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A sharper estimate for a generalization of the sum-of-divisors function

I am interested in the function $f_n(m)$ which can be defined by the Dirichlet generating function $$\zeta(s)\zeta(s - 1) \cdots\zeta(s - n + 1) = \sum\limits_{m = 1}^\infty \frac{f_n(m)}{m^s} $$ This ...
Bear's user avatar
  • 31
-1 votes
1 answer
106 views

Analyzing a Dirichlet series with log-oscillating terms via Fourier methods

I am investigating the series $S(z)$ defined as follows: $$ S(z) = \sum_{n=1}^{\infty} n^{-a}\cos(b\ln(n)), $$ where $z = a + bi \in \mathbb{C}$, with $0 < a < 1$, and $b \in \mathbb{R}$. I want ...
swami's user avatar
  • 375
1 vote
1 answer
200 views

Classification of L functions and Dirichlet series by poles

I am interested in the connection between particular Dirichlet series' abscissa of convergence and the poles of L-functions. Let $D(z) = \sum_{n=1}^\infty\frac{a_n}{n^z}$ be a Dirichlet series ...
asdfasdgasdfgasfgasdf's user avatar
2 votes
2 answers
219 views

Conditional convergence of exponential sums related to a Hecke modular form

Definition Consider the Fourier coefficients $\psi(n)$ of the modular form $\eta^4(6\tau)$, which are defined in terms of $q=\exp(i2\pi\tau)$ by the identity: $$\eta^4(6\tau) = q \prod_1^\infty (1-q^{...
Christopher-Lloyd Simon's user avatar
0 votes
1 answer
149 views

A question about the setup of zero density estimates for Dirichlet $L$-functions

For $L(s,\chi)= \sum_{n \geq 1}\frac{\chi(n)}{n^s}$, where $s = \sigma + it$, we define the function $N(\sigma, T, \chi)$ which counts the zeros $\rho = \beta + i\gamma$ for which $L(\rho, \chi) =0$ ...
Josh's user avatar
  • 1
6 votes
0 answers
277 views

Approximating $\zeta'/\zeta$ (and its derivatives) by a finite sum

Let $A(s) = (-\zeta'/\zeta)^{(r)}(s) = \sum_n a_n n^{-s}$, where $r\geq 0$. (We can consider $r=0$ first for simplicity.) Say I want to approximate $A(s)$ for $s=1+it$ by a finite sum - preferably a ...
H A Helfgott's user avatar
  • 19.3k
14 votes
1 answer
721 views

Euler's proof of $\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$

Euler proved $$\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$$ where the reasoning of the signs thus is prepared, so that of the second may be had as $-$, prime ...
Nomas2's user avatar
  • 367
1 vote
0 answers
112 views

Question on recursive formulas for $\eta(2 n+1)$ and $\beta(2 n)$ where $n\in\mathbb{N}$

This question is a refinement of my related MSE question which was asked over 2 years ago and no answers have yet been posted. Consider the following formulas for the Dirichlet eta function $\eta(s)$ ...
Steven Clark's user avatar
  • 1,091
0 votes
0 answers
111 views

Are $\zeta'(0)$ and $\beta'(0)$ algebraic numbers?

Let $\zeta$ be the Riemann zeta function and $\beta$ the Dirichlet beta function. We know that $\zeta (0)=-1/2$ and $\beta (0)=1/2$ are algebraic numbers over $\mathbb{Q}$. This led me to the ...
Nomas2's user avatar
  • 367
4 votes
1 answer
204 views

Abscissa of convergence of the $\tau$ Dirichlet series

Define the $\tau$ Dirichlet series $L$ by $$L(s)=\sum_{n=1}^\infty \frac{\tau (n)}{n^s}$$ where $\tau$ is defined by $$q\prod_{n=1}^\infty (1-q^n)^{24}=\sum_{n=1}^\infty \tau (n)q^n$$ where $|q|\lt 1$....
Nomas2's user avatar
  • 367
2 votes
0 answers
126 views

Limit of scaled infinite sum with Dirichlet characters modulo 4: is it zero?

I am trying to get an asymptotic formula such as $$ L_4(s, n) \sim L_4(s) + \rho_n(s)\Lambda_n + \frac{\alpha(s)}{\sqrt{n}} + \frac{\beta(s)}{\sqrt{n\log n}}+\cdots$$ where $L_4(s, n)$ is the first $n$...
Vincent Granville's user avatar
6 votes
1 answer
339 views

The Dirichlet series of the harmonic numbers

I'm curious about the Dirichlet series $$F(s) = \sum_{n = 1}^\infty \frac{H_n}{n^s}$$ of the sequence $H_n = \sum_{k = 1}^n \frac{1}{k}$ of harmonic numbers. Its abscissa of convergence is $1$. ...
Jesse Elliott's user avatar
0 votes
0 answers
75 views

Could the patterns in the roots of the Riemann-Liouville differintegral of $(s-1)\,L(s, \chi_{q,j})$ be explained?

