow the Gaussian law. It is also shown that fluctuations of two zeros, $\gamma _{k}$ and $\gamma _{k+x},$ with $x\sim \left( \log k\right) ^{\beta }, \beta >0,$ for large $k$ follow the two-variate Gaussian distribution with correlation $\left( 1-\beta \right) _{+}\! .$      

Jacob’s ladders,the structure of the Hardy-Littlewood integral and some new class of nonlinear integral equations

In this paper we obtain new formulae for short and microscopic parts of the Hardy-Littlewood integral, and the first asymptotic formula for the sixth-order expression $\left| {\zeta \left( {\tfrac{1} {2} + i\phi _1 \left( t \right)} \right)} \right|^4 \left| {\zeta \left( {\tfrac{1} {2} + it} \right)} \right|^2$\left| {\zeta \left( {\tfrac{1} {2} + i\phi _1 \left( t \right)} \right)} \right|^4 \left| {\zeta \left( {\tfrac{1} {2} + it} \right)} \right|^2. These formulae cannot be obtained in the theories of Balasubramanian, Heath-Brown and Ivić.     

Numerical evaluation of Goursat’s infinite integral

The infinite integral ò₀^￥x dx/(1+x⁶sin²x)\int_0^{\infty}x\,dx/(1+x^6\sin^2x) converges but is hard to evaluate because the integrand f(x) = x/(1 + x ⁶sin² x) is a non-convergent and unbounded function, indeed f(kπ) = kπ→ ∞ (k→ ∞). We present an efficient method to evaluate the above integral in high accuracy and actually obtain an approximate value in up to 73 significant digits on an octuple precision system in C++.     

Fast algorithms for implementation of the Green’s function

This paper is devoted to new fast algorithms for implementation of the Green’s function for the Helmholtz operator in high-frequency regions in periodic and helical structures.     

Jacob’s ladders and the almost exact asymptotic representation of the Hardy-Littlewood integral

In this paper we introduce a nonlinear integral equation such that the system of global solutions to this equation represents the class of a very narrow beam as T → ∞ (an analog of the laser beam) and this sheaf of solutions leads to an almost-exact representation of the Hardy-Littlewood integral (1918). The accuracy of our result is essentially better than the accuracy of related results of Balasubramanian, Heath-Brown, and Ivic.     

Improving the accuracy of multiple integral evaluation by applying Romberg’s method

Romberg’s method, which is used to improve the accuracy of one-dimensional integral evaluation, is extended to multiple integrals if they are evaluated using the product of composite quadrature formulas. Under certain conditions, the coefficients of the Romberg formula are independent of the integral’s multiplicity, which makes it possible to use a simple evaluation algorithm developed for one-dimensional integrals. As examples, integrals of multiplicity two to six are evaluated by Romberg’s method and the results are compared with other methods.     

Representation of the Green’s function of the exterior Neumann problem for the Laplace operator

We represent the Green’s function of the classical Neumann problem for the exterior of the unit ball of arbitrary dimension. We show that the Green’s function can be expressed through elementary functions. The explicit form of the function is written out.     

An asymptotic formula for the primitive of Hardy’s function

Let Z(t) be the classical Hardy function in the theory of Riemann’s zeta-function. An asymptotic formula with an error term O(T ^1/6log?T) is given for the integral of Z(t) over the interval [0,T], with special attention paid to the critical cases when the fractional part of \(\sqrt{T/2\pi }\) is close to \(\frac{1}{4}\) or \(\frac{3}{4}\).     

Combinatorial interpretations of Ramanujan’s tau function

We use a $q$-series identity by Ramanujan to give a combinatorial interpretation of Ramanujan’s tau function which involves $t$-cores and a new class of partitions which we call $(m,k)$-capsids. The same method can be applied in conjunction with other related identities yielding alternative combinatorial interpretations of the tau function.     

Discretized Newman-Shapiro Operators and Jackson’s Inequality on the Sphere

Applying a quadrature rule with positive weights to some integral operators introduced by Newman and Shapiro we obtain generalized hyperinterpolation operators on the sphere, whose approximation error can be estimated — inspite of the discretisation — by means of the modulus of continuity. The main reason is that the weight distribution satisfies necessarily some regularity condition, which has been used before in hyperinterpolation by Sloan and Womersley, and which turned out to hold always by its own.