# Publishing house / Journals and Serials / Acta Arithmetica / All issues

## Acta Arithmetica

PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

## On weak Mellin transforms, second degree characters and the Riemann hypothesis

### Volume 177 / 2017

Acta Arithmetica 177 (2017), 219-275 MSC: Primary 11M26; Secondary 11F27, 33C15, 33C55. DOI: 10.4064/aa8240-7-2016 Published online: 31 January 2017

#### Abstract

Let $f$ be a function defined on $\mathbb{R}$ or $\mathbb{Q}_p$ and suppose that the integral defining the Mellin transform (or zeta integral) of $f$ does not converge. We can however say that $f$ has a “weak Mellin transform” $M_f(s)$ if ${\mathop{\rm Mell}(\phi \star f,s) = \mathop{\rm Mell} (\phi,s)M_f(s)}$ for all test functions $\phi$ in $C_c^\infty(\mathbb{R}^*)$ or $C_c^\infty(\mathbb{Q}_p^*)$. We show that if $f$ is of the form $f(x) = \psi\bigl(\frac a2x^2+bx\bigl)$, where $\psi$ is an additive character on $\mathbb{R}$ or $\mathbb{Q}_p$ and $a$ is invertible, then the weak Mellin transform of $f$ exists for $\Re(s) \gt 0$, satisfies a functional equation and vanishes only for $\Re(s) = 1/2$.

#### Authors

• Bruno SauvalleMINES ParisTech, PSL – Research University
60 Bd Saint-Michel
75006 Paris, France
e-mail

## Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

## Rewrite code from the image
{"ft_lang_label":"__label__en","ft_lang_prob":0.67946506,"math_prob":0.98799324,"size":816,"snap":"2020-34-2020-40","text_gpt3_token_len":275,"char_repetition_ratio":0.13423645,"word_repetition_ratio":0.016806724,"special_character_ratio":0.3272059,"punctuation_ratio":0.056962024,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9967613,"pos_list":[0,1,2],"im_url_duplicate_count":[null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-08T06:04:27Z\",\"WARC-Record-ID\":\"<urn:uuid:5745f956-f432-447d-9477-ac28cf67b897>\",\"Content-Length\":\"43667\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b3968971-51f2-4402-a8c3-4a6cce55ad6f>\",\"WARC-Concurrent-To\":\"<urn:uuid:53b84eda-7a2a-4803-a43d-eedaa7580fef>\",\"WARC-IP-Address\":\"195.187.71.110\",\"WARC-Target-URI\":\"https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/177/3/92045/on-weak-mellin-transforms-second-degree-characters-and-the-riemann-hypothesis\",\"WARC-Payload-Digest\":\"sha1:G6DDIQRD2GRTPQJWG5XQNOHWIPC6BXLY\",\"WARC-Block-Digest\":\"sha1:ODU2IYZX6NAYZB6LG5CKXKDNSZOT3EEA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439737289.75_warc_CC-MAIN-20200808051116-20200808081116-00061.warc.gz\"}"}