This question aims to extend this question to (automorphic) Dirichlet L-functions. Take: $$f(s,q,j) = (s-1)\,L(s, \chi_{q,j})$$ with $q$ the modulus and $j$ the index of a character $\chi$. A fast way ...
Agno's user avatar
  • 4,169
6 votes
1 answer
120 views

Finding the continued fraction for the "tails" of $\eta(3)$

I am interested in the continued fractions for the "tails" or "correction term" of the series sum of specific constants. For example, the Madhava's correction term for $\pi/4$: $$ \...
Nanhui Lee's user avatar
2 votes
0 answers
55 views

Dirichlet series solution to Poisson Point Process question (repost from math.SE)

Reposting here because the bounty on the original math.SE post expired, with no solutions or comments received. For any discrete subset $S$ of $\mathbb{R}^d$, consider a digraph formed by placing an ...
Jim Ferry's user avatar
  • 121
0 votes
0 answers
50 views

A Riemann-Liouville differintegral for all entire Dirichlet L-series. Could it be simplified further?

It appears that the well-known relation between entire Dirichlet L-series and the Hurwitz zeta function $\zeta(s,a)$, with $k$ = modulus, $j$ = index of the Dirichlet character $\chi$: $$(s-1)\,L\left(...
Agno's user avatar
  • 4,169
2 votes
1 answer
166 views

Exceptional zeros of a convolutional inverse

Let $\kappa:\lbrace 1,2,3,\ldots\rbrace\longrightarrow \mathbb Z$ be the convolutional inverse in the Dirichlet ring of $n\longmapsto {n+1\choose 2}$. It is defined by $\kappa(1)=1$ and by the ...
Roland Bacher's user avatar
2 votes
1 answer
160 views

Is this a valid method of extending convergence of the Maclaurin series for $\frac{x}{x+1}$ from $|x|<1$ to $\Re(x)>-1$?

I originally asked this question on Math StackExchange a few months ago and no answers or even comments have yet been posted, so I'm asking this question again here on Math OverFlow. This Math ...
Steven Clark's user avatar
  • 1,091
1 vote
2 answers
662 views

An inequality related to Catalan's constant and $\zeta(3)$

Problem : Show that : $$\frac{1}{\zeta(3)}<2C-1$$ Where we can see the zeta function and the Catalan's constant . After a bounty on Maths Stack Exchange there is no satisfying answer . See https://...
DesmosTutu's user avatar
7 votes
2 answers
375 views

Proving a series for $\pi$ by Plouffe

Simon Plouffe found experimentally a series for $\pi$ that can be written as $$\frac{\pi}{24} = \sum_{n=1}^\infty \frac{1}{n} \left( \frac{3}{e^{\pi n}-1} -\frac{4}{e^{2\pi n}-1} +\frac{1}{e^{4\pi n}-...
Jaume Oliver Lafont's user avatar
0 votes
0 answers
101 views

Prime races in two competing arithmetic progressions - error bound

I read an article by Andrew Granville on the subject, there's actually quite a bit of recent literature on the topic. My problem is as follows. I have two sequences of primes: $(p_{1,n})$ and $(p_{3,n}...
Vincent Granville's user avatar
4 votes
1 answer
320 views

Zeros of Dirichlet function $L(s,\chi_4)$

I am wondering if there are some know results for the non-trivial roots at ${\rm Re}(s) = \frac{1}{2}$, even maybe a table of the first few roots with $t>0$. This sister function $$ L_4^* (s,\chi_4)...
Vincent Granville's user avatar
3 votes
1 answer
232 views

Positivity of partial Dirichlet series for a quadratic character?

Let $\chi\colon(\mathbb{Z}/N\mathbb{Z})^\times\rightarrow\{\pm1\}$ be a primitive quadratic Dirichlet character of conductor $N$. For any integer $m=1,2,\cdots,\infty$, consider the partial Dirichlet ...
Zhang's user avatar
  • 81
-2 votes
1 answer
132 views

Convergence of scrambled product for Dirichlet-$L$ function with modulo 4 character

A Dirichlet-$L$ function is typically defined by its series, and its Euler product is a consequence of the definition. Here my approach is the other way around. I define the function $$ L_4^*(s) = \...
Vincent Granville's user avatar
2 votes
1 answer
95 views

A sum related to the first moment of quadratic $L$-functions at $s=1$

Let $(\frac{m}{n})$ be the Jacobi quadratic symbol defined for positive squarefree odd integers $n,m$. Does the following sum go to infinity? $$ \sum_{1\leq n \leq (\log x)^{100} } \mu^2(2n) \sum_{(\...
Dr. Pi's user avatar
  • 2,999
2 votes
0 answers
149 views

