Non-random behavior in sums of modular symbols

We give explicit expressions for the Fourier coefficients of Eisenstein series twisted by Dirichlet characters and modular symbols on [Formula: see text] in the case where [Formula: see text] is prime and equal to the conductor of the Dirichlet character. We obtain these expressions by computing the spectral decomposition of automorphic functions closely related to these Eisenstein series. As an application, we then evaluate certain sums of modular symbols in a way which parallels past work of Goldfeld, O’Sullivan, Petridis, and Risager. In one case we find less cancelation in this sum than would be predicted by the common phenomenon of “square root cancelation”, while in another case we find more cancelation.

Riemann Hypothesis and Random Walks: The Zeta Case

In previous work, it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its L-function is valid to the right of the critical line ℜ(s)>12, and the Riemann hypothesis for this class of L-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet L-functions. We apply these results to the study of the argument of the zeta function. In another application, we define and study a one-point correlation function of the Riemann zeros, which leads to the construction of a probabilistic model for them. Based on these results we describe a new algorithm for computing very high Riemann zeros, and we calculate the googol-th zero, namely 10100-th zero to over 100 digits, far beyond what is currently known. Of course, use is made of the symmetry of the zeta function about the critical line.

Dirichlet series expansions of p-adic L-functions

We study p-adic L-functions $$L_p(s,\chi )$$ L p ( s , χ ) for Dirichlet characters $$\chi $$ χ . We show that $$L_p(s,\chi )$$ L p ( s , χ ) has a Dirichlet series expansion for each regularization parameter c that is prime to p and the conductor of $$\chi $$ χ . The expansion is proved by transforming a known formula for p-adic L-functions and by controlling the limiting behavior. A finite number of Euler factors can be factored off in a natural manner from the p-adic Dirichlet series. We also provide an alternative proof of the expansion using p-adic measures and give an explicit formula for the values of the regularized Bernoulli distribution. The result is particularly simple for $$c=2$$ c = 2 , where we obtain a Dirichlet series expansion that is similar to the complex case.

Hankel determinants of sequences related to Bernoulli and Euler polynomials

We evaluate the Hankel determinants of various sequences related to Bernoulli and Euler numbers and special values of the corresponding polynomials. Some of these results arise as special cases of Hankel determinants of certain sums and differences of Bernoulli and Euler polynomials, while others are consequences of a method that uses the derivatives of Bernoulli and Euler polynomials. We also obtain Hankel determinants for sequences of sums and differences of powers and for generalized Bernoulli polynomials belonging to certain Dirichlet characters with small conductors. Finally, we collect and organize Hankel determinant identities for numerous sequences, both new and known, containing Bernoulli and Euler numbers and polynomials.

Dirichlet characters of the rational polynomials

<abstract><p>Denote by $ \chi $ a Dirichlet character modulo $ q\geq 3 $, and $ \overline{a} $ means $ a\cdot\overline{a} \equiv 1 \bmod q $. In this paper, we study Dirichlet characters of the rational polynomials in the form</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \sum\limits^{q}_{a = 1}'\chi(ma+\overline{a}), $\end{document} </tex-math></disp-formula></p> <p>where $ \sum\limits_{a = 1}^{q}' $ denotes the summation over all $ 1\le a\le q $ with $ (a, q) = 1 $. Relying on the properties of character sums and Gauss sums, we obtain W. P. Zhang and T. T. Wang's identity ^{[<xref ref-type="bibr" rid="b6">6</xref>]} under a more relaxed situation. We also derive some new identities for the fourth power mean of it by adding some new ingredients.</p></abstract>

Efficient computation of the Euler–Kronecker constants of prime cyclotomic fields

We introduce a new algorithm, which is faster and requires less computing resources than the ones previously known, to compute the Euler–Kronecker constants $${\mathfrak {G}}_q$$ G q for the prime cyclotomic fields $$ {\mathbb {Q}}(\zeta _q)$$ Q ( ζ q ) , where q is an odd prime and $$\zeta _q$$ ζ q is a primitive q-root of unity. With such a new algorithm we evaluated $${\mathfrak {G}}_q$$ G q and $${\mathfrak {G}}_q^+$$ G q + , where $${\mathfrak {G}}_q^+$$ G q + is the Euler–Kronecker constant of the maximal real subfield of $${\mathbb {Q}}(\zeta _q)$$ Q ( ζ q ) , for some very large primes q thus obtaining two new negative values of $${\mathfrak {G}}_q$$ G q : $${\mathfrak {G}}_{9109334831}= -0.248739\dotsc $$ G 9109334831 = - 0.248739 ⋯ and $${\mathfrak {G}}_{9854964401}= -0.096465\dotsc $$ G 9854964401 = - 0.096465 ⋯ We also evaluated $${\mathfrak {G}}_q$$ G q and $${\mathfrak {G}}^+_q$$ G q + for every odd prime $$q\le 10^6$$ q ≤ 10 6 , thus enlarging the size of the previously known range for $${\mathfrak {G}}_q$$ G q and $${\mathfrak {G}}^+_q$$ G q + . Our method also reveals that the difference $${\mathfrak {G}}_q - {\mathfrak {G}}^+_q$$ G q - G q + can be computed in a much simpler way than both its summands, see Sect. 3.4. Moreover, as a by-product, we also computed $$M_q=\max _{\chi \ne \chi _0} \vert L^\prime /L(1,\chi ) \vert $$ M q = max χ ≠ χ 0 | L ′ / L ( 1 , χ ) | for every odd prime $$q\le 10^6$$ q ≤ 10 6 , where $$L(s,\chi )$$ L ( s , χ ) are the Dirichlet L-functions, $$\chi $$ χ run over the non trivial Dirichlet characters mod q and $$\chi _0$$ χ 0 is the trivial Dirichlet character mod q. As another by-product of our computations, we will provide more data on the generalised Euler constants in arithmetic progressions.