Error or gap in "Modular Functions and Dirichlet Series", by Apostol

My question concerns Apostol's Chapter 7, Kronecker's Theorem with Applications. It's Theorem 7.11, page 156. I’m attaching the proof in question. There is a lot going on, but I’ve highlighted the ...
Lawrence Paulson's user avatar
2 votes
0 answers
176 views

How to best approximate $1/\zeta(s)$ by a finite sum

I would like to approximate $1/\zeta(s)$ for $s=1+it$ by a finite sum: $$\frac{1}{\zeta(1+it)} = \sum_n \frac{\mu(n)}{n} \eta\left(\frac{n}{x}\right) + \epsilon(t)$$ with $\eta$ a function of compact ...
H A Helfgott's user avatar
  • 19.3k
6 votes
0 answers
203 views

Mean value theorem for Dirichlet series of prime support?

Let $\{a_n\}_{1\leq n\geq N}$, $a_n\in \mathbb{C}$. Let $F(s) = \sum_{n=1}^N a_n n^{-s}$. By a mean-value theorem (Montgomery-Vaughan, 1973), $$\int_0^T |F(i t)|^2 = \sum_{n=1}^N |a_n|^2 (T + O(n)).$$ ...
H A Helfgott's user avatar
  • 19.3k
4 votes
0 answers
126 views

Converse theorem for zeta universality

Voronin's Universality Theorem for $\zeta(s)$ is that the zeta function can uniformly approximate any non-vanishing holomorphic function to any degree of accuracy in the right-half of the critical ...
modperspec's user avatar
1 vote
0 answers
111 views

If the Dirichlet series $L(z,\chi)$ diverges for $\sigma< 1$, does its alternating version converge for some $\sigma_0 < 1$, and conversely?

Here $\chi$ is a completely multiplicative function with $\chi(p)\in\{-1,+1\}$ for any prime $p$, but not necessarily a character. Also $s=\sigma + it$ as usual. The series corresponding to $L(s,\chi)$...
Vincent Granville's user avatar
1 vote
2 answers
307 views

Abscissa of convergence for a very specific Dirichlet series / Euler product

I am interested in the convergence of the following Euler product: $$ \prod_p \frac{1}{1-\chi(p)\cdot p^{-s}}. $$ The product is over all primes (in increasing order), with $\chi(p)=+1$ if $p \bmod 4 =...
Vincent Granville's user avatar
1 vote
0 answers
96 views

Bounds and repulsion domains for the Dirichlet eta function $\eta(\sigma+it)$, for fixed $\sigma$

Let $\eta(\sigma+it)$ be the Dirichlet eta function, with $t>0$ (the variable) and $\sigma$ be fixed, with $\frac{1}{2}\leq \sigma <2$. I define the hole $\Omega_T =\Omega_T(\sigma)$ as the ...
Vincent Granville's user avatar
3 votes
0 answers
63 views

Number of cosets by quaternion order

Let $D$ be a quaternion division algebra over $\mathbb Q$ and let $\Lambda\subset D$ be a maximal order. For $n\in\mathbb Z$ let $\Lambda^{(n)}$ denote the set of all $\lambda\in\Lambda$ with reduced ...
user avatar
8 votes
1 answer
393 views

Is $\frac{1}{L(1+it)}$ unbounded?

Let $\chi$ be a Dirichlet character and $L(s, \chi)$ be the corresponding L-function. Is $$\frac{1}{L(1+it, \chi)}$$ unbounded for $t \in \mathbb{R}$? I'm aware that this is true if $L=\zeta$, but I'm ...
Holomorphic manifold's user avatar
14 votes
1 answer
556 views

Mean value theorem for Dirichlet series - optimize?

Let $a_n\in \mathbb{C}$. We can prove a mean value theorem, meaning an inequality $$\int_0^T \left|\sum_{n=1}^\infty a_n n^{-i t}\right|^2 dt \leq \sum_{n=1}^\infty (c_0 T + c_1 n + c_2) |a_n|^2.$$ ...
H A Helfgott's user avatar
  • 19.3k
12 votes
2 answers
526 views

Computation of modified Gauss sums

Let $\chi$ be a primitive Dirichlet character of conductor $q$. I want to compute numerically $$G(k)=\sum_{n\bmod q}\chi(n)e^{2\pi i n(n-k)/(2q)}$$ for all $k$ with $0\le k<2q$ with $k\equiv q\...
Henri Cohen's user avatar
  • 11.9k
0 votes
0 answers
135 views

Dirichlet series associated to divisor function

How can I express the following Dirichlet series $$\sum_{n=1}^{\infty}\frac{d_{k}^{2}(n)}{n^{s}}$$ (where $d(n)$-divisor function;$k\geq 1$) in terms of the zeta function?
user155294's user avatar
1 vote
0 answers
177 views

Zeta functions of schemes of finite type over $\mathbb{Z}$

Let $X$ be a scheme of finite type over $\mathbb{Z}$. In Section 11 of my 2008 paper in J. Number Theory, "Ring structures on groups of arithmetic functions," I define an additive analogue $...
Jesse Elliott's user avatar
0 votes
1 answer
69 views

What is the critical exponent for irregular function in the Sobolev scale?

When I first saw the definition of general Sobolev spaces with real exponent I immediately got interested in the following problem: pick several of your favourite irregular functions/distributions and ...
truebaran's user avatar
  • 9,210
5 votes
1 answer
167 views

Frequency of large values of the Mertens function

It is known that with $M(x) = \sum_{n\le x}\mu(n)$, there are infinitely many $x$ s.t. $|M(x)|\ge x^{\frac{1}{2} - \varepsilon}$ (see Chapter 15 of Montgomery-Vaughan, for example). Is there any way ...
Mayank Pandey's user avatar
9 votes
0 answers
520 views

Modern treatment of Delange's Tauberian Theorem

Tauberian theorems abound in the literature. One of the most general, powerful, and versatile is due to Delange, and appears as Theorem I of the paper: H. Delange - Généralisation du théorème de ...
Daniel Loughran's user avatar
1 vote
1 answer
252 views

Generalizing closed form representations related to conjectured analytic formulas for $f_a(x)=\sum\limits_{n=1}^x a(n)$

Consider the summatory function $f_a(x)$ defined in formula (1) below where the related Dirichlet series $F_a(s)$ defined in formula (2) below converges for $\Re(s)\ge 2$. $$f_a(x)=\sum\limits_{n=1}^...
Steven Clark's user avatar
  • 1,091
6 votes
0 answers
630 views

Generalized prime number theorem and Riemann Hypothesis for non-number math objects

My question is about some math objects (matrices, polynomials) and operators that satisfy a number of properties which can lead to a theory similar to PNT, RH, Dirichlet functions, abscissa of ...
Vincent Granville's user avatar
0 votes
1 answer
485 views

Questions on analytic representations of the Kronecker delta function $\delta(x-1)$ and the Moebius function $\mu(n)$

This question is related to analytic formulas for $a(n)$ where $f_a(x)$ and $F_a(s)$ defined in formulas (1) and (2) below are the summatory function and Dirichlet series associated with $a(n)$. $$...
Steven Clark's user avatar
  • 1,091
0 votes
0 answers
144 views

Abscissa of convergence of transformed Dirichlet series

Let $$F(s)=\sum_{k=1}^\infty \frac{f(k)}{k^s} \mbox{ and }F^*(s)=\sum_{k=2}^\infty \frac{f(k)g(k)}{k^s},$$ where the infinite sum $\sum f(k)$ diverges, $f(k)$ and $g(k)$ are real numbers, $s$ a ...
Vincent Granville's user avatar
7 votes
0 answers
138 views

Are there natural Dirichlet series whose completions have poles in the region of absolute convergence?

The Selberg class of $L$-functions are Dirichlet series $$ L(s, f) = \sum_{n \geq 1} \frac{a(n)}{n^s}, $$ satisfying certain properties that can be abbreviated as analyticity, a Ramanujan conjecture, ...
davidlowryduda's user avatar
1 vote
0 answers
344 views

Analytic continuation of Euler product $\phi(s)=\prod_p(1+p^{-s})^{-1}$

I am actually interested in the analytic continuation of $\phi_w(s)=\prod_p(1+w\cdot p^{-s})^{-1}$. Here $w$ is rational, or the imaginary unit multiplied by a rational. Consider for now that $w=1$. ...
Vincent Granville's user avatar
6 votes
4 answers
417 views

Reference for behavior of Artin $L$-functions at $\Re(s) = 1$

Would anyone know a reference that proves the basic facts about Artin $L$-functions at $\Re(s) = 1$? Namely, the non-vanishing and holomorphicity for non-trivial characters. I assume this was done in ...
Kim's user avatar
  • 4,134
2 votes
0 answers
111 views

An analytic function, asymptotically expandable in a Dirichlet series, is the sum of this series

Let there be a function $F(s)$ that is analytic in some half-plane $\sigma>\sigma_0$ (where $s=\sigma + it $). Let the function $F(s)$ have an asymptotic expansion of the form $F(s)\sim\sum\limits_{...
Anton Devyatkov's user avatar
1 vote
1 answer
292 views

Continuing an analytic continuation of the Dirichlet $\eta$-function?

The Dirichlet $\eta$-function is defined as: $$\eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s} \qquad \Re(s) > 0$$ and has the full analytical continuation: $$\eta(s) = \sum_{n=1}^N \frac{(-1)^{...
Agno's user avatar
  • 4